Vladimir E Zakharov
 Professor, Mathematics
 Regents Professor
 Professor, Applied Mathematics  GIDP
Contact
 (520) 6214841
 Mathematics, Rm. 518
 Tucson, AZ 85721
 zakharov@math.arizona.edu
Awards
 Fellow of the American Mathematical Society
 Fall 2012
 Fellow of the Optical Society of America
 Fall 2012
Interests
No activities entered.
Courses
201819 Courses

Adv Applied Mathematics
MATH 422 (Spring 2019) 
Adv Applied Mathematics
MATH 522 (Spring 2019) 
Independent Study
MATH 599 (Spring 2019) 
Independent Study
PHYS 599 (Spring 2019)
201718 Courses

Appl Partial Diff Eq
MATH 456 (Spring 2018) 
Appl Partial Diff Eq
MATH 556 (Spring 2018) 
Dissertation
MATH 920 (Spring 2018) 
Independent Study
PHYS 599 (Spring 2018) 
Dissertation
MATH 920 (Fall 2017)
201617 Courses

Appl Partial Diff Eq
MATH 456 (Spring 2017) 
Appl Partial Diff Eq
MATH 556 (Spring 2017) 
Independent Study
MATH 499 (Spring 2017) 
Tpc Mathematical Physics
MATH 488 (Spring 2017) 
Tpc Mathematical Physics
MATH 588 (Spring 2017)
Scholarly Contributions
Chapters
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2015). Compact Equation vs Fully Nonlinear One. In Extreme Ocean Waves(pp 2344). Springer.
Journals/Publications
 Zakharov, V. E. (2018). V. Zakharov, Analytic theory of winddriven sea, IUTAM Symposium Wind Waves, 48 September 2017, London, UK; Procedia IUTAM 26 (2018) 4358.. Procedia IUTAM Symposium Wind Waves, 48 September 2017, London, UK, 26, 4358.
 Zakharov, V. E., Resio, D., & Pushkarev, A. (2018). On the tuningfree statistical model of ocean surface waves. Proceedings of the ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2018, June 1722, 2018, Madrid, Spain.. doi:doi:10.1115/OMAE201878417
 Zakharov, V. E., & Badulin, S. (2017). Ocean swell within the kinetic equation for water waves. Nonlin. Processes Geophys., 24, 237253. doi:https://doi.org/10.5194/npg242372017
 Zakharov, V. E., & Geogjaev, V. (2017). Numerical and analytical calculations of the parameters of powerlaw spectra for deep water gravity waves. JETP Letters, 106(3), 184187. doi:DOI: 10.1134/S0021364017150012
 Zakharov, V. E., & Zakharov, A. (2017). Production, Redistribution, and Inequality. Modeling and Analysis of Information Systems, 24(1), 512. doi:DOI: 10.18255/1818101520171512
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2017). Envelope equation for water waves. J. Ocean Engineering & Marine Energy, 3(4), 409415. doi:https://doi.org/10.1007/s407220170100z
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2017). Statistics of freak waves in numerical tank. Lobachevskii J. Math., 38(5), 888892. doi:https://doi.org/10.1134/S1995080217050080
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2017). Super compact equation for water waves. J. Fluid Mech., 828, 661679. doi:https://doi.org/10.1017/jfm.2017.529
 Zakharov, V. E., Gelash, A., & Lvov, V. (2017). Complete Hamiltonian formalism for inertial waves in rotating fluids. J. Fluid Mech., 831, 128150. doi:doi:10.1017/jfm.2017.611
 Zakharov, V. E., Resio, D., & Pushkarev, A. (2017). Balanced source terms for wave generation within the Hasselmann equation. Nonlin. Processes Geophys., 24, 581597. doi:https://doi.org/10.5194/npg245812017
 Zakharov, V. E., & Agafontsev, D. (2016). Integrable turbulence generated from modulational instability of cnoidal waves. Nonlinearity, 29(11), 3551 3578. doi:DOI: 10.1088/09517715/29/11/3551
 Zakharov, V. E., & Dyachenko, A. (2016). Spatial equation for water waves. JETP Letters, 103(3), 181184. doi:doi:10.1134/S0021364016030048
 Zakharov, V. E., & Pushkarev, A. (2016). Limited fetch revisited: Comparison of wind input terms in surface wave modeling. Ocean Modelling, 103, 1837.
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2016). About compact equations for water waves. NATURAL HAZARDS, 84, S529S540. doi:DOI: 10.1007/s1106901624787
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2016). New compact equation for numerical simulation of freak waves on deep water. J. Phys.: Conf. Ser., 681, 012028. doi:doi:10.1088/17426596/681/1/012028
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2016). Probability distribution functions of freak waves: nonlinear versus linear model. Stud. Appl. Math., 137(2), 189198. doi:DOI: 10.1111/sapm.12116
 Zakharov, V. E., Dyachenko, S., & Zakharov, D. (2016). Primitive potentials and bounded solutions of the KdV equation. Physica D, 333, 148156. doi:10.1016/j.physd.2016.04.002
 Zakharov, V. E., Korotkevich, A., & Dyachenko, A. (2016). Numerical simulation of surface waves instability on a discrete grid. PhysicaD: Nonlinear Phenomena, 321, 5166. doi:DOI: 10.1016/j.physd.2016.02.017
 Zakharov, V. E., Zakharov, D., & Dyachenko, S. (2016). Bounded solutions of KdV and nonperiodic onegap potentials in quantum mechanics. Letters in Mathematical Physics, 106(6), 731740. doi:DOI: 10.1007/s1100501608386
 Zakharov, V. E., Zakharov, D., & Dyachenko, S. (2016). Nonperiodic onedimensional ideal conductors and integrable turbulence. Phys. Lett A, 380(46), 38813885. doi:DOI: 10.1016/j.physleta.2016.09.040
 Zakharov, V. E., & Agafontsev, D. (2015). Integrable turbulence and formation of rogue waves. Nonlinearity, 28(8), 27912823.
 Zakharov, V. E., & Agafontsev, D. (2015). Intermittency in generalized NLS equation with focusing sixwave interactions. Physics Letters A, 379(4142), 25862590. doi:doi:10.1016/j.physleta.2015.05.042
 Zakharov, V. E., & Korotkevich, A. (2015). Evaluation of a spectral line width for the Phillips spectrum by means of numerical simulation. Nonlin. Processes Geophys., 22, 325335. doi:doi:10.5194/npg223252015
 Zakharov, V. E., Arkhipov, D., & Khabakhpashev, G. (2015). Describing dynamics of nonlinear axisymmetric waves in dispersive media with new equation. Phys. Letters A, 379(2223), 14141417. doi:DOI: 10.1016/j.physleta.2015.03.010
 Zakharov, V. E., Badulin, S., Hwang, P., & Caulliez, G. (2015). Universality of Sea Wave Growth and Its Physical Roots. J. Fluid Mech., 780, 503535. doi:doi:10.1017/jfm.2015.468
 Zakharov, V. E., Dyachenko, A., & Kachulin, D. (2015). Evolution of onedimensional winddriven sea spectra. JETP Letters, 102(8), 513517. doi:10.1134/S0021364015200035
 Zakharov, V. E., Erokhin, N., & Zolnikova, N. (2015). Exactly solvable model of resonance tunneling of an electromagnetic wave in plasma containing shortscale inhomogeneities. Plasma Physics Reports, 41(2), 182187. doi:10.1134/S1063780X14120022
 Zakharov, V. E., Karas, V., & Vlasenko, A. (2015). Nonequilibrium kinetics of the electronphonon sybsystem of a crystal in a strong electric field as a base of the electroplastic effect. JETP, 121(3), 499508. doi:10.7868/S004445101509014X
 Zakharov, V. E., Kibler, B., Gelash, A., Akhmediev, N., & Chabchoub, A. (2015). Omnipresent modulation instability beyond simple periodicity. Phys. Rev. X, 5, 041026. doi:DOI: 10.1103/PhysRevX.5.041026
 Zakharov, V. E., & Dyachenko, A. (2014). Freak waves at the surface of deep water. J. Phys. Conf. Ser., 510, 012050.More infoWe apply canonical transformation to water wave equation not only to remove cubic nonlinear terms but to simplify drastically fourth order terms inHamiltonian. After the transformation wellknown but cumbersome Zakharov equation is drastically simplifed and can be written in Xspace in compact way. This new equation is very suitable for analytic study as well as for numerical simulation. At the same time one of the important issues concerning this system is the question of its integrability. The first part of the work is devoted to numerical and analytical study of the integrability of the equation obtained in. In the second part we present generalization of the improved Zakharov equation for the "almost" 2D water waves at the surface of deep water. When considering waves slightly inhomogeneous in transverse direction, one can think in the spirit of KadomtsevPetviashvili equation for KortevegdeVries equation taking into account weak transverse direction. Equation can be written instead of classical variables _(x; y; t) and (x; y; t) in terms of canonical normal variable b(x; y; t). This equation is very suitable for robust numerical simulation. Due to specifc structure of nonlinearity in the Hamiltonian the equation can be effectively solved on the computer. It was applied for simulation of sea surface waving including freak waves appearing.
 Zakharov, V. E., & Gelash, A. (2014). Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability. Nonlinearity, 27(4), 139.More infoWe describe a general Nsolitonic solution of the focusing nonlinearSchr¨odinger equation in the presence of a condensate by using the dressingmethod. We give the explicit form of one and twosolitonic solutions andstudy them in detail as well as solitonic atoms and degenerate solutions. Wedistinguish a special class of solutions that we call regular solitonic solutions.Regular solitonic solutions do not disturb phases of the condensate at infinityby coordinate. All of them can be treated as localized perturbations of thecondensate. We find a broad class of superregular solitonic solutions whichare small perturbations at a certain moment of time. Superregular solitonicsolutions are generated by pairs of poles located on opposite sides of the cut.They describe the nonlinear stage of the modulation instability of the condensateand play an important role in the theory of freak waves.
 Zakharov, V. E., & Hadad, Y. (2014). Transparency of Strong Gravitational Waves. Journal of Geometry and Physics, 80, 37–48.More infoThis paper studies diagonal spacetime metrics. It is shown that the overdetermined Einstein vacuum equations are compatible if one Killing vector exists. The stability of plane gravitational waves of the Robinson type is studied. This stability problem bares a fantastic mathematical resemblance to the stability of the Schwarzschild black hole studied by Regge and Wheeler. Just like for the Schwarzschild black hole, the Robinson gravitational waves are proven to be stable with respect to small perturbations. We conjecture that a bigger class of vacuum solutions are stable, among which are all gravitational solitons. Moreover, the stability analysis reveals a surprising fact: a wave barrier will be transparent to the Robinson waves, which therefore passes through the barrier freely. This is a hint of integrability of the 1 + 2 vacuum Einstein equations for diagonal metrics.
 Zakharov, V. E., & Shamin, R. (2014). Energy Portrait of Rogue Waves. JETP Letters, 99(9), 514–517.More infoProcesses of the concentration of energy at the formation of rogue waves have been studied in computer experiments based on the exact hydrodynamic equations for an ideal fluid. The distribution of anomalies of waves both in height and in energy has been found in the computer experiment. Correlation between the energy concentration and height of anomalously large surface waves has been revealed. The results can be used to estimate the danger of anomalously large surface waves.
 Zakharov, V. E., Odesskii, A., & Onorato, M. (2014). FIVEWAVE CLASSICAL SCATTERING MATRIX AND INTEGRABLE EQUATIONS. Theoretical and Mathematical Physics, 180(1), 759–764.More infoWe study the fivewave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the fivewave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg–de Vries equation, the Benjamin–Ono equation, and the intermediate longwave equation. In two dimensions, we find a new equation that satisfies our requirement.
 Dyachenko, A. I., Kachulin, D. I., & Zakharov, V. E. (2013). Collisions of two breathers at the surface of deep water. Natural Hazards and Earth System Sciences, 13(12), 32053210.More infoAbstract: We present results of numerical experiments on longterm evolution and collisions of breathers (which correspond to envelope solitons in the NLSE approximation) at the surface of deep ideal fluid. The collisions happen to be nonelastic. In the numerical experiment it can be observed only after many acts of interactions. This supports the hypothesis of "deep water nonintegrability". The experiments were performed in the framework of the new and refined version of the Zakharov equation free of nonessential terms in the quartic Hamiltonian. Simplification is possible due to exact cancellation of nonelastic fourwave interaction. © 2013 Author(s).
 Dyachenko, A. I., Kachulin, D. I., & Zakharov, V. E. (2013). On the nonintegrability of the free surface hydrodynamics. JETP Letters, 98(1), 4347.More infoAbstract: The integrability of the compact 1D Zakharov equation has been analyzed. The numerical experiments show that the multiple collisions of breathers (which correspond to envelope solitons in the NLSE approximation) are not pure elastic. The amplitude of sixwave interactions for the compact 1D Zakharov equation has also been analyzed. It has been found that the sixwave amplitude is not canceled for this equation. Thus, the 1D Zakharov equation is not integrable. © 2013 Pleiades Publishing, Ltd.
 Pushkarev, A., & Zakharov, V. E. (2013). Quasibreathers in the MMT model. Physica D: Nonlinear Phenomena, 248(1), 5561.More infoAbstract: We report numerical detection of a new type of localized structures in the frame of MajdaMcLaughlinTabak (MMT) model adjusted for description of essentially nonlinear gravity waves on the surface of ideal deep water. These structuresquasibreathers or oscillating quasisolitonscan be treated as groups of freak waves closely resembling experimentally observed "Three Sisters" wave packets on the ocean surface. The MMT model has quasisolitonic solutions. Unlike NLSE solitons, MMT quasisolitons are permanently backward radiating energy, but nevertheless do exist during thousands of carrier wave periods. Quasisolitons of small amplitude are regular and stable, but largeamplitude ones demonstrate oscillations of amplitude and spectral shape. This effect can be explained by periodic formation of weak collapses, carrying out negligibly small amount of energy. We call oscillating quasisolitons "quasibreathers".© 2013 Elsevier B.V. All rights reserved.
 Zakharov, V. E., & Gelash, A. A. (2013). Nonlinear stage of modulation instability. Physical Review Letters, 111(5).More infoAbstract: We study the nonlinear stage of the modulation instability of a condensate in the framework of the focusing nonlinear Schrödinger equation (NLSE). We find a general Nsolitonic solution of the focusing NLSE in the presence of a condensate by using the dressing method. We separate a special designated class of "regular solitonic solutions" that do not disturb phases of the condensate at infinity by coordinate. All regular solitonic solutions can be treated as localized perturbations of the condensate. We find an important class of "superregular solitonic solutions" which are small perturbations at a certain moment of time. They describe the nonlinear stage of the modulation instability of the condensate. © 2013 American Physical Society.
 Zakharov, V. E., & Karas, V. I. (2013). Nonequilibrium Kolmogorovtype particle distributions and their applications. PhysicsUspekhi, 56(1), 4978.More infoAbstract: The paper reviews the current state of research on nonequilibrium (Kolmogorov type) stationary and nonstationary distributions of particles with statically screened Coulomb interaction that are exact solutions of the Boltzmann or Landau collision integral with a source and a sink ensuring the energy flow along the spectrum in momentum space. Analysis is made of the advantages of the new process (based on nonequilibrium distributions) of energy conversion and of the timedependent nonequilibrium kinetics of an electronphonon system of a crystal in a strong electric field (electroplastic effect). © 2013 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences.
 Zakharov, V., & Gelash, A. (2013). Freak waves as a result of modulation instability. Procedia IUTAM, 9, 165175.More infoAbstract: We describe a general Nsolitonic solution of the focusing NLSE in the presence of a condensate by using the dressing method. We find a broad class of superregular solitonic solutions which are small perturbations at certain a moment of time. They describe the nonlinear stage of the modulation instability of the condensate and play an important role in the theory of freak waves. © 2013 The Authors.
 Dyachenko, A. I., & Zakharov, V. E. (2012). A dynamic equation for water waves in one horizontal dimension. European Journal of Mechanics, B/Fluids, 32(1), 1721.More infoAbstract: We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourthorder terms in the Hamiltonian. This transformation explicitly uses the vanishing exact fourwave interaction for water gravity waves for a 2D potential fluid. After transformation, the wellknown but cumbersome Zakharov equation is drastically simplified and can be written in Xspace in a compact form. This new equation is very suitable for analytical studies and numerical simulations. © 2011 Elsevier Masson SAS. All rights reserved.
 Zakharov, V. E., & Kuznetsov, E. A. (2012). Solitons and collapses: Two evolution scenarios of nonlinear wave systems. PhysicsUspekhi, 55(6), 535556.More infoAbstract: Two alternative scenarios pertaining to the evolution of nonlinear wave systems are considered: solitons and wave collapses. For the former, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of smallamplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrödinger equation and the threewave system is used to show how the boundedness of the Hamiltonian  and hence the stability of the soliton minimizing it  can be proved rigorously using the integral estimate method based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of smallamplitude waves promotes collapse in this case. © 2012 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences.
 Zakharov, V. E., & Shamin, R. V. (2012). Statistics of rogue waves in computer experiments. JETP Letters, 96(1), 6669.More infoAbstract: Rogue waves have been studied in the exact simulation of complete hydrodynamic equations for an ideal fluid with a free surface. The statistical characteristics of rogue waves such as the occurrence intensity, average existence time, and maximum energy dissipation at collapse have been obtained in computer experiments. © 2012 Pleiades Publishing, Ltd.
 Zakharov, V., Newell, A. C., Rumpf, B., & Zakharov, V. E. (2012). Spontaneous breaking of the spatial homogeneity symmetry in wave turbulence. Physical review letters, 108(19).More infoWe report a surprising new result for wave turbulence which may have broader ramifications for general turbulence theories. Spatial homogeneity, the symmetry property that all statistical moments are functions only of the relative geometry of any configuration of points, can be spontaneously broken by the instability of the finite flux KolmogorovZakharov spectrum in certain (usually one dimensional) situations. As a result, the nature of the statistical attractor changes dramatically, from a sea of resonantly interacting dispersive waves to an ensemble of coherent radiating pulses.
 Dyachenko, A. I., & Zakharov, V. E. (2011). Compact equation for gravity waves on deep water. JETP Letters, 93(12), 701705.More infoAbstract: Using the fact of vanishing four waves interaction for water gravity waves for 2D potential fluid, we have significantly simplified the wellknown but cumbersome Zakharov equation. The Hamiltonian of the obtained equation is very simple and includes only the fourth order nonlinear term. It raises the question of the integrability of free surface hydrodynamics. This new equation is very suitable both for analytic study and for numerical simulation. © 2011 Pleiades Publishing, Ltd.
 Erokhin, N. S., & Zakharov, V. E. (2011). Generation of strong splashes of the field of an electromagnetic wave during reflectionless blooming of a layer of a nonuniform medium. Doklady Physics, 56(7), 362365.More infoAbstract: The generation of strong splashes of the field of an electromagnetic wave during reflectionless blooming of a layer of a nonuniform medium is reported. The wave amplitude can increase by an order of magnitude and more depending on the starting parameters in the region of splashes. The amplitude of the electromagnetic wave considerably increases in the region of field splashes so it is necessary to take into account the effect of nonlinearity. It is observed that the thickness of the splash region substantially decreases, while the amplitude of variations of nonlinear permittivity substantially increases. The substantial qualitative difference for subwave splashes of the profiles of the wave vector and effective permittivity should be noted. The relation of the amplitude of the field of the electromagnetic wave to the effective permittivity is described by a nonlinear equation.
 Erokhin, N. S., & Zakharov, V. E. (2011). Reflectionless passage of an electromagnetic wave through an inhomogeneous plasma layer. Plasma Physics Reports, 37(9), 762767.More infoAbstract: An exactly solvable model is used as a basis to study the reflectionless passage of a transverse electromagnetic wave through an inhomogeneous plasma containing largeamplitude, smallscale (subwavelength) structures (in particular, opaque regions) that cannot be correctly described by approximate methods. It is shown that, during the reflectionless passage of an electromagnetic wave, strong wave field splashes can occur in certain plasma sublayers. The nonuniform spatial plasma density profile is characterized by a number of free parameters describing the modulation depth of the dielectric function, the characteristic sizes of the structures and their number, the thickness of the inhomogeneous plasma region, and so on. Such plasma density structures are shown to be very diverse when, e.g.,a wave that is incident from vacuum propagates without reflection through a plasma layer (wave barrier transillumination). With the cubic nonlinearity taken into account, a onedimensional problem of the nonlinear transillumination of an inhomogeneous plasma can be solved exactly. © 2011 Pleiades Publishing, Ltd.
 Nakoryakov, V. E., Zakharov, V. E., Gasenko, V. G., & Tsoi, A. N. (2011). A new method of gas hydrate production. Journal of Engineering Thermophysics, 20(4), 335337.
 Zakharov, V. E., & Badulin, S. I. (2011). On energy balance in winddriven seas. Doklady Earth Sciences, 440(2), 14401444.More infoAbstract: We show that in the energy balance in the winddriven sea the process of fourwave nonlinear interaction plays the leading role. This process surpasses competing mechanismsinput energy from wind and dissipation of energy due to white capping at least in order of magnitude. This result is supported by analytical calculations and numerical simulations. © Pleiades Publishing, Ltd., 2011.
 Zakharov, V. E. (2010). Dynamics of vortex line in presence of stationary vortex. Theoretical and Computational Fluid Dynamics, 24(14), 377382.More infoAbstract: The motion of a thin vortex with infinitesimally small vorticity in the velocity field created by a steady straight vortex is studied. The motion is governed by nonintegrable PDE generalizing the Nonlinear Schrodinger equation (NLSE). Situation is essentially different in a corotating case, which is analog of the defocusing NLSE and a counterrotating case, which can be compared with the focusing NLSE. The governing equation has special solutions shaped as rotating helixes. In the counterrotating case all helixes are unstable, while in the corotating case they could be both stable and unstable. Growth of instability of counterrotating helix ends up with formation of singularity and merging of vortices. The process of merging goes in a selfsimilar regime. The basic equation has a rich family of solitonic solutions. Analytic calculations are supported by numerical experiment. © 2009 SpringerVerlag.
 Zakharov, V. E. (2010). Energy balance in a winddriven sea. Physica Scripta T, T142.More infoAbstract: In this paper, we offer the answers to certain questions extremely important for the development of a selfconsistent analytical theory for the winddriven sea. (i) We discuss the separation into 'resonant' and 'slave' harmonics in an ensemble of weakly nonlinear gravity waves on the surface of deep water, and we construct an explicit form of the generation function for canonical transformation that eliminates the slave harmonics. (ii) When two waves compiling a quadruple are short in comparison with two others, we find an asymptotic form for the fourwave coupling coefficient. This result makes it possible to reduce the Hasselmann equation to the nonlinear diffusion equation, whose solution describes the wellknown effect of angular spreading of wave spectra on its rear face. (iii) Studying the isotropic KolmogorovZakharov solution of the Hasselmann equation, we find numerically the values of Kolmogorov constants. (iv) We calculate the nonlinear damping of surface waves appearing due to fourwave interaction and compare the damping with the growth rate of the instability of the wave surface induced by the wind. It is found that for all known models of wind input, the nonlinear damping surpasses the instability at least in order of magnitude. This result, supported by numerical simulation of the Hasselmann equation, leads to the conclusion: in a real sea, except for the case of very young waves, fourwave interaction is the dominant process. This statement opens the way for the development of a welljustified analytical theory for the winddriven sea. © 2010 The Royal Swedish Academy of Sciences.
 Zakharov, V. E. (2010). On domination of nonlinear wave interaction in the energy balance of winddriven sea. Fizika Nizkikh Temperatur (Kharkov), 36(89), 971985.More infoAbstract: Here some aspects of the physics of winddriven sea are investigated theoretically. It is demonstrated that the effective fourwave nonlinear interaction plays the leading role in the formation of the spectra of turbulent waves. In particular this interaction leads to nonlinear damping which exceeds standard data at least by the order of magnitude. The theory developed here is compared with the available experimental data. © V.E. Zakharov, 2010.
 Zakharov, V. E. (2010). The dominant nonlinear wave interaction in the energy balance of a winddriven sea. Low Temperature Physics, 36(8), 772784.More infoAbstract: Here some aspects of the physics of a winddriven sea are investigated theoretically. It is demonstrated that an effective fourwave nonlinear interaction plays a leading role in the formation of the spectra of turbulent waves. In particular, this interaction leads to nonlinear damping which exceeds standard observations at least by an order of magnitude. The theory developed here is compared with available experimental data. © 2010 American Institute of Physics.
 Zakharov, V. E., & Dyachenko, A. I. (2010). About shape of giant breather. European Journal of Mechanics, B/Fluids, 29(2), 127131.More infoAbstract: The pulse of freak waves on the surface of deep water can be a breathertype solution of the Euler equation. The shape of surface is periodic function of time in a moving frame. Only in the limit of very small steepness its shape is described by the Nonlinear Shredinger Equation (NLSE). For moderately small steepness we derived more exact envelope nonlocal equation similar to wellknown Dysthe equation (DE), which is more convenient for analytical and numerical study. We have found approximate solution of this equation by the use of the variational approach. © 2009.
 Zakharov, V. E., & Karas, V. I. (2010). Nonequilibrium kolmogorov type particle distributions and their applications. Problems of Atomic Science and Technology, 204209.More infoAbstract: The modern condition of researches nonequilibrium (Kolmogorov type) stationary and nonstationary particle distributions with a flux on a spectrum is submitted. On the basis of Boltzmann collision integrals and in Landau form for not relativistic charged particles interacting under Coulomb law in view of static shielding, in homogeneous and isotropic matter by analytical methods and with the help completely conservative finite difference schemes consider formation of nonequilibrium particle distribution functions. It is shown, as the received results can be used for a prediction of behaviour of conductors and semiconductors both with own and with impurity conductivity irradiated with fastion beams or laser radiation.
 Zakharov, V. E., & Shamin, R. V. (2010). Probability of the occurrence of freak waves. JETP Letters, 91(2), 6265.More infoAbstract: The statistics of the occurrence of freak waves on a surface of an ideal heavy liquid is studied. The freak (rogue, extreme) waves arise in the course of evolution of a statistically homogeneous random Gaussian wave field. The mean steepness of initial data varies from small (μ2 = 1.54 × 103) to moderate (μ2 = 3.08 × 103) values. The frequency of the occurrence of extreme waves decreases with an increase in the spectral width of the initial distribution, but remains relatively high even for broad spectra (Δk/Δ ~ 1). © Pleiades Publishing, Inc., 2010.
 Zakharov, V. E., Dyachenko, A. I., & Shamin, R. V. (2010). How probability for freak wave formation can be found. European Physical Journal: Special Topics, 185(1), 113124.More infoAbstract: The statistics of arising of freak waves on a surface of an ideal heavy fluid is studied. The freak waves arise during the evolution of statistically homogeneous initial conditions with a Gaussian probability. Simple equation describing evolution of "almost" 1D water waves is derived using important property of vanishing fourwave interaction. © 2010 EDP Sciences and Springer.
 Balk, A. M., & Zakharov, V. E. (2009). Cascade generation of zonal flows by the drift wave turbulence. Physics Letters, Section A: General, Atomic and Solid State Physics, 373(44), 40494052.More infoAbstract: The purpose of this research is to investigate the formation of zonal flows that can lead to the enhanced confinement of plasma in tokamaks. We show that zonal flows can be effectively formed by resonance triad interactions in the process of the inverse cascade. We discuss what energy sources are more effective for the formation of zonal flows. © 2009 Elsevier B.V. All rights reserved.
 Zakharov, V. E. (2009). Turbulence in integrable systems. Studies in Applied Mathematics, 122(3), 219234.More infoAbstract: Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence  Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP1, threewave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of RayleyJeanstype stationary solutions, though their general stationary solutions are not explored yet. © 2009 No claim to original US government works.
 Zakharov, V. E., & Ostrovsky, L. A. (2009). Modulation instability: The beginning. Physica D: Nonlinear Phenomena, 238(5), 540548.More infoAbstract: We discuss the early history of an important field of "sturm and drang" in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given. © 2008 Elsevier B.V. All rights reserved.
 Zakharov, V. E., Korotkevich, A. O., & Prokofiev, A. O. (2009). On dissipation function of ocean waves due to whitecapping. AIP Conference Proceedings, 1168, 12291231.More infoAbstract: The Hasselmann kinetic equation [1] provides a statistical description of waves ensemble. Several catastrophic events are beyond statistical model. In the case of gravity waves on the surface of the deep fluid may be the most frequent and important events of such kind are whitecapping and wave breaking. It was shown earlier that such effects leads to additional dissipation in the energy contaning region around waves spectral peak, which can be simulated by means of empiric dissipative term in kinetic equation. In order to find dependence of this term with respect to nonlinearity in the system (steepness of the surface) we preformed two numerical experiments: weakly nonlinear one in the framework of 3D hydrodynamics and fully nonlinear one for 2D hydrodynamic. In spite of significantly different models and initial conditions, both these experiments yielded close results. Obtained data can be used to define analytical formula for dependence of the dissipative term of dissipation coefficient with respect to mean steepness of the surface. © 2009 American Institute of Physics.
 Badulin, S. I., Babanin, A. V., Zakharov, V. E., & Resio, D. T. (2008). Numerical verification of weakly turbulent law of wind wave growth. Solid Mechanics and its Applications, 6, 211226.More infoAbstract: Numerical solutions of the kinetic equation for deep water wind waves (the Hasselmann equation) for various functions of external forcing are analyzed. For wave growth in spatially homogeneous sea (the socalled durationlimited case) the numerical solutions are related with approximate selfsimilar solutions of the Hasselmann equation. These selfsimilar solutions are shown to be considered as a generalization of the classic KolmogorovZakharov solutions in the theory of weak turbulence. Asymptotic law of wave growth that relates total wave energy with net total energy input (energy flux at high frequencies) is proposed. Estimates of selfsimilarity parameter that links energy and spectral flux and can be considered as an analogue of KolmogorovZakharov constants are obtained numerically. © 2008 Springer.
 Dias, F., Dyachenko, A. I., & Zakharov, V. E. (2008). Theory of weakly damped freesurface flows: A new formulation based on potential flow solutions. Physics Letters, Section A: General, Atomic and Solid State Physics, 372(8), 12971302.More infoAbstract: Several theories for weakly damped freesurface flows have been formulated. In this Letter we use the linear approximation to the NavierStokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deepwater gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification. © 2007 Elsevier B.V. All rights reserved.
 Dyachenko, A. I., & Zakharov, V. E. (2008). On the formation of freak waves on the surface of deep water. JETP Letters, 88(5), 307311.More infoAbstract: Numerical simulation of the fully nonlinear water equations demonstrates the existence of giant breathers on the surface of deep water. The numerical analysis shows that this breather (or soliton of envelope) does not loose energy. The existence of such a breather can explain the appearance of freak waves. © 2008 Pleiades Publishing, Ltd.
 Korotkevich, A. O., Pushkarev, A., Resio, D., & Zakharov, V. E. (2008). Numerical verification of the weak turbulent model for swell evolution. European Journal of Mechanics, B/Fluids, 27(4), 361387.More infoAbstract: The purpose of this article is to numerically verify the theory of weak turbulence. We have performed numerical simulations of an ensemble of nonlinearly interacting free gravity waves (a swell) by two different methods: by solving the primordial dynamical equations describing the potential flow of an ideal fluid with a free surface, and by solving the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. In both cases we have observed effects predicted by this theory: frequency downshift, angular spreading and formation of a ZakharovFilonenko spectrum Iω ∼ ω4. To achieve quantitative coincidence of the results obtained by different methods, we have to augment the Hasselmann kinetic equation by an empirical dissipation term Sdiss modeling the coherent effects of whitecapping. Using the standard dissipation terms from the operational wave predicting model (WAM) leads to a significant improvement on short times, but does not resolve the discrepancy completely, leaving the question about the optimal choice of Sdiss open. In the long run, WAM dissipative terms essentially overestimate dissipation. © 2007 Elsevier Masson SAS. All rights reserved.
 Zakharov, V. E., & Dyachenko, A. I. (2008). Freak Waves and giant breathers. Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering  OMAE, 2, 10191024.More infoAbstract: It is assumed the solitons could propagate only on the surface of finite depth fluid. We show numerically that the strong localized perturbation of free fluid surface could propagate on the surface of deep fluid also. They are not solitons in a "classical" sense of this term; they are "breathers". It means that the motion of surface is a periodic function on time in a certain moving framework. This framework moves with the group velocity. Breathers of small steepness(ka < 0.07) are known for years as solitonic solutions of the Nonlinear Schrodinger Equation (NLSE) (Do not confuse with the "breather" solution of the Nonlinear Schrodinger Equation). Our new result is following: The breathers exits without decay up to very high steepness (at least to ka=0.5). These steep giant breathers can be identified with Freak Waves. The most plausible explanation of the giant breather stability is integrability of the Euler equation describing a potential flow of deep ideal fluid with free surface. So far we don't have a direct proof of this extremely strong statement, but we have a whole string of indirect evidences of this integrability. Copyright © 2008 by ASME.
 Zakharov, V. E., Dyachenko, A. I., & Prokofiev, A. O. (2008). Freak waves: Peculiarities of numerical simulations. Extreme Ocean Waves, 129.More infoAbstract: Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using twodimensional code, based on conformal mapping of the fluid to the lower halfplane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of nonlinear Shrödinger equation solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves. © 2008 Springer Netherlands.
 Badulin, S. I., Babanin, A. V., Zakharov, V. E., & Resio, D. (2007). Weakly turbulent laws of windwave growth. Journal of Fluid Mechanics, 591, 339378.More infoAbstract: The theory of weak turbulence developed for winddriven waves in theoretical works and in recent extensive numerical studies concludes that nondimensional features of selfsimilar wave growth (i.e. wave energy and characteristic frequency) have to be scaled by internal wavefield properties (fluxes of energy, momentum or wave action) rather than by external attributes (e.g. wind speed) which have been widely adopted since the 1960s. Based on the hypothesis of dominant nonlinear transfer, an asymptotic weakly turbulent relation for the total energy ε and a characteristic wave frequency ω* was derived ε ω*4/g2 = αss (ω*3 dε/dt/g2)1/3. The selfsimilarity parameter αss was found in the numerical durationlimited experiments and was shown to be naturally varying in a relatively narrow range, being dependent on the energy growth rate only. In this work, the analytical and numerical conclusions are further verified by means of known field dependencies for wave energy growth and peak frequency downshift. A comprehensive set of more than 20 such dependencies, obtained over almost 50 years of field observations, is analysed. The estimates give αss very close to the numerical values. They demonstrate that the weakly turbulent law has a general value and describes the wave evolution well, apart from the earliest and full wave development stages when nonlinear transfer competes with wave input and dissipation. © 2007 Cambridge University Press.
 Erokhin, N. S., & Zakharov, V. E. (2007). On transillumination of wave barriers for electromagnetic radiation in an inhomogeneous plasma. Doklady Physics, 52(9), 485487.More infoAbstract: A study was conducted to analyze nonreflection nonlinear transmission of an electromagnetic wave through the region of an inhomogeneous plasma nonlinear wave barrier. The mathematical model used to analyze transillumination of a barrier during the interaction of a wave with the inhomogeneous medium was based on solving the Helmholtz equation, with cubic nonlinearity. It was found that other versions of nonreflection nonlinear transillumination of inhomogeneous wave barriers in the layered inhomogeneous plasmas were also possible in other media. The effect was possible for quasiperiodic inhomogeinities, with a number of layers. The results show that nonreflection transillumination and transmission of barriers is also possible for other types of waves in an inhomogeneous plasma.
 Korotkevich, A. O., Pushkarev, A. N., Resio, D., & Zakharov, V. E. (2007). Numerical verification of the hasselmann equation. Tsunami and Nonlinear Waves, 135172.More infoAbstract: The purpose of this article is numerical verification of the thory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates pretty good applicability of the weak turbulent approach. In both cases we observed effects predicted by this theory: frequency downshift, angular spreading as well as formation of ZakharovFilonenko spectrum I ω ∼ ω 4. To achieve quantitative coincidence of the results obtained by different methods one has to accomplish the Hasselmann kinetic equation by an empirical dissipation term S diss modeling the coherent effects of whitecapping. Adding of the standard dissipation terms used in the industrial wave predicting model (WAM) leads to significant improvement but not resolve the discrepancy completely, leaving the question about optimal choice of S diss open. Numerical modeling of swell evolution in the framework of the dynamical equations is affected by the side effect of resonances sparsity taking place due to finite size of the modeling domain. We mostly overcame this effect using fine integration grid of 512 × 4096 modes. The initial spectrum peak was located at the wave number k = 300. Similar conditions can be hardly realized in the laboratory wave tanks. One of the results of our article consists in the fact that physical processes in finite size laboratory wave tanks and in the ocean are quite different, and the results of such laboratory experiments can be applied to modeling of the ocean phenomena with extra care. We also present the estimate on the minimum size of the laboratory installation, allowing to model open ocean surface wave dynamics. © 2007 Springer Berlin Heidelberg.
 Zakharov, V. E., Korotkevich, A. O., Pushkarev, A., & Resio, D. (2007). Coexistence of weak and strong wave turbulence in a swell propagation. Physical Review Letters, 99(16).More infoAbstract: By performing two parallel numerical experimentssolving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equationwe examined the applicability of the theory of weak turbulence to the description of the time evolution of an ensemble of free surface waves (a swell) on deep water. We observed qualitative coincidence of the results. To achieve quantitative coincidence, we augmented the kinetic equation by an empirical dissipation term modeling the strongly nonlinear process of white capping. Fitting the two experiments, we determined the dissipation function due to wave breaking and found that it depends very sharply on the parameter of nonlinearity (the surface steepness). The onset of white capping can be compared to a secondorder phase transition. The results corroborate the experimental observations of Banner, Babanin, and Young. © 2007 The American Physical Society.
 Zakharov, V. E., Dyachenko, A. I., & Prokofiev, A. O. (2006). Freak waves as nonlinear stage of Stokes wave modulation instability. European Journal of Mechanics, B/Fluids, 25(5), 677692.More infoAbstract: Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using twodimensional code, based on conformal mapping of the fluid to the lower halfplane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of NLSE solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves. © 2006 Elsevier SAS. All rights reserved.
 Badulin, S. I., Pushkarev, A. N., Resio, D., & Zakharov, V. E. (2005). Selfsimilarity of winddriven seas. Nonlinear Processes in Geophysics, 12(6), 891945.More infoAbstract: The results of the theoretical and numerical study of the Hasselmann kinetic equation for deep water waves in presence of wind input and dissipation are presented. The guideline of the study: nonlinear transfer is the dominating mechanism of windwave evolution. In other words, the most important features of winddriven sea could be understood in a framework of conservative Hasselmann equation while forcing and dissipation determine parameters of a solution of the conservative equation. The conservative Hasselmann equation has a rich family of selfsimilar solutions for durationlimited and fetchlimited windwave growth. These solutions are closely related to classic stationary and homogeneous weakturbulent Kolmogorov spectra and can be considered as nonstationary and nonhomogeneous generalizations of these spectra. It is shown that experimental parameterizations of windwave spectra (e.g. JONSWAP spectrum) that imply selfsimilarity give a solid basis for comparison with theoretical predictions. In particular, the selfsimilarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age Cp/U10. This comparison is detailed in the extensive numerical study of durationlimited growth of wind waves. The study is based on algorithm suggested by Webb (1978) that was first realized as an operating code by Resio and Perrie (1989, 1991). This code is now updated: the new version is up to one order faster than the previous one. The new stable and reliable code makes possible to perform massive numerical simulation of the Hasselmann equation with different models of wind input and dissipation. As a result, a strong tendency of numerical solutions to selfsimilar behavior is shown for rather wide range of wave generation and dissipation conditions. We found very good quantitative coincidence of these solutions with available results on durationlimited growth, as well as with experimental parametrization of fetchlimited spectra JONSWAP in terms of windwave age Cp/U10. © 2005 Author(s). This work is licensed under a Creative Commons License.
 Dyachenko, A. I., & Zakharov, V. E. (2005). Modulation instability of stokes wave → freak wave. JETP Letters, 81(6), 255259.More infoAbstract: Formation of waves of large amplitude (freak waves, killer waves) at the surface of the ocean is studied numerically. We have observed that freak waves have the same ratio of the wave height to the wave length as limiting Stokes waves. When a freak wave reaches this limiting state, it breaks. The physical mechanism of freak wave formation is discussed. © 2005 Pleiades Publishing, Inc.
 Lushnikov, P. M., & Zakharov, V. E. (2005). Erratum: "On optimal canonical variables in the theory of ideal fluid with free surface" by P.M. Lushnikov and V.E. Zakharov (Physica D: Nonlinear Phenomena (2005) 203 (929) DOI:10.1016/j.physd.2005.02.010). Physica D: Nonlinear Phenomena, 206(34), 275.
 Lushnikov, P. M., & Zakharov, V. E. (2005). On optimal canonical variables in the theory of ideal fluid with free surface. Physica D: Nonlinear Phenomena, 203(12), 929.More infoAbstract: Dynamics of ideal fluid with free surface can be effectively solved by perturbing the Hamiltonian in weak nonlinearity limit. However it is shown that perturbation theory, which includes third and fourth order terms in the Hamiltonian, results in the illposed equations because of short wavelength instability. To fix that problem we introduce the canonical Hamiltonian transformation from original physical variables to new variables for which instability is absent. © 2005 Published by Elsevier B.V.
 Zakharov, V. E. (2005). Theoretical interpretation of fetch limited winddriven sea observations. Nonlinear Processes in Geophysics, 12(6), 10111020.More infoAbstract: We show that the results of major fetch limited field studies of windgenerated surface gravity waves on deep water can be explained in the framework of simple analytical model. The spectra measured in these experiments are described by selfsimilar solutions of "conservative" Hasselmann equation that includes only advective and nonlinear interaction terms. Interaction with the wind and dissipation due to the wave breaking indirectly defines parameters of the selfsimilar solutions. © 2005 Author(s). This work is licensed under a Creative Commons License.
 Zakharov, V. E., & Nazarenko, S. V. (2005). Dynamics of the BoseEinstein condensation. Physica D: Nonlinear Phenomena, 201(34), 203211.More infoAbstract: BoseEinstein condensation is a very general physical phenomenon which takes place not only in the systems of bosonic atoms, but also in optical wave systems. Many important features of the condensation in such diverse systems can be captured by the nonlinear Schroedinger model. Within this model we develop a statistical description in which the condensate is nonlinearly coupled to wave turbulence described by a kinetic equation. Our focus will be on the strongcondensate regime in which the threewave interaction replaces the fourwave process operating on the preceding stages of an explosive condensate formation and its initial growth. In the strongcondensate regime, the condensate growth accelerates and becomes quadratic in time. This regime will proceed until the wave dispersion drops below a critical value and the state of dispersionless acoustic turbulence forms. © 2004 Elsevier B.V. All rights reserved.
 Zakharov, V. E., Korotkevich, A. O., Pushkarev, A. N., & Dyachenko, A. I. (2005). Mesoscopic wave turbulence. JETP Letters, 82(8), 487491.More infoAbstract: We report results of simulation of wave turbulence. Both inverse and direct cascades are observed. The definition of "mesoscopic turbulence" is given. This is a regime when the number of modes in a system involved in turbulence is high enough to qualitatively simulate most of the processes but significantly smaller than the threshold, which gives us quantitative agreement with the statistical description, such as the kinetic equation. Such a regime takes place in numerical simulation, in essentially finite systems, etc. © 2005 Pleiades Publishing, Inc.
 Dyachenko, A. I., Korotkevich, A. O., & Zakharov, V. E. (2004). Weak turbulent Kolmogorov spectrum for surface gravity waves. Physical Review Letters, 92(13), 1345011.More infoPMID: 15089618;Abstract: The evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in small wave number was studied. The primitive Eular equations were implemented for the potential flow of an ideal fluid with free surface written in hamiltonian canonical varibles. The growth of wave energy due to stochastic external force was arrested by nonlinear resonant fourwave processes, which leads to formation of powerlike Kolmogorov spectrum. It was found that, for the different pumping level surface elevation spectra differ only due to the different energy flux.
 Pushkarev, A., Resio, D., & Zakharov, V. (2004). Second generation diffusion model of interacting gravity waves on the surface of deep fluid. Nonlinear Processes in Geophysics, 11(3), 329342.More infoAbstract: We propose a second generation phenomenological model for nonlinear interaction of gravity waves on the surface of deep water. This model takes into account the effects of nonlocality of the original Hasselmann diffusion equation still preserving important properties of the first generation model: physically consistent scaling, adherence to conservation laws and the existence of KolmogorovZakharov solutions. Numerical comparison of both models with the original Hasselmann equation shows that the second generation models improves the angular distribution in the evolving wave energy spectrum. © European Geosciences Union 2004.
 Zakharov, V., Dias, F., & Pushkarev, A. (2004). Onedimensional wave turbulence. Physics Reports, 398(1), 165.More infoAbstract: The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of wave turbulence has been less studied, partly because it developed later. Wave turbulence takes place in physical systems of nonlinear dispersive waves. In most applications nonlinearity is small and dispersive wave interactions are weak. The weak turbulence theory is a method for a statistical description of weakly nonlinear interacting waves with random phases. It is not surprising that the theory of weak wave turbulence began to develop in connection with some problems of plasma physics as well as of wind waves. The present review is restricted to onedimensional wave turbulence, essentially because finer computational grids can be used in numerical computations. Most of the review is devoted to wave turbulence in various wave equations, and in particular in a simple onedimensional model of wave turbulence introduced by Majda, McLaughlin and Tabak in 1997. All the considered equations are model equations, but consequences on physical systems such as ocean waves are discussed as well. The main conclusion is that the range in which the theory of pure weak turbulence is valid is narrow. In general, wave turbulence is not completely weak. Together with the weak turbulence component, it can include coherent structures, such as solitons, quasisolitons, collapses or broad collapses. As a result, weak and strong turbulence coexist. In situations where coherent structures cannot develop, weak turbulence dominates. Even though this is primarily a review paper, new results are presented as well, especially on selforganized criticality and on quasisolitonic turbulence. © 2004 Elsevier B.V. All rights reserved.
 Dyachenko, A. I., Korotkevich, A. O., & Zakharov, V. E. (2003). Decay of the monochromatic capillary wave. JETP Letters, 77(9), 477481.More infoAbstract: It was demonstrated by direct numerical simulation that, in the case of weakly nonlinear capillary waves, one can get resonant waves interaction on the discrete grid when resonant conditions are never fulfilled exactly. The waves's decay pattern was obtained. The influence of the mismatch of resonant condition was studied as well. © 2003 MAIK "Nauka/Interperiodica".
 Dyachenko, A. I., Korotkevich, A. O., & Zakharov, V. E. (2003). Weak turbulence of gravity waves. JETP Letters, 77(10), 546550.More infoAbstract: For the first time weak turbulent theory was demonstrated for surface gravity waves. Direct numerical simulation of the dynamical equations shows Kolmogorov turbulent spectra as predicted by analytical analysis [1] from kinetic equation. © 2003 MAIK "Nauka/Interperiodica".
 Pushkarev, A., Resio, D., & Zakharov, V. (2003). Weak turbulent approach to the windgenerated gravity sea waves. Physica D: Nonlinear Phenomena, 184(14), 2963.More infoAbstract: We performed numerical simulation of the kinetic equation describing behavior of an ensemble of randomphase, spatially homogeneous gravity waves on the surface of the infinitely deep ocean. Results of simulation support the theory of weak turbulence not only in its basic points, but also in many details. The weak turbulent theory predicts that the main physical processes taking place in the wave ensemble are downshift of spectral peak and "leakage" of energy and momentum to the region of very small scales where they are lost due to local dissipative processes. Also, the spectrum of energy right behind the spectral peak should be close to the weak turbulent Kolmogorov spectrum which is the exact solution of the stationary kinetic (Hasselmann) equation. In a general case, this solution is anisotropic and is defined by two parameters  fluxes of energy and momentum to high wave numbers. Even in the anisotropic case the solution in the high wave number region is almost proportional to the universal form ω4. This result should be robust with respect to change of the parameters of forcing and damping. In all our numerical experiments, the ω4 Kolmogorov spectrum appears in very early stages and persists in both stationary and nonstationary stages of spectral development. A very important aspect of the simulations conducted here was the development of a quasistationary wave spectrum under wind forcing, in absence of any dissipation mechanism in the spectral peak region. This equilibrium is achieved in the spectral range behind the spectral peak due to compensation of wind forcing and leakage of energy and momentum to high wave numbers due to nonlinear fourwave interaction. Numerical simulation demonstrates slowing down of the shift of the spectral peak and formation of the bimodal angular distribution of energy in the agreement with field and laboratory experimental data. A more detailed comparison with the experiment can be done after developing of an upgraded code making possible to model a spatially inhomogeneous ocean. © 2003 Elsevier B.V. All rights reserved.
 Bogdanov, L. V., & Zakharov, V. E. (2002). The Boussinesq equation revisited. Physica D: Nonlinear Phenomena, 165(34), 137162.More infoAbstract: The continuous spectrum and soliton solutions for the Boussinesq equation are investigated using the ∂̄dressing method. Solitons demonstrate quite extraordinary behavior; they may decay or form a singularity in a finite time. Formation of singularity (collapse of solitons) for the Boussinesq equation was discovered several years ago. Systematic study of the solitonic sector is presented. © 2002 Elsevier Science B.V. All rights reserved.
 Onorato, M., Osborne, A. R., Serio, M., Resio, D., Pushkarev, A., Zakharov, V. E., & Brandini, C. (2002). Freely decaying weak turbulence for sea surface gravity waves. Physical Review Letters, 89(14), 144501/1144501/4.More infoAbstract: An overview is given on the observation that the Kolmogorov spectra predicted by weak turbulence theory for surface gravity waves agree with numerical simulations of the Euler equations. This result is important for many fields in physics because it supports the idea that the formation of power laws in the power spectrum of ensembles of stochastic weakly nonlinearly interacting waves may generally be explained using weak turbulence theory.
 Zakharov, V. E., Dyachenko, A. I., & Vasilyev, O. A. (2002). New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. European Journal of Mechanics, B/Fluids, 21(3), 283291.More infoAbstract: A method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface was presented. The method was based on the combination of conformal mapping and Fourier transform. The method was found to be efficient for the study of strongly nonlinear effects in gravity waves including wave breaking and the formation of rogue waves.
 Dias, F., Guyenne, P., & Zakharov, V. E. (2001). Kolmogorov spectra of weak turbulence in media with two types of interacting waves. Physics Letters, Section A: General, Atomic and Solid State Physics, 291(23), 139145.More infoAbstract: A system of onedimensional equations describing media with two types of interacting waves is considered. This system can be viewed as an alternative to the model introduced by Majda, McLaughlin and Tabak in 1997 for assessing the validity of weak turbulence theory. The predicted Kolmogorov solutions are the same in both models. The main difference between both models is that coherent structures such as wave collapses and quasisolitons cannot develop in the present model. As shown recently these coherents structures can influence the weakly turbulent regime. It is shown here that in the absence of coherent structures weak turbulence spectra can be clearly observed numerically. © 2001 Elsevier Science B.V. All rights reserved.
 Zakharov, V. E. (2001). Integration of the GaussCodazzi equations. Theoretical and Mathematical Physics, 128(1), 946956.More infoAbstract: The GaussCodazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in ℝ3 are integrable by the dressing method. This method allows constructing classes of Combescureequivalent surfaces with the same "rotation coefficients." Each equivalence class is defined by a function of two variables ("master function of a surface"). Each class of Combescureequivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.
 Zakharov, V. E., & Meister, C. . (2001). Proton cyclotron instability in the drifting magnetoplasma. Astronomische Nachrichten, 322(1), 5764.More infoAbstract: A general expression for the tensor of the dielectrical susceptibility in an anisotropic plasma with particle drifts is derived and the dispersion equation is found for waves propagating in arbitrary direction with respect to the mean magnetic field. The dispersion equation is solved for the case of electromagnetic ioncyclotron waves. It is found that in the plasma of the auroral magnetosphere strong plasma instability may occur so that the value of the growth rate of the waves is of the order of the wave frequency. Besides, the plasma instability is excited at less values of the wave number if the magnetospheric altitude becomes larger.
 Zakharov, V. E., Guyenne, P., Pushkarev, A. N., & Dias, F. (2001). Wave turbulence in onedimensional models. Physica D: Nonlinear Phenomena, 152153, 573619.More infoAbstract: A twoparameter nonlinear dispersive wave equation proposed by Majda, McLaughlin and Tabak is studied analytically and numerically as a model for the study of wave turbulence in onedimensional systems. Our ultimate goal is to test the validity of weak turbulence theory. Although weak turbulence theory is independent on the sign of the nonlinearity of the model, the numerical results show a strong dependence on the sign of the nonlinearity. A possible explanation for this discrepancy is the strong influence of coherent structures  wave collapses and quasisolitons  in wave turbulence. © 2001 Published by Elsevier Science B.V.
 Zakharov, V. E., Vasilyev, O. A., & Dyachenko, A. I. (2001). Kolmogorov spectra in onedimensional weak turbulence. JETP Letters, 73(2), 6365.More infoAbstract: In this article, we report the results of our numerical simulation of a onedimensional modified MMT model, which includes the processes of "onetothree" wave interactions. We show that this model, with properly chosen parameters, behaves according to the weakturbulence theory. In particular, it demonstrates the validity of the Kolmogorov spectrum over a wide range of wave numbers. © 2001 MAIK "Nauka/Interperiodica".
 Guyenne, P., Zakharov, V., Pushkarev, A., & Dias, F. (2000). Wave turbulence in onedimensional models. Comptes Rendus de l'Academie de Sciences  Serie IIb: Mecanique, 328(10), 757762.More infoAbstract: In this note, we assess the validity of weak turbulence theory by numerically investigating a threeparameter nonlinear dispersive wave equation. The predicted powerlaw solutions are explicitly determined and then compared with the numerical results. For both signs of nonlinearity, we show that the weakly turbulent regime is strongly influenced by the presence of coherent structures. These are wave collapses and quasisolitons. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.
 Pushkarev, A. N., & Zakharov, V. E. (2000). Turbulence of capillary waves  theory and numerical simulation. Physica D: Nonlinear Phenomena, 135(1), 98116.More infoAbstract: An ensemble of weakly interacting capillary waves on a free surface of deep ideal fluid is described statistically by methods of weak turbulence. The stationary kinetic equations for capillary waves have an exact Kolmogorov solution which gives for the spatial spectrum of elevations asymptotics Ik = C(P1/2/σ3/4)k19/4. The Kolmogorov constant C is found analytically together with the interval of locality in Kspace. Direct numerical simulation of the dynamical equations in the approximation of small surface angles confirms the presence of almost isotropic Kolmogorov spectrum in the large k region. Besides, at small amplitudes of the pumping, an essentially new phenomenon is found: 'frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k. This phenomenon is connected with the finiteness of the region (or, in other words, discreteness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.
 Perrie, W., & Zakharov, V. (1999). Equilibrium range cascades of windgenerated waves. European Journal of Mechanics, B/Fluids, 18(3), 365371.More infoAbstract: It was established theoretically by Zakharov and Filonenko (1966) that the direct cascade region of the equilibrium range of the spectrum should follow an f4 variation. This has since been verified from experimental data, by Toba (1973), Donelan et al (1985) and others. In this study we present a numerical verification of this f4 variation, assuming physically realistic parameterizations for nonlinear wavewave interactions, Snl, for energy input to waves by the wind, Sin and removed by wave  breaking dissipation, Sds.
 Zakharov, V. (1999). Statistical theory of gravity and capillary waves on the surface of a finitedepth fluid. European Journal of Mechanics, B/Fluids, 18(3), 327344.More infoAbstract: In many physical situations, the oscillations of the free surface of a fluid are a random process in space and time. This is equally correct for ripples in a tea cup as well as for large ocean waves. In both cases, the situation must be described by the averaged equations imposed on a certain set of correlation functions. The derivation of such equations is not a simple problem even on a `physical' level of rigor. It is especially important to determine correctly the conditions of applicability for a given statistical description. For some physical reasons they might happen to be narrow. In this context, the statistical description of potential surface waves on the surface of an ideal fluid of finite depth is described.
 Zakharov, V. E., & Manakov, S. V. (1999). On propagation of short pulses in strong dispersion managed optical lines. JETP Letters, 70(9), 578582.More infoAbstract: We show that the propagation of short pulses in optical lines with strong dispersion management is described by an integrable Hamiltonian system. The leading nonlinear effect is the formation of a collective dispersion which is a result of the interaction of all pulses propagating along the line. © 1999 American Institute of Physics.
 Zakharov, V. E., & Meister, C. . (1999). Currentdriven plasma turbulence in a magnetic flux tube. Astronomische Nachrichten, 320(6), 425436.More infoAbstract: The behaviour of a multicomponent anisotropic plasma in a magnetic flux tube is studied in the presence of currentdriven electrostatic ioncyclotron turbulence. The plasma transport is considered in both parallel and perpendicular directions with respect to the given tube. As one of the sources of the parallel electric field, the anomalous resistivity of the plasma caused by the turbulence is taken into account. The acceleration and heating processes of the plasma are simulated numerically. It is found that at the upper boundary of the nightside auroral ionosphere, the resonant waveparticle interactions are most effective in the case of upward fieldaligned currents with densities of a few 106 A/m2. The occurring anomalous resistivity may cause differences of the electric potential along the magnetic field lines of some kV. Further it is shown that the thickness of the magnetic flux tube and the intensity of the convection strongly influence the turbulent plasma heating.
 Zakharov, V. E., & Meister, C. . (1999). Transport of thermal plasma above the auroral ionosphere in the presence of electrostatic ioncyclotron turbulence. Annales Geophysicae, 17(1), 2736.More infoAbstract: The electron component of intensive electric currents flowing along the geomagnetic field lines excites turbulence in the thermal magnetospheric plasma. The protons are then scattered by the excited electromagnetic waves, and as a result the plasma is stable. As the electron and ion temperatures of the background plasma are approximately equal each other, here electrostatic ioncyclotron (EIC) turbulence is considered. In the nonisothermal plasma the ionacoustic turbulence may occur additionally. The anomalous resistivity of the plasma causes largescale differences of the electrostatic potential along the magnetic field lines. The presence of these differences provides heating and acceleration of the thermal and energetic auroral plasma. The investigation of the energy and momentum balance of the plasma and waves in the turbulent region is performed numerically, taking the magnetospheric convection and thermal conductivity of the plasma into account. As shown for the quasisteady state, EIC turbulence may provide differences of the electric potential of ΔV ≈ 110 kV at altitudes of 500 < A < 10000 km above the Earth's surface. In the turbulent region, the temperatures of the electrons and protons increase only a few times in comparison with the background values.
 Zakharov, V. E., & Pushkarev, A. N. (1999). Diffusion model of interacting gravity waves on the surface of deep fluid. Nonlinear Processes in Geophysics, 6(1), 110.More infoAbstract: A simple phenomenological model for nonlinear interactions of gravity waves on the surface of deep water is developed. The S(nl) nonlinear interaction term in the kinetic equation for wave action in replaced by the nonlinear secondorder diffusiontype operator. Analytical and numerical studies show that the new model gives a reasonably good description of a real situation, consuming three order of magnitude less computer time.
 Zakharov, V. E. (1998). Description of the northogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type, I: Integration of the lamé equations. Duke Mathematical Journal, 94(1), 103139.
 Zakharov, V. E., & Kuznetsov, E. A. (1998). Optical solitons and quasisolitons. Journal of Experimental and Theoretical Physics, 86(5), 10351046.More infoAbstract: Optical solitons and quasisolitons are investigated in reference to Cherenkov radiation. It is shown that both solitons and quasisolitons can exist, if the linear operator specifying their asymptotic behavior at infinity is signdefinite. In particular, the application of this criterion to stationary optical solitons shifts the soliton carrier frequency at which the first derivative of the dielectric constant with respect to the frequency vanishes. At that point the phase and group velocities coincide. Solitons and quasisolitons are absent, if the thirdorder dispersion is taken into account. The stability of a soliton is proved for fourth order dispersion using the signdefiniteness of the operator and integral estimates of the Sobolev type. This proof is based on the boundedness of the Hamiltonian for a fixed value of the pulse energy. © 1998 American Institute of Physics.
 Zakharov, V. E., & Manakov, S. V. (1998). On reduction in systems integrable by method of inverse problem dispersion. Doklady Akademii Nauk, 360(3), 324327.
 Zakharov, V. E., & Manakov, S. V. (1998). Reductions in systems integrated by the method of the inverse scattering problem. Doklady Mathematics, 57(3), 471474.
 Pushkarev, A. N., & Zakharov, V. E. (1997). Turbulence of capillary wavestheory and numerical simulation. Advances in Fluid Mechanics, 17, 111131.More infoAbstract: An ensemble of weaklyinteracting capillary waves on a free surface of deep ideal fluid is described statistically by the methods of weak turbulence. The stationary kinetic equation for capillary waves has an exact Kolmogorov solution which gives, for a spatial spectrum of elevations, asymptotics Ik = C (P 1/2 /σ3/4 k19/4. The Kolmogorov constant C is found analytically together with the interval of locality in k̄space. Direct numerical simulation of dynamical equations, in the small surface angles approximation, confirms the presence of an almost isotropic Kolmogorov spectrum in the large k̄ region. In the pumping region, the spectrum is defined by nonresonant processes of nonlinear damping. This fact can be explained by the narrowness of the inertial interval. Moreover, at small amplitudes of the pumping, an essentially new phenomenon is found: `frozen' turbulence, in which, despite the big number of interacting waves (of the order of 100) there is no energy flux toward high k̄. This phenomenon is connected with the finiteness of the region (or, in other words, the discreetness of the spectrum in Fourier space). This is believed to be universal for different sorts of nonlinear systems.
 Zakharov, V. E., & Kuznetsov, E. A. (1997). Hamiltonian formalism for nonlinear waves. PhysicsUspekhi, 40(11), 10871116.More infoAbstract: The Hamiltonian description of hydrodynamic type systems in application to plasmas, hydrodynamics, and magnetohydrodynamics is reviewed with emphasis on the problem of introducing canonical variables. The relation to other Hamiltonian approaches, in particular naturalvariable Poisson brackets, is pointed out. It is shown that the degeneracy of noncanonical Poisson brackets relates to a special type of symmetry, the relabeling transformations of fluidparticle Lagrangian markers, from which all known vorticity conservation theorems, such as Ertel's, Cauchy's, Kelvin's, as well as vorticity frozenness and the topological Hopf invariant, are derived. The application of canonical variables to collisionless plasma kinetics is described. The Hamiltonian structure of Benney's equations and of the Rossby wave equation is discussed. DaveyStewartson's equation is given the Hamiltonian form. A general method for treating weakly nonlinear waves is presented based on classical perturbation theory and the Hamiltonian reduction technique. ©1997 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences.
 Zakharov, V. E., & Meister, C. . (1997). Largescale transport of plasma in the Earth's plasma sheet: Comparative analysis for adiabatic and nonadiabatic cases. Astronomische Nachrichten, 318(1), 5162.More infoAbstract: A system of multifluid MHDequations is used to compare adiabatic and nonadiabatic transport of the energetic particles in the magnetospheric plasma sheet. A "slowflow" approximation is considered to study largescale transport of the anisotropic plasma consisting of energetic electrons and protons. Nonadiabatic transport of the energetic plasma is caused by scattering of the particles in the presence of both wave turbulence and arbitrary timevarying electric fields penetrating from the solar wind into the magnetosphere. The plasma components are devided into particle populations defined by their given initial effective values of the magnetic moment per particle. The spatial scales are also given to estimate the nonuniformity of the geomagnetic field along the chosen mean path of a particle. The latters are used to integrate approximately the system of MHDequations along each of these paths. The behaviour of the magnetic moment mentioned above and of the parameter which characterizes the pitchangle distribution of the particles are studied selfconsistently in dependence on the intensity of nonadiabatic scattering of the particles. It is shown that, in the inner magnetosphere, this scattering influences the particles in the same manner as pitchangle diffusion does. It reduces the pitchangle anisotropy in the plasma. The state of the plasma may be unstable in the current sheet of the magnetotail. If the initial state of the plasma does not correspond to the equilibrium one, then, in this case, scattering influences the particles so as to remove the plasma further from the equilibrium state. The coefficient of the particle diffusion across the geomagnetic field lines is evaluated. This is done by employing the Langevin approach to take the stochastic electric forces acting on the energetic particles in the turbulent plasma into account. The behaviour of the energy density of electrostatic fluctuations in the magnetosphere is estimated.
 Dyachenko, A. I., & Zakharov, V. E. (1996). Toward an integrable model of deep water. Physics Letters, Section A: General, Atomic and Solid State Physics, 221(12), 8084.More infoAbstract: Using the combination of the canonical formalism and conformal mapping, a theory of the free surface of deep water in the approximation of a high curvature is developed. It is shown that the numerical simulation is in excellent agreement with the analytical description.
 Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D., & Zakharov, V. E. (1996). Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Physics Letters, Section A: General, Atomic and Solid State Physics, 221(12), 7379.More infoAbstract: Using the combination of the canonical formalism for freesurface hydrodynamics and conformal mapping to the halfplane we obtain a simple system of pseudodifferential equations for the surface shape and hydrodynamic potential. The system is welladjusted for a numerical simulation. Some typical results of such a simulation are presented.
 Dyachenko, A. I., Zakharov, V. E., & Kuznetsov, E. A. (1996). Nonlinear dynamics of the free surface of an ideal fluid. Plasma Physics Reports, 22(10), 829840.More infoAbstract: On the basis of a combination of the canonical formalism for describing freesurface fluid of finite depth and conformal mapping to a horizontal strip, a simple system of pseudodifferential equations is obtained for the surface shape η and the hydrodynamic velocity potential Ψ. This system can be effectively studied in the case of smallangle deviations of the fluid surface from a plane surface (without gravity and surface tension) and, in the opposite case, when the value of the Jacobian of the conformai mapping is very large in the vicinity of some point on the surface. In the first case, the complex velocity potential obeys a Hopftype differential equation. In this approximation, singularities on the surface appear as a result of violation of the analyticity of the potential in the strip due to the motion of singularities (i.e., the branch points) towards the strip boundaries. These singularities are shown to be mainly of the root type. For such singularities, the curvature becomes infinite, ηxx∼x1/2, but the surface remains smooth. In the other limiting case, in a firstorder expansion in inverse powers of the Jacobian, the whole system of equations can be reduced to a single equation that coincides with the wellknown Laplacian Growth Equation. Within this model, remarkable special solutions of the system can be constructed describing such physical phenomena as the formation of fingertype configurations or the generation of separate droplets changing the surface topology.
 Pushkarev, A. N., & Zakharov, V. E. (1996). Turbulence of capillary waves. Physical Review Letters, 76(18), 33203323.More infoAbstract: A numerical model for direct simulation of the surface of ideal fluid based on the expansion of the Hamiltonian of the surface up to terms of fourth order is developed. For the case of capillary wave we observe the formation of powerlike spectrum of spatial elevations close to one predicted by weakturbulent theory Ik ≃ k19/4, which previously was not confirmed either experimentally or numerically.
 Zakharov, V. E., & Dyachenko, A. I. (1996). HighJacobian approximation in the free surface dynamics of an ideal fluid. Physica D: Nonlinear Phenomena, 98(24), 652664.More infoAbstract: Using a combination of the canonical formalism for freesurface hydrodynamics and conformal mapping to a horizontal strip we obtain a simple system of pseudodifferential equations for the surface shape and hydrodynamic velocity potential. The system is wellsuited for numerical simulation. It can be effectively studied in the case when the Jacobian of the conformal mapping takes very high values in the vicinity of some point on the surface. At first order in an expansion in inverse powers of the Jacobian one can reduce the whole system of equations to a single equation which coincides with the wellknown Laplacian Growth Equation (LGE). In the framework of this model one can construct remarkable special solutions of the system describing such physical phenomena as formation of fingertype configurations or changing of the surface topology  generation of separate droplets.
 Zakharov, V. E., & Pudovkin, M. I. (1996). Electrodynamic coupling between ionospheric convection patterns in the northern and southern hemispheres. Annales Geophysicae, 14(4), 419430.More infoAbstract: A numerical model of the highlatitude ionospheric electric field is presented. To perform the calculations, a model of the fieldaligned current source is proposed. The electric field patterns are calculated consistently both in the northern and southern hemispheres. Effects of season, universal time, solar and geomagnetic activity, the neutral atmosphere winds, and of the IMF sector structure are considered. In particular, dynamics of the parameters of convection cells are investigated that depend on the action of these factors. Comparison of the results with experimental data is carried out.
 Dyachenko, A. I., Lvov, Y. V., & Zakharov, V. E. (1995). Fivewave interaction on the surface of deep fluid. Physica D: Nonlinear Phenomena, 87(14), 233261.More infoAbstract: This article deals with the studying of the interaction of gravity waves propagating on the surface of an ideal fluid of infinite depth. The system of the corresponding equation is proven to be integrable up to the fourth order in power of steepness of the waves, but to be nonintegrable in the next, fifth, order. An exact formula for the fivewave scattering matrix element is obtained using diagram technique on the resonant surface. The stationary solutions of the fivewave kinetic equation are studied as well. © 1995.
 Musher, S. L., Rubenchik, A. M., & Zakharov, V. E. (1995). Weak Langmuir turbulence. Physics Report, 252(4), 177274.More infoAbstract: Weak turbulence theory presents a regular method for a statistical description of nonlinear wave interactions. The present review deals with an application of weak turbulence theory to Langmuir wave turbulence. Our main attention is devoted to a plasma with comparable ion and electron temperatures, both magnetized and unmagnetized. In this practically important situation ionsound motions are heavily damped, which simplifies the physics of nonlinear phenomena. We will demonstrate that the turbulence spectra are highly anisotropic and take the form of "jets" in kspace, and that the onset of a steady state is nontrivial and sometimes does not occur at all. On the base of the jetlike spectra approach it is possible to find the turbulence spectra, to evaluate the anomalous absorption rate and to determine the comparable role of the different absorption mechanisms for a number of practical problems: the excitation of waves by powerful electromagnetic radiation or by electron and ion beams. We demonstrate also that the range in which pure weak turbulence is valid is pretty narrow. The jetlike spectra structure stimulates a modulation instability and after that wave selffocusing and collapse. Then, weak and strong turbulence coexist. The final part of the review deals with the turbulence of nonisothermal plasmas when additional degrees of freedom are excited. We demonstrate that the ideas, models and methods, presented in this review, give us a chance to advance greatly in the understanding of turbulence patterns.
 Bogdanov, L. V., & Zakharov, V. E. (1994). Integrable (1+1)dimensional systems and the Riemann problem with a shift. Inverse Problems, 10(4), 817835.More infoAbstract: We study (1+1)dimensional integrable systems considering them as special cases of the more general (2+1)dimensional systems. Using the nonlocal delta problem approach in (2+1) dimensions, we show that the delta problem with a shift and (for the decreasing solutions) the Riemann problem with a shift arise naturally in (1+1) dimensions. The Boussinesq equation and the firstorder relativisticallyinvariant systems are investigated. The approach developed also allows us to investigate the structure of the continuous spectrum and the inverse scattering problem for an arbitraryorder ordinary differential operator on the infinite line.
 Dyachenko, A. I., & Zakharov, V. E. (1994). Is freesurface hydrodynamics an integrable system?. Physics Letters A, 190(2), 144148.More infoAbstract: A strong argument is found in support of the integrability of freesurface hydrodynamics in the onedimensional case. It is shown that the first term in the perturbation series in powers of nonlinearity is identically equal to zero, the consequences of which are discussed as well. © 1994.
 Kuznetsov, E. A., Spector, M. D., & Zakharov, V. E. (1994). Formation of singularities on the free surface of an ideal fluid. Physical Review E, 49(2), 12831290.More infoAbstract: It is shown that the equations of motion of an ideal fluid with a free surface in the absence of both gravitational and capillary forces can be effectively solved in the approximation of small surface angles. It can be done by means of an analytical continuation of both the velocity potential on the surface and its elevation. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible. The first one is of the root character provided by the analytical behavior of the velocity potential. In this case the process of the singularity formation, representing some analog of the wave breaking, is described as a motion of branch points in the complex plane towards the real axis. The second type can be obtained as a result of the interaction of two movable branch points leading to the formation of wedges on the free surface. The third kind is associated with a motion in the complex plane of the singular points of the analytical continuation of the elevation, resulting in the appearance of strong singularities for the surface profile. © 1994 The American Physical Society.
 Nazarenko, S. V., & Zakharov, V. E. (1994). The role of the convective modes and sheared variables in the Hamiltonian dynamics of uniformshearflow perturbations. Physics Letters A, 191(56), 403408.More infoAbstract: The convective modes in shear flows are associated with the algebraic time evolution of perturbations. The sheared variables, corresponding to the amplitudes of the convective modes, are shown to diagonalize the quadratic Hamiltonian. In other words, the sheared variables are the normal variables of the Hamilton equations, and the convective modes are the normal modes in the Hamiltonian dynamics. © 1994.
 Kuznetsov, E. A., Spector, M. D., & Zakharov, V. E. (1993). Surface singularities of ideal fluid. Physics Letters A, 182(46), 387393.More infoAbstract: We show that the equations of motion of an ideal fluid with a free surface due to inertial forces only can be effectively solved in the approximation of small surface angles. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible: the root ones for which the process of the singularity formation represents some analog of the wave breaking; singularities in the form of wedges on the interface; the floating ones associated with motion in the complex plane of the singular points of the analytical continuation of the surface shape. © 1993.
 Yakhot, V., & Zakharov, V. (1993). Hidden conservation laws in hydrodynamics; energy and dissipation rate fluctuation spectra in strong turbulence. Physica D: Nonlinear Phenomena, 64(4), 379394.More infoAbstract: The Hamiltonian formulation of hydrodynamics in Clebsch variables is used for construction of a statistical theory of turbulence. The statistics of the Clebsch field is assumed to be close to Gaussian. It is shown that the energy spectrum consists of two ranges with E(k)≈k 5 3 and E(k)≈k1. The spectrum of the dissipation rate fluctuations has three scaling regimes: Eε{lunate}(k)≈k1; k 1 3 and k0 at the large, intermediate and small scales, respectively. The origin of the exponential distribution of velocity differences is discussed. The new scaling regime corresponds to a hidden conservation law, discovered in the Clebsch formulation of hydrodynamics. It is shown that viscous effects are responsible for production of the conserved quantity. The theoretical predictions are compared with results of numerical simulations of decaying turbulence. © 1993.
 Dyachenko, A. I., Nazarenko, S. V., & Zakharov, V. E. (1992). Wavevortex dynamics in drift and βplane turbulence. Physics Letters A, 165(4), 330334.More infoAbstract: For the theory of drift plasma and βplane geophysical dynamics both largescale vortex and smallscale wave components are important: linear excitation and dissipation occur mainly at small scales, while concentration of the energy spectrum takes place (through the inverse cascade) at large vortices. Based on the time and space separation of these scales averaged evolution equations are derived. The equation for the small scales describes the propagation of highfrequency quanta on the background of a flow produced by largescale vortices; this equation provides the conservation of the spectral density of the potential enstrophy of small scales. The equation for the largescale component is the CharneyHasegawaMima equation with a source term having the form of the ponderomotive force and providing the inverse energy cascade from small to large scales. A new computational approach for the modeling of drift and βplane turbulence is proposed on the basis of the equations obtained  the quantum in the cell method. © 1992.
 Fokas, A. S., & Zakharov, V. E. (1992). The dressing method and nonlocal RiemannHilbert problems. Journal of Nonlinear Science, 2(1), 109134.More infoAbstract: We consider equations in 2+1 solvable in terms of a nonlocal RiemannHilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using the Nwave interactions, the DaveyStewartson I, and the KadomtsevPetviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of the Nwave interactions. © 1992 SpringerVerlag New York Inc.
 Nazarenko, S. V., & Zakharov, V. E. (1992). Kinetic equation for point vortices in a shear flow. Physica D: Nonlinear Phenomena, 56(4), 381388.More infoAbstract: The kinetic equation for a set of Stewart point vortices moving on the background of a shear flow with uniform vorticity is derived. Besides the total number of vortices with certain amplitude values, this equation provides conservation of the total vorticity of vortex gas on each shear flow stream line. Such conservation laws prevent, in the general case, relaxation of the system to the Maxwell equilibrium distribution of vortices. In fact, the system is shown to tend to an equilibrium state, in which vortex gas vorticity on each stream line of shear flow is formed by vortices with equal amplitude signs, only the most intense vortices being concentrated in more exited regions, while less intensive vortices are concentrated at the periphery of perturbed domains. Applicability of the kinetic equation obtained for the description of Helmholtz vortices sets is discussed. © 1992.
 Pudovkin, M. I., & Zakharov, V. E. (1992). Numerical simulations of magnetosphereionosphere coupling, including kinetic effects in the plasma sheet. Planetary and Space Science, 40(8), 10711080.More infoAbstract: A new version of a selfconsistent model of ionospheremagnetosphere coupling is presented. It is based on a kinetic description of the transport of hot protons and electrons in the Earth's magnetosphere, including their precipitation into the auroral ionosphere. Anisotropy of the pitchangle distribution is also taken into account for these particles. The influence of spatial inhomogeneity of the heightintegrated Pedersen and Hall ionospheric conductivities on the electric fields and currents in the magnetosphere and ionosphere is studied. These conductivities are calculated as functions of the precipitated auroral electron fluxes and solar zenith angle. Comparison of the results with experimental data is carried out. © 1992.
 Balk, A. M., Nazarenko, S. V., & Zakharov, V. E. (1991). New invariant for drift turbulence. Physics Letters A, 152(56), 276280.More infoAbstract: A new invariant for drift wave (or Rossby wave) turbulence in two cases, (1) in zonal flow and (2) in the large scale range, is discovered. This invariant is proved to be a unique additional invariant (besides energy and momentum). Thus the first examples of wave systems with a finite number of additional invariants are obtained. A new Kolmogorovtype spectrum with the flux of the additional invariant through the scales is derived and the structure of fluxes in the kspace of invariants is analyzed. © 1991.
 Duncan, D. B., Eilbeck, J. C., Walshaw, C. H., & Zakharov, V. E. (1991). Solitary waves on a strongly anisotropic KP lattice. Physics Letters A, 158(34), 107111.More infoAbstract: We consider a strongly anisotropic 2D atomic lattice model which in a continuum limit becomes the KadomtsevPetviashvili equation. Solitary waves on this lattice are studied by a variety of analytic and numerical tecniques. © 1991.
 Dyachenko, A. I., Pushkarev, A. N., Rubenchik, A. M., Sagdeev, R. Z., Shvets, V. F., & Zakharov, V. E. (1991). Computer simulation of Langmuir collapse. Physica D: Nonlinear Phenomena, 52(1), 78102.More infoAbstract: The problem of Langmuir wave collapse in 2D and 3D plasma is considered. A new approach for computer simulation of this phenomenon is proposed, which includes two different theoretical models: averaged dynamical equations and Vlasov's set of equations. It allows to take into account all essential effects during the whole process of collapse and, hence, to get a reliable picture of the collapse in detail and to save markedly computer resources. Peculiarities of the numeric methods are also discussed. © 1991.
 Kosmatov, N. E., Shvets, V. F., & Zakharov, V. E. (1991). Computer simulation of wave collapses in the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 52(1), 1635.More infoAbstract: The selftuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear Schrödinger equation is developed. The scheme allows to reach extremely high field amplitudes ( ψ2 ψ02∼1015 without lack of validity. Numerical results are compared with analytical predictions. © 1991.
 Balk, A. M., Nazarenko, S. V., & Zakharov, V. E. (1990). On the nonlocal turbulence of drift type waves. Physics Letters A, 146(4), 217221.More infoAbstract: Two new effects the drift turbulence can display are disclosed: (1) the turbulence spectrum in kspace separates into unconnected components of large and small scales, (2) the very presence of weak smallscale turbulence imposes rigid restrictions on powerful largescale components. © 1990.
 Zakharov, V. E. (1989). The algebra of integrals of motion of twodimensional hydrodynamics in clebsch variables. Functional Analysis and Its Applications, 23(3), 189196.
 Zakharov, V. E., & Piterbarg, L. I. (1988). Canonical variables for rossby waves and plasma drift waves. Physics Letters A, 126(89), 497500.More infoAbstract: The Poisson bracket defining a hamiltonian formulation of the Rossby wave equation is transformed to the Gardner bracket via a special functional change. The diagonal form of the bracket enables us to introduce the normal canonical variables in the considered hamiltonian system. The first terms of the hamiltonian expansion in powers of the canonical variables are calculated. The proposed method of the Poisson bracket diagonalization is relevant for other physically significant problems: barotropic waves above an uneven bottom, waves in the presence of a scalar nonlinearity and quasigeostrophic flow of a vertically stratified fluid, including the baroclinic effects of topography as dynamical boundary conditions. © 1988.
 Zakharov, V. E., & Schulman, E. I. (1988). On additional motion invariants of classical Hamiltonian wave systems. Physica D: Nonlinear Phenomena, 29(3), 283320.More infoAbstract: It is shown that the existence of an analytic invariant in addition to the natural ones (momentum, energy and, in some cases, "number of particles") leads to the existence of infinitely many such invariants. Nevertheless, the existence of the additional motion invariant does not guarantee complete integrability. Complete integrability follows from the existence of an additional invariant only if the dispersion law is nondegenerative with respect to decays. If the dispersion law is degenerative, the "number of" motion invariants is insufficient for complete integrability and the Smatrix is factorized via decay processes "one into two" with real intermediate particles. In this paper we present also our results concerning enumeration of degenerative dispersion laws. © 1988.
 Burtsev, S. P., Zakharov, V. E., & Mikhailov, A. V. (1987). Inverse scattering method with variable spectral parameter. Theoretical and Mathematical Physics, 70(3), 227240.
 Kuznetsov, E. A., Rubenchik, A. M., & Zakharov, V. E. (1986). Soliton stability in plasmas and hydrodynamics. Physics Reports, 142(3), 103165.More infoAbstract: The stability of solitons is reviewed for nonlinear conservative media. The main attention is paid to the description of the methods: perturbation theory, inverse scattering transform, Lyapunov method. Its applications are demonstrated in detail for the nonlinear Schrödinger equation, the KdV equation, and their generalizations. Applications to problems in plasma physics and hydrodynamics are considered. © 1986.
 Zakharov, V. E. (1986). Shock waves propagating on solitons on the surface of a fluid. Radiophysics and Quantum Electronics, 29(9), 813817.
 Zakharov, V. E., & Kuznetsov, E. A. (1986). Multiscale expansions in the theory of systems integrable by the inverse scattering transform. Physica D: Nonlinear Phenomena, 18(13), 455463.More infoAbstract: It is shown that using multiscale expansions conventionally employed in the theory of nonlinear waves one can transform systems integrable by the IST method into other systems of this type. © 1986.
 Gabitov, I. R., Zakharov, V. E., & Mikhailov, A. V. (1985). MaxwellBloch equation and the inverse scattering method. Theoretical and Mathematical Physics, 63(1), 328343.
 Zakharov, V. E., & Lavrova, O. (1985). Rarefaction Zones on a Background of Onedimensional Wave Motion.. Izvestia Akademii nauk SSSR. Fizika atmosfery i okeana, 21(12), 12951298.More infoAbstract: A study is made of the possibility of the existence of transverse 'rarefaction zones' on a background of onedimensional wave motion. The onedimensional wave motion is described by an equation, which is a generalization of the wellknown Schroedinger nonlinear equation, for a complex function representing the time envelope of the KTh Fourier component of the wave motion. In two model cases it is shown that such solutions of 'rarefaction soliton' type can exist.
 Zakharov, V. E., & Manakov, S. V. (1985). Construction of higherdimensional nonlinear integrable systems and of their solutions. Functional Analysis and Its Applications, 19(2), 89101.
 Zakharov, V. E., & Manakov, S. V. (1985). Multidimensional nonlinear integrable systems and methods for constructing their solutions. Journal of Soviet Mathematics, 31(6), 33073316.More infoAbstract: A new method for constructing multidimensional nonlinear integrable systems and their solutions by means of a nonlocal Riemann problem is presented. This is the natural generalization of the method of the local Riemann problem to the case of several space variables and includes the wellknown ZakharovShabat method of dressing by Volterra operators. © 1985 Plenum Publishing Corporation.
 Zakharov, V. E., Musher, S. L., & Rubenchik, A. M. (1985). Hamiltonian approach to the description of nonlinear plasma phenomena. Physics Reports, 129(5), 285366.More infoAbstract: This review discusses the methods of description of nonlinear processes. We demonstrate the usefulness of the Hamiltonian approach to these problems. We show the existence of Hamiltonian structures for a number of plasma situations. The choice of normal variables results in a standard form of equations for all kinds of problems. The actual physics involved changes only dispersion laws and the structure of the matrix elements. This approach makes it possible to consider a number of problems in a unique way. We discuss the stability of monochromatic waves and the statistical description of a plasma. The connection between decay and modulational instability growth rates and matrix elements is demonstrated. The standard form of the equations enables us to introduce a statistical description in a very simple way. We discuss the usual kinetic wave equations and their generalization for inhomogeneous turbulence and turbulence excited by a coherent pump. We pay special attention to the problem of Langmuir turbulence. The average dynamical equations are deduced in a consistent way and we present a detailed discussion of the limits of this description. © 1985.
 Zakharov, V. E., & Konopelchenko, B. G. (1984). On the theory of recursion operator. Communications in Mathematical Physics, 94(4), 483509.More infoAbstract: The general structure and properties of recursion operators for Hamiltonian systems with a finite number and with a continuum of degrees of freedom are considered. Weak and strong recursion operators are introduced. The conditions which determine weak and strong recursion operators are found. In the theory of nonlinear waves a method for the calculation of the recursion operator, which is based on the use of expansion into a power series over the fields and the momentum representation, is proposed. Within the framework of this method a recursion operator is easily calculated via the Hamiltonian of a given equation. It is shown that only the onedimensional nonlinear evolution equations can posses a regular recursion operator. In particular, the KadomtsevPetviashvili equation has no regular recursion operator. © 1984 SpringerVerlag.
 Zakharov, V. E., & Mikhailov, A. V. (1983). Method of the inverse scattering problem with spectral parameter on an algebraic curve. Functional Analysis and Its Applications, 17(4), 247251.
 Zakharov, V. E., & Zaslavskii, M. M. (1983). Dependence of the Wave Parameters on the Wind Velocity, the Duration of Its Action and the Fetch in the Weakturbulence Theory of Wind Waves.. Izvestia Akademii nauk SSSR. Fizika atmosfery i okeana, 19(4), 406415.More infoAbstract: The equation relating the frequency of the spectral maximum omega //n and the wave action flux p, which are the main parameters of the spectrum of sufficiently developed waves in the energycontaining range, is derived from the kinetic equation for the wind wave spectrum in the horizontally inhomogeneous and unsteady case. The closure of this equation is obtained using experimental data on the dependence of p equals pg**2/U//a**4 on omega //m equals omega //mU//a/g, and it is shown that for a description of wind waves an even simpler approximation of a constant action flux can be used, in which all the processes of interaction of the waves with the wind and the drift current can be taken into account by the sole parameterization constant p equals const equals 1. 4 multiplied by 10** minus **8.
 Anisimov, S. I., Berezovskiǐ, M., Ivanov, M. F., Petrov, I. V., Rubenchick, A. M., & Zakharov, V. E. (1982). Computer simulation of the Langmuir collapse. Physics Letters A, 92(1), 3234.More infoAbstract: Computer simulation of the collapse of Langmuir waves is carried out, using the PIC method. The highfrequency electric field energy has been shown to transform almost completely into electron energy at the final stage of the collapse. © 1982.
 Zakharov, V. E., & Sapronov, Y. (1982). AXIALLY SYMMETRICAL NUMERICAL MODEL FOR A TROPICAL CYCLONE.. Soviet meteorology and hydrology, 2126.More infoAbstract: The model is based on a LaxWendroff difference scheme having a second order of accuracy in space and time. This enables one to avoid artificialviscosity effects to a considerable extent and to perform experiments on the effects of the turbulent viscosity on the evolution of a model tropical cyclone (TC). The results show that the evolution of the TC is very sensitive to the ratio of the dissipation to the generation energy. This indicates that a very vigorous approach is necessary to the physical and numerical apsects of the description of these processes in TC models.
 Zakharov, V. E., & Schulman, E. I. (1982). To the integrability of the system of two coupled nonlinear Schrödinger equations. Physica D: Nonlinear Phenomena, 4(2), 270274.
 Zakharov, V. E., & Zaslavskii, M. M. (1982). Generation and Dissipation Ranges in the Kinetic Equation of the Weakly Turbulent Theory of Wind Waves.. Izvestia Akademii nauk SSSR. Fizika atmosfery i okeana, 18(10), 10661076.More infoAbstract: The possibilities of applying the philosophy of weakly turbulent Kolmogorov spectra to real wind waves are investigated. On the basis of theoretical estimates of the wavewind interaction and experimental data, it is shown that the generation ranges in the kinetic equation for a wind wave spectrum which make the principal contribution to the integral values of the momentum, energy and action flows to waves from the atmosphere are localized in a region of fairly small scales. The dissipation related to wave breaking is concentrated in an even more finescaled region, and therefore in the case of fairly developed wave motion one may expect that the scales of the energycontaining vicinity of the spectral maximum and the scales of the generation and dissipation ranges are far apart.
 Zakharov, V. E., & Zaslavskii, M. M. (1982). Kinetic Equation and Kolmogorov Spectra in the Weak Turbulence Theory of Wind Waves.. Izvestia Akademii nauk SSSR. Fizika atmosfery i okeana, 18(9), 970979.More infoAbstract: On the basis fo experimental data concerning the similarity between the energycarrying components of fairly developed wind waves and linear free waves, a kinetic equation of the weak turbulence approximation is used to describe them, which takes into account nonlinear interactions, runup from the wind, and dissipation. The special features of this approximation are discussed which are related to the appearance of a law of conservation of wave action specific to weakly nonlinear gravity waves in addition to the ordinary dynamic laws of conservation of momentum and energy. It is shown that in the isotropic case (when the wave momentum is equal to zero) the exact steady solutions of the kinetic equation outside the runup and dissipation ranges are two alternative Kolmogorov power spectra determined, respectively, by energy and action flows.
 Manakov, S. V., & Zakharov, V. E. (1981). Threedimensional model of relativisticinvariant field theory, integrable by the Inverse Scattering Transform. Letters in Mathematical Physics, 5(3), 247253.More infoAbstract: In the spacetime with signature (2,2) the selfduality equations in the specific case of potentials independent of one of the coordinates are reduced to a relativisticinvariant system in the (21)dimensional spacetime. A general solution of this system is constructed by means of IST. A soliton solution, finite in all directions, is discussed. It is found that there is no classical scattering of both solitons and continuous spectrum waves. © 1981 D. Reidel Publishing Company.
 Zakharov, V. E. (1981). On the Benney equations. Physica D: Nonlinear Phenomena, 3(12), 193202.
 Zakharov, V. E. (1980). Benney equations and quasiclassical approximation in the method of the inverse problem. Functional Analysis and Its Applications, 14(2), 8998.
 Zakharov, V. E., & Mikhailov, A. V. (1980). On the integrability of classical spinor models in twodimensional spacetime. Communications in Mathematical Physics, 74(1), 2140.More infoAbstract: Well known classical spinor relativisticinvariant twodimensional field theory models, including the GrossNeveu, VaksLarkinNambuJonaLasinio and some other models, are shown to be integrable by means of the inverse scattering problem method. These models are shown to be naturally connected with the principal chiral fields on the symplectic, unitary and orthogonal Lie groups. The respective technique for construction of the solitonlike solutions is developed. © 1980 SpringerVerlag.
 Zakharov, V. E., & Mikhailov, A. V. (1980). Variational principle for equations integrable by the inverse problem method. Functional Analysis and Its Applications, 14(1), 4344.
 Zakharov, V. E., & Schulman, E. I. (1980). Degenerative dispersion laws, motion invariants and kinetic equations. Physica D: Nonlinear Phenomena, 1(2), 192202.
 Zakharov, V. E., & Semenov, V. A. (1980). DYNAMIC BOUNDARY LAYER OF A MATURE HURRICANE.. Soviet meteorology and hydrology, 4147.More infoAbstract: The problem of the turbulent boundary layer of a stationary mature hurricane with an assigned distribution of the tangential velocity is examined. The system of equations of dynamics was closed by the equation for turbulence energy. The mixing length was profiled over the height and radius as applied to the problem. The formulated system of equations was solved by a numerical method based on the use of the momentum theorem. The results obtained correspond to known experimental data. The effect of various types of approximation in equations of dynamics on the results obtained is shown.
 Zakharov, V. E., & Shabat, A. B. (1979). Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Functional Analysis and Its Applications, 13(3), 166174.
 Zakharov, V. E., & Takhtadzhyan, L. A. (1979). Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theoretical and Mathematical Physics, 38(1), 1723.
 Belavin, A. A., & Zakharov, V. E. (1978). YangMills equations as inverse scattering problem. Physics Letters B, 73(1), 5357.More infoAbstract: Nonlinear selfduality equations are shown to be conditions of compatibility of two linear equations. All the Ninstanton fields are constructed explicitly. © 1978.
 Zakharov, V. E., & Sapronov, Y. (1978). PARAMETERIZATION OF THE CLOUD SYSTEM IN A HURRICANE.. Sov Meteorol Hydrol, 1622.More infoAbstract: Use of a model of a single stationary cloud is proposed for parameterization of cloudgroup activity. The boundary and background characteristics for the cloud are obtained by integration of a physicomathematical hurricane model. The possibility in principle of using the proposed parameterization to model the evolution of the hurricane is demonstrated. The parameterization model derived can be used to obtain averaged characteristics for a cloud group, such as are needed for simulation of hurricane modification.
 Manakov, S. V., Zakharov, V. E., Bordag, L. A., Its, A. R., & Matveev, V. B. (1977). Twodimensional solitons of the KadomtsevPetviashvili equation and their interaction. Physics Letters A, 63(3), 205206.More infoAbstract: Explicit analytic formulae for twodimensional solitons are given. It is proved that, unlike onedimensional solitons, twodimensional ones do not interact at all. © 1977.
 Zakharov, V. E., & Manakov, S. V. (1976). Generalization of the inverse scattering problem method. Theoretical and Mathematical Physics, 27(3), 485487.
 Budneva, O. B., Zakharov, V. E., & Synakh, V. S. (1975). CERTAIN MODELS FOR WAVE COLLAPSE.. Sov J Plasma Phys, 1(4), 335338.More infoAbstract: Numerical experiments on spherically symmetric wave collapse are reported and discussed. The experiments were based on equations describing the timedependent selffocusing of light and the dynamics of plasma waves in a plasma.
 Zakharov, V. E., & L'vov, V. (1975). Statistical description of nonlinear wave fields. Radiophysics and Quantum Electronics, 18(10), 10841097.
 Zakharov, V. E., Mastryukov, A. F., & Synakh, V. S. (1975). DYNAMICS OF PLASMAWAVE COLLAPSE IN A HOT PLASMA.. Sov J Plasma Phys, 1(4), 339343.More infoAbstract: A collapse phenomenon occurs according to the dynamical equations of a hot plasma averaged over the period of the electron plasma waves. The evolution of the collapse is analyzed, and its selfsimilar nature is demonstrated. The theoretical results have been checked and confirmed in numerical experiments for two cavity configurations: planar and axisymmetric.
 Zakharov, V. E. (1974). The Hamiltonian Formalism for waves in nonlinear media having dispersion. Radiophysics and Quantum Electronics, 17(4), 326343.
 Zakharov, V. E., & Manakov, S. V. (1974). On the complete integrability of a nonlinear Schrödinger equation. Theoretical and Mathematical Physics, 19(3), 551559.
 Zakharov, V. E., & Rubenchik, A. M. (1974). Nonlinear interaction of highfrequency and lowfrequency waves. Journal of Applied Mechanics and Technical Physics, 13(5), 669681.More infoAbstract: The interaction of highfrequency waves with lowfrequency (acoustic) waves is investigated. The analysis is carried out in the Hamiltonian formalism in the interest of generality. The instability problem is investigated for the highfrequency wave. The general results obtained in the article are applied to the stability analysis of electromagnetic waves in plasmas and dielectrics. Wave propagation in weakly dispersive media is considered. It is shown that the waves are unstable. The possibility of selffocusing of the waves is studied. © 1974 Consultants Bureau.
 Zakharov, V. E., & Shabat, A. B. (1974). A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Functional Analysis and Its Applications, 8(3), 226235.
 Zakharov, V. E., Sobolev, V. V., & Synakh, V. S. (1974). Breakdown of a monochromatic wave in a medium with an inertiafree nonlinearity. Journal of Applied Mechanics and Technical Physics, 13(1), 8084.More infoAbstract: The nonlinear instability mode of a monochromatic wave in a medium with an inertiafree nonlinearity is analyzed theoretically and simulated numerically. It is shown that, if longitudinal and transverse instabilities occur simultaneously, the wave is splitinto threedimensional clusters containing amplitude singularities. As a result, the monochromatic wave "breaks down," which is accompanied by a considerable widening of its spectrum and angular divergence. © 1974 Consultants Bureau.
 Sobolev, V. V., Synakh, V. S., & Zakharov, V. E. (1973). Some numerical investigations in nonlinear optics. Computer Physics Communications, 5(1), 4850.More infoAbstract: The problem of the selffocusing of a light beam in nonlinear media is the central problem in nonlinear optics. The powerful laser beam propagation through a real medium under certain conditions is accompanied with such a phenomenon. Mathematically the problem deals with the investigation of the asymptotic behaviour of the solution of the parabolic equation 2i ∂u ∂z= ∂2u ∂r2+ 1 r ∂u ∂r+f{hook}({divides}u{divides}2)u with given initial distribution u(r,0) and boundary condition u(∞, z) = 0 where u is the electromagnetic field amplitude, f is a function which describes the refractive index deviation from its constant value in the linear medium. It is complex in the case of nonconservative media. In our investigation we combine analytical and numerical methods. The computational study of the selffocusing problem is complicated due to the boundary condition at infinity and the abrupt light amplitude behaviour in the paraxial region. We managed to overcome these difficulties by introducing the socalled quasiuniform grid for the radial variable and by using the special technique of the correct transfer of the boundary condition from infinity. The main physical results are: (1) the conditions for the light selftrapping and waveguide creation are found, (2) the selffocusing mechanism and the law of increasing beam amplitude when approaching the collapse point are discovered; (3) the influence of the different kinds of absorption is investigated and the process of light "turbulence" is explained;. All the analytical and numerical results are comparable with the experimental situation as well as with treatments by other authors. © 1973.
 Zakharov, V. E., & Kharitonov, V. G. (1973). Instability of monochromatic waves on the surface of a liquid of arbitrary depth. Journal of Applied Mechanics and Technical Physics, 11(5), 747751.More infoAbstract: A study is made of the stability, with respect to the spontaneous appearance of modulation, of steady periodic waves of small amplitude on the surface of an ideal liquid depth. The wellknown instability of waves on the surface of a liquid of infinite depth disappears on making the transition to small depths. © 1973 Consultants Bureau.
 Zakharov, V. E. (1972). Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2), 190194.More infoAbstract: We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface η(r, t) and the hydrodynamic potential ψ(r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case. In section 2, using a method similar to van der Pohl's method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude. In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negativepressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d2ω/dk2) is different from the sign of the frequency shift due to the nonlinearity. As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves. © 1972 Consultants Bureau.
 Zakharov, V. E., & Faddeev, L. D. (1971). Kortewegde Vries equation: A completely integrable Hamiltonian system. Functional Analysis and Its Applications, 5(4), 280287.
 Zakharov, V. E., & Filonenko, N. N. (1971). Weak turbulence of capillary waves. Journal of Applied Mechanics and Technical Physics, 8(5), 3740.More infoAbstract: In recent years the theory of weak turbulence, i.e. the stochastic theory of nonlinear waves [I, 9], has been intensively developed. In the theory of weak turbulence nonlinearity of waves is assumed to be small; this enables us, using the hypothesis of the random nature of the phases of individual waves, to obtain the kinetic equation for the mean squares of the wave aplitudes. In many cases of weak turbulence a situation arises where damping is considerable in the region of large wave numbers and is separated from the region where the basic energy of the waves is concentrated (as a result either of pumping or of the initial conditions) with a wide region of transparency. In [3,4] the hypothesis was stated that weak turbulence in these eases is completely analogous to hydrodynamic turbulence for large Reynolds numbers in the sense that in the region of transparency a univelsal spectrum is established which is determined onIy by the flow of energy into the region of large wave numbers. The spectrum of hydrodynamic turbulence Sk  k 5/s was obtained by A. N. Kolmogorov and A. M. Obukhov [5,6] from dimensional considerations. In the case of weak turbulence the spectrum  obtained as an exact solution of the stationary kinetic equation. Below the ease of weak turbulence of capillary waves on the surface of a liquid is considered. A kinetic equation is obtained for capillary waves. It is significant that in this case the basic contribution to interaction is provided by the process of the decomposition of a wave into two and by the process of two waves merging into one. It is shown that the collision term of the kinetic equation vanishes with the solution ek  k 7/4. Arguments are advanced in favor of the fact that this solution can be interpreted as a universal spectrum in the region of transparency. © 1971 Consultants Bureau.
 Zakharov, V. E. (1967). Weak turbulence in media with a decay spectrum. Journal of Applied Mechanics and Technical Physics, 6(4), 2224.More infoAbstract: The theory of weak turbulence of a plasma has been investigated in many papers [15]. It has been established that weak turbulence may be described by means of the kinetic wave equations. Here the collision term in the kinetic equation is the sum of two substantially different components. The first of these has the character of nonlinear wave damping and differs from zero in those cases where interaction between waves and particles is significant. It has a comparatively simple mathematical nature and can be analyzed. The second component is specifically a collision term, it depends closely on the form of the spectrum in the medium and describes the exchange of energy between different groups of waves. The case when the second component plays the principal role in the collision term has scarcely been studied. The present paper is devoted to a study of this case. The analysis is carried out for a simple isotropic model of a medium with an almost linear dispersion law, but with a positive second derivative; we shall call such a spectrum a decay spectrum. This model is much closer to reality than the model considered in [6]. The results obtained from this model are evidently fairly general in character and express substantially the regularity of behavior of weak turbulence in media with a weak decay spectrum. The basic result of the paper is as follows: apart from the RayleighJeans solution, there exists another solution which reduces the collision term to zero. This solution corresponds to a process which is substantially nonequilibrium, and may be realized in actual problems, where there are always wave sources or transfer terms playing the same part, only in cases where there is wave damping in the medium with a coefficient which increases fairly rapidly into the region of large k. Here the universal character, as it were, of the nonequilibrium process is realized. © 1967 The Faraday Press, Inc.
 Zakharov, V. E. (1965). A solvable model of weak turbulence. Journal of Applied Mechanics and Technical Physics, 6(1), 1016.More infoAbstract: A model of weak turbulence amenable to solution is investigated. There are three modes of interacting waves whose laws of dispersion are so chosen that it is possible to go over to the diffusion approximation in kspace. The steadystate spectrum of the turbulence in the presence of regions of instability, transparency and damping is found. The formal apparatus of nonlinear wave dynamics is also discussed. © 1966 The Faraday Press, Inc.
 Alikhanov, S. G., Zakharov, V. E., & Khorasanov, G. L. (1963). Diffusion of a plasma in a magnetic field due to collisions. Soviet Atomic Energy, 14(2), 127132.
 Alikhanov, S. G., Zakharov, V. E., & Khorasanov, G. L. (1963). Plasma diffusion in a magnetic field due to coulomb collisions. Journal of Nuclear Energy. Part C, Plasma Physics, Accelerators, Thermonuclear Research, 5(5), 309313.More infoAbstract: The diffusion of a decaying helium plasma has been studied by a v.h.f. method. The investigations were carried out at parameters for which the diffusion flux across the magnetic field was due to electronion collisions. The measured diffusion coefficients in magnetic fields of 700 to 800 Oe are in accordance with those derived on the basis of the classical theory of binary Coulomb collisions. In the range of magnetic fields from 1000 to 5000 Oe a considerable deviation from the theory is observed, the diffusion coefficient being proportional to 1/H. The asymptotic solution is found for the diffusion equation which describes a decaying fully ionized plasma in the case of axial symmetry.
Presentations
 Zakharov, V. E. (2018, December). Weak turbulent theory of ocean waves. AGU 100, Fall Meeting. Washington DC.
 Zakharov, V. E. (2018, September). Primitive potentials and bounded solutions of the KdV equation. Geometric Structures in Integrable Systems. Lecce, Italy.
 Zakharov, V. E. (2017, June 1217). , Plenary talk: Unresolved problems in the theory of integrable systems. FrenchAmerican Conference on Nonlinear Dispersive PDEs. Marseille, France.
 Zakharov, V. E. (2017, June 1724). Plenary talk: Rayleigh statistics of waves in integrable nonlinear systems and Integrability of deep water equations. Physics and Mathematics of Nonlinear Phenomena: 50 years of IST. Gallipoli, Italy.
 Zakharov, V. E. (2017, March 2024). Rogue waves and modulational instability. The 10th International Nonlinear Wave and Chaos Workshop (NWCW17). La Jolla, San Diego.
 Zakharov, V. E. (2017, September 48). Plenary talk: Analytic theory of the Winddriven Sea. IUTAM Symposium on Wind Waves. University College, London, UK.
 Zakharov, V. E. (2016, December 1216). Lorenz Lecture: Analytic Theory of WindDriven Sea. AGU Fall Meeting. San Francisco.