Alan C Newell

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• Newell, A. C. (2013). Advances in Wave Turbulence: 83 (World Scientific Series on Nonlinear Science Series A). World Scientific Series on Nonlinear Science Series A: Volume 83.
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  Co-author- Wave Turbulence, A Story Far from Over (Alan C Newell and Benno Rumpf)

Journals/Publications

• Newell, A. C. (2014). Phyllotaxis: A review of papers, but a story far from over. Physica D.
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  Under review. This is 57 page review article.W / M. Pennybacher, P.D. Shipman
• Venkataramani, S. C., & Newell, A. C. (2016). Elastic sheets, phase surfaces and pattern universes. Studies in Applied Mathematics. doi:10.1111/sapm.12184
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  We connect the theories of the deformation of elastic surfaces and phasesurfaces arising in the description of almost periodic patterns. In particular,we show striking parallels between expansions for the energy of elasticsurfaces in powers of the thickness $h$ and the free energy averaged over aperiod of an almost periodic pattern expanded in powers of $\epsilon$, theinverse aspect ratio of the pattern field. In both cases, the resulting energycan be expressed in terms of the first and second fundamental forms of thesurfaces involved, the elastic surface in the former case and the phase surfacein the latter. We discuss various results that are obtained by exploiting thisanalogy and also address some of the outstanding questions. One common resultof particular interest concerns the condensation of the Gaussian curvature ontoisolated point defects in two dimensions and onto loop filaments in threedimensions. We also lay out an ambitious and somewhat speculative program to build amulti-scale model of the universe inspired by patterns, in which the short(spatial and temporal) scale structure on the Planck scales is given by anearly periodic microstructure, and macroscopic/slowly varying/averagedbehaviors on scales much larger than the Planck scale leads to a hierarchy ofstructures and features including analogues of quarks, leptons, dark matter,dark energy and inflationary cosmology.[Journal_ref: ]
• Newell, A. C. (2014). 'Quarks' and 'leptons' in three dimensional patterns. European Journal of Mechanics, B/Fluids.
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  Abstract: The following questions are addressed: How did the symmetries which lead to fractional invariants arise? Can one start with systems with much simpler symmetries, say space translation and rotation, and, by stressing such systems, give rise via phase transitions to objects with natural fractional invariants? Such systems are manifold in nature. They are called pattern forming systems. One has to go no further than one's own fingertips to see examples of two dimensional cross sections of the objects which are the centerpieces of this paper. © 2014 Elsevier Masson SAS. All rights reserved.
• Newell, A. C. (2013). Fibonacci patterns; Common or Rare. Science Direct.
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  Using a partial differential equation model derived from the ideas of the Meyerowitz and Traas groups on the role of the growth hormone auxin and those of Green and his group on the role compressive stresses can play in plants, we demonstrate how all features of spiral phyllotaxis can be recovered by the passage of a pushed pattern forming front. The front is generated primarily by a PIN1 mediated instability of a uniform auxin concentration and leaves in its wake an auxin fluctuation field at whose maxima new primordia are assumed to be initiated. Because it propagates through a slowly changing metric, the patterns have to make transitions between spirals enumerated by decreasing parastichy numbers. The point configurations of maxima coincide almost exactly with those configurations generated by the use of discrete algorithms based on optimal packing ideas which suggests that pushed pattern forming fronts may be a general mechanism by which natural organisms can follow optimal strategies.
• Newell, A. C. (2013). Phyllotaxis, Pushed Pattern-Forming Fronts and Optimal Packing. Physical Review Letters.
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  We demonstrate that the pattern forming partial differential equation derived from the auxin distribution model proposed by Meyerowitz, Traas, and others gives rise to all spiral phyllotaxis properties observed on plants. We show how the advancing pushed pattern front chooses spiral families enumerated by Fibonacci sequences with all attendant self-similar properties, a new amplitude invariant curve, and connect the results with the optimal packing based algorithms previously used to explain phyllotaxis. Our results allow us to make experimentally testable predictions.
• Newell, A. C. (2013). Wave instability under short wave amplitude perturbations. Physics Letters A.
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  The instabilities of nonlinear waves with a square-root dispersion ω ∼|k| are studied. We present anew type of instability that affects wavelengths of the order of the carrier wave. This instability caninitiate the formation of collapses and of narrow pulses.
• Newell, A. C., & Pennybacker, M. (2013). Fibonacci patterns: Common or rare?. Procedia IUTAM, 9, 86-109.
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  Abstract: Using a partial differential equation model derived from the ideas of the Meyerowitz and Traas groups on the role of the growth hormone auxin and those of Green and his group on the role compressive stresses can play in plants, we demonstrate how all features of spiral phyllotaxis can be recovered by the passage of a pushed pattern forming front. The front is generated primarily by a PIN1 mediated instability of a uniform auxin concentration and leaves in its wake an auxin fluctuation field at whose maxima new primordia are assumed to be initiated. Because it propagates through a slowly changing metric, the patterns have to make transitions between spirals enumerated by decreasing parastichy numbers. The point configurations of maxima coincide almost exactly with those configurations generated by the use of discrete algorithms based on optimal packing ideas which suggests that pushed pattern forming fronts may be a general mechanism by which natural organisms can follow optimal strategies. © 2013 The Authors.
• Pennybacker, M., & Newell, A. C. (2013). Phyllotaxis, pushed pattern-forming fronts, and optimal packing. Physical Review Letters, 110(24).
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  Abstract: We demonstrate that the pattern forming partial differential equation derived from the auxin distribution model proposed by Meyerowitz, Traas, and others gives rise to all spiral phyllotaxis properties observed on plants. We show how the advancing pushed pattern front chooses spiral families enumerated by Fibonacci sequences with all attendant self-similar properties, a new amplitude invariant curve, and connect the results with the optimal packing based algorithms previously used to explain phyllotaxis. Our results allow us to make experimentally testable predictions. © 2013 American Physical Society.
• Rumpf, B., & Newell, A. C. (2013). Wave instability under short-wave amplitude modulations. Physics Letters, Section A: General, Atomic and Solid State Physics, 377(18), 1260-1263.
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  Abstract: The instabilities of nonlinear waves with a square-root dispersion ω∼|k| are studied. We present a new type of instability that affects wavelengths of the order of the carrier wave. This instability can initiate the formation of collapses and of narrow pulses. © 2013 Elsevier B.V.
• Glasner, K., Kolesik, M., Moloney, J. V., & Newell, A. C. (2012). Canonical and singular propagation of ultrashort pulses in a nonlinear medium. International Journal of Optics, 2012.
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  Abstract: We examine the two types of singular behaviors of ultrashort pulses in a nonlinear medium, pulse steepening if the weak longitudinal dispersion is normal and collapse if it is anomalous. Connections with analogous behaviors of wave packets of almost monochromatic waves in strongly dispersive media are discussed.
• Newell, A. C. (2012). Pattern quarks and leptons. Applicable Analysis, 91(2), 213-223.
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  Abstract: Disclinations, concave and convex, are the canonical point defects of two-dimensional planar patterns in systems with translational and rotational symmetries. From these, all other point defects (vortices, dislocations, targets, saddles and handles) can be built. Moreover, handles, coupled concave-convex disclination pairs arise as instabilities, symmetry breaking events. The purpose of this article is to show that embedded in three or more dimensions, concave and convex disclination strings, two-dimensional disclinations with loop backbones, have interesting and suggestive invariant indices which are integer multiples of. © 2012 Copyright Taylor and Francis Group, LLC.
• Newell, A. C., Rumpf, B., & Zakharov, V. E. (2012). Spontaneous breaking of the spatial homogeneity symmetry in wave turbulence. Physical Review Letters, 108(19).
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  Abstract: We 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 Kolmogorov-Zakharov 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. © 2012 American Physical Society.
• Whalen, P., Moloney, J. V., Newell, A. C., Newell, K., & Kolesik, M. (2012). Optical shock and blow-up of ultrashort pulses in transparent media. Physical Review A - Atomic, Molecular, and Optical Physics, 86(3).
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  Abstract: Ultrashort pulses can exhibit two distinctive types of singularity: self-focusing collapse and self-steepening shock. We examine various ultrashort pulse propagation models and their relative effectiveness in explaining these phenomena. In particular, the modified Kadomtsev-Petviashvilli equation of type 1 (MKP1) is examined in some detail. We show that MKP1 is not simply a few-cycle pulse model but is valid in a more general broad spectrum setting. Furthermore, we emphasize that the dispersion of the MKP1 model can result in poor estimation of frequency-dependent phenomena, such as harmonic generation, which occur far away from the carrier frequency. Some of this loss of accuracy can be removed by using a more general MKP1 dispersion relation. © 2012 American Physical Society.
• Newell, A. C., & Rumpf, B. (2011). Wave turbulence. Annual Review of Fluid Mechanics, 43, 59-78.
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  Abstract: In this article, we state and review the premises on which a successful asymptotic closure of the moment equations of wave turbulence is based, describe how and why this closure obtains, and examine the nature of solutions of the kinetic equation. We discuss obstacles that limit the theory's validity and suggest how the theory might then be modified. We also compare the experimental evidence with the theory's predictions in a range of applications. Finally, and most importantly, we suggest open challenges and encourage the reader to apply and explore wave turbulence with confidence. The narrative is terse but, we hope, delivered at a speed more akin to the crisp pace of a Hemingway story than the wordjumblingtumbling rate of a Joycean novel. © 2011 by Annual Reviews. All rights reserved.
• Shipman, P. D., Sun, Z., Pennybacker, M., & Newell, A. C. (2011). How universal are Fibonacci patterns?. European Physical Journal D, 62(1), 5-17.
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  Abstract: Pattern patterns, or phyllotaxis, the arrangements of phylla (flowers, leaves, bracts, florets) in the neighborhood of growth tips, have intrigued natural scientists for over four hundred years. Prominent amongst the observed features is the fact that phylla lie on families of alternately oriented spirals and that the numbers in these families belong to subsets {m j } of the integers defined by the Fibonacci rule m j + 1 = m j + m j - 1. The corresponding patterns, which we call Fibonacci patterns, are widespread and universal on plants. Our goal in this paper is to ask if they may also be seen in other physical structures and to try to quantify the circumstances under which one may expect Fibonacci patterns to occur. © 2010 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.
• Connaughton, C., & Newell, A. C. (2010). Dynamical scaling and the finite-capacity anomaly in three-wave turbulence. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 81(3).
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  Abstract: We present a systematic study of the dynamical scaling process leading to the establishment of the Kolmogorov-Zakharov (KZ) spectrum in weak three-wave turbulence. In the finite-capacity case, in which the transient spectrum reaches infinite frequency in finite time, the dynamical scaling exponent is anomalous in the sense that it cannot be determined from dimensional considerations. As a consequence, the transient spectrum preceding the establishment of the steady state is steeper than the KZ spectrum. Constant energy flux is actually established from right to left in frequency space after the singularity of the transient solution. From arguments based on entropy production, a steeper transient spectrum is heuristically plausible. © 2010 The American Physical Society.
• Rumpf, B., Newell, A. C., & Zakharov, V. E. (2009). Turbulent transfer of energy by radiating pulses. Physical Review Letters, 103(7).
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  Abstract: We propose a new mechanism for turbulent transport in systems which support radiating nonlinear solitary wave packets or pulses. The direct energy cascade is provided by adiabatically evolving pulses, whose widths and carrier wavelengths decrease. The inverse cascade is due to the excitation of radiation. The spectrum is steeper than the Kolmogorov-Zakharov spectrum of wave turbulence. © 2009 The American Physical Society.
• Newell, A. C., & Zakharov, V. E. (2008). The role of the generalized Phillips' spectrum in wave turbulence. Physics Letters, Section A: General, Atomic and Solid State Physics, 372(23), 4230-4233.
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  Abstract: We suggest the generalized Phillips' spectrum, which we define as that spectrum for which the statistical properties of wave turbulence inherit the symmetries of the original governing equations, is, in many circumstances, the spectrum which obtains in those regions of wavenumber space in which the Kolmogorov-Zakharov (KZ) spectra are no longer valid. This spectrum has many very special properties. We discuss its connection with the singularities which are associated with the whitecap events observed in windblown seas. © 2008 Elsevier B.V. All rights reserved.
• Newell, A. C., Shipman, P. D., & Sun, Z. (2008). Phyllotaxis as an example of the symbiosis of mechanical forces and biochemical processes in living tissue. Plant Signaling and Behavior, 3(8), 586-589.
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  PMID: 19704477;PMCID: PMC2634505;Abstract: Phyllotaxis, the arrangement of a plant's phylla (flowers, bracts, stickers) near its shoot apical meristem (SAM), has intrigued natural scientists for centuries. Even today, the reasons for the observed patterns and their special properties, the physical and chemical mechanisms which give rise to strikingly similar configurations in a wide variety of plants, the almost-constant golden divergence angle, the almost constant plastichrone ratio, the choices of parastichy numbers and the prevalence of Fibonacci sequences to which these numbers belong, are at best only partially understood. Our goals in this Addendum are: (i) To give a brief overview of current thinking on possible mechanisms for primordia (the bumps on the plant surface which eventually mature into fully developed structures such as leaves or florets) formation and give a descriptive narrative of the mathematical models which encode various hypotheses. (ii) To emphasize the point that patterns, whether they be phyllotactic configurations on plant surfaces or convection cells on the sun's surface, are macroscopic objects whose behaviors are determined more by symmetries of the proposed model and less by microscopic details. Because of this, the identification of observations with the predications of a particular model can only be made with confidence when the match coincides over a range of circumstances and parameters. (iii) To discuss some of the key results of the proposed models and, in particular, introduce the prediction of a new and, in principle, measurable invariant in plant phyllotaxis. (iv) To introduce a new model of primordia formation which is more in keeping with the pictures and paradigms of Hofmeister,1 Snow & Snow,2 and Douady and Couder3,4, which see primordia as forming in a fairly narrow annular zone surrounding the plant's SAM separating a region of undifferentiated cells from a fully developed patterned state. (v) To consider the challenge of phyllotaxis in the broader context of pattern formation in biological tissue which responds to both mechanical and biochemical processes. ©2008 Landes Bioscience.
• Newell, A. C., Shipman, P. D., & Sun, Z. (2008). Phyllotaxis: Cooperation and competition between mechanical and biochemical processes. Journal of Theoretical Biology, 251(3), 421-439.
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  PMID: 18207165;Abstract: Current theories and models of the formation of phyllotactic patterns at plant apical meristems center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partial differential equations are similar in form. The implications of this universality in the form of the equations on interpreting the results of simulations are discussed. We develop a combined model that incorporates the coupling of biochemistry and mechanics. The combined model is accessible to analysis by reduction to a set of ordinary differential equations for the amplitudes of shapes associated with both the auxin concentration field and plant surface deformation. Analysis of these amplitude equations reveals the parameter choices under which the two mechanisms may cooperate in determining the pattern, or under which one or the other mechanism may dominate. © 2007 Elsevier Ltd. All rights reserved.
• Korotkevich, A. O., Newell, A. C., & Zakharov, V. E. (2007). Communication through plasma sheaths. Journal of Applied Physics, 102(8).
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  Abstract: We wish to transmit messages to and from a hypersonic vehicle around which a plasma sheath has formed. For long distance transmission, the signal carrying these messages must be necessarily low frequency, typically 2 GHz, to which the plasma sheath is opaque. The idea is to use the plasma properties to make the plasma sheath appear transparent. © 2007 American Institute of Physics.
• Roberts, D. C., & Newell, A. C. (2007). Erratum: Finite time collapse of N classical fields described by coupled nonlinear Schrödinger equations (Physical Review E (2006) 74 (047602)). Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 75(1).
• Roberts, D. C., & Newell, A. C. (2006). Finite-time collapse of N classical fields described by coupled nonlinear Schrödinger equations. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74(4).
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  Abstract: We prove the finite-time collapse of a system of N classical fields, which are described by N coupled nonlinear Schrödinger equations. We derive the conditions under which all of the fields experiences this finite-time collapse. Finally, for two-dimensional systems, we derive constraints on the number of particles associated with each field that are necessary to prevent collapse. © 2006 The American Physical Society.
• Kücken, M., & Newell, A. C. (2005). Fingerprint formation. Journal of Theoretical Biology, 235(1), 71-83.
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  PMID: 15833314;Abstract: Fingerprints (epidermal ridges) have been used as a means of identifications for more than 2000 years. They have also been extensively studied scientifically by anthropologists and biologists. However, despite all the empirical and experimental knowledge, no widely accepted explanation for the development of epidermal ridges on fingers, palms and soles has yet emerged. In this article we argue that fingerprint patterns are created as the result of a buckling instability in the basal cell layer of the fetal epidermis. Analysis of the well-known von Karman equations informs us that the buckling direction is perpendicular to the direction of greatest stress in the basal layer. We propose that this stress is induced by resistance of furrows and creases to the differential growth of the basal layer and regression of the volar pads during the time of ridge formation. These ideas have been tested by computer experiments. The results are in close harmony with observations. Specifically, they are consistent with the well-known observation that the pattern type is related to the geometry of the fingertip surface when fingerprint patterns are formed. © 2005 Elsevier Ltd. All rights reserved.
• Kücken, M., Newell, A. C., & Shipman, P. D. (2005). Mean creep: The soft mode in elastic sheet buckling. Physica D: Nonlinear Phenomena, 205(1-4), 181-188.
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  Abstract: In this note, we introduce the equations for the order parameters describing the buckling of thin, elastic sheets. What is new is the realization that mean creep, namely in-plane displacements, are soft (Goldstone) modes which can be driven by variations in the pattern intensity and which, in turn, affect how the buckling pattern develops. The order-parameter equations are canonical and belong to the universal classes of equations for pattern order parameters to which Yoshiki Kuramoto has contributed so much. We are very pleased to be part of this special issue of Physica D in honoring this remarkable colleague. © 2005 Elsevier B.V. All rights reserved.
• Newell, A. C., & Shipman, P. D. (2005). Plants and fibonacci. Journal of Statistical Physics, 121(5-6), 937-968.
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  Abstract: The universality of many features of plant patterns and phyllotaxis has mystified and intrigued natural scientists for at least four hundred years. It is remarkable that, to date, there is no widely accepted theory to explain the observations. We hope that the ideas explained below lead towards increased understanding. © 2005 Springer Science+Business Media, Inc.
• Shipman, P. D., & Newell, A. C. (2005). Polygonal planforms and phyllotaxis on plants. Journal of Theoretical Biology, 236(2), 154-197.
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  PMID: 16005308;Abstract: We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the ribbed, hexagonal, or parallelogram planforms on plants can be understood as the energy-minimizing buckling pattern of a compressed sheet (the plant's tunica) on an elastic foundation. The key idea is that the elastic energy is minimized by configurations consisting of special triads of periodic deformations. We study the conditions that lead to continuous or discontinuous transitions between patterns, state testable predictions, and suggest experiments to test the theory. © 2005 Elsevier Ltd. All rights reserved.
• Kücken, M., & Newell, A. C. (2004). A model for fingerprint formation. Europhysics Letters, 68(1), 141-146.
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  Abstract: The uniqueness of fingerprints (epidermal ridges) has been recognized for over two thousand years. They have been studied scientifically for more than two hundred years. Yet, in spite of the accumulation of a wealth of empirical and experimental knowledge, no widely accepted explanation for the development of epidermal ridges on fingers, palms and soles has yet emerged. Informed by an extensive literature study we suggest that fingerprint patterns are created as the result of a buckling instability in the basal cell layer of the fetal epidermis. Analysis of the well-known von Karman equations informs us that the buckling direction is perpendicular to the direction of greatest stress in the basal layer. We propose that this stress is induced by resistance of furrows and creases to the differential growth of the basal layer and regression of the volar pads during the time of ridge formation. These theories have been tested by computer experiments. The results are in close harmony with observations. Specifically, they are consistent with the well-known observation that the pattern type depends on the geometry of the fingertip surface when fingerprint patterns are formed.
• Rumpf, B., & Newell, A. C. (2004). Intermittency as a consequence of turbulent transport in nonlinear systems. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 69(2 2), 026306-1-026306-4.
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  PMID: 14995558;Abstract: The turbulent flows that transfer energy from a stirring range at large scales to the dissipation range at small scales were investigated. It was shown that a single cascade of weakly interacting waves that transport particles from a source to a link leads to a steady loss of energy in the system. It was observed that coherent structures are necessary for any solution that has statistically stationary transport properties. The results show that coherent structures are an essential component of the transport of particles from large scales to small dissipation scales and that the particle flow leads to a steady loss of coupling energy, so the fluctuations have a low ratio of energy per particle.
• Shipman, P. D., & Newell, A. C. (2004). Phyllotactic patterns on plants. Physical Review Letters, 92(16), 168102-1.
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  PMID: 15169264;Abstract: An approach to minimize the elastic energy of the plant surface posticts spiral and whorl patterns was described. The strain energy was minimized by configurations consiting of special triads of almost periodic deformations. It was found that when the stress was larger than critical value, the constant deformation state was found to be linearly unstable and certain shapes and configurations were amplified. A wide spectrum of plant patterns were reproduced with the divergence angles observed in nature, to show the occurrences of Fibonacci-like sequences and the golden angle as natural consequencies.
• Biven, L. J., Connaughton, C., & Newell, A. C. (2003). Structure functions and breakdown criteria for wave turbulence. Physica D: Nonlinear Phenomena, 184(1-4), 98-113.
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  Abstract: We study the structure functions of wave turbulence at small separations. We show that the criteria for breakdown obtained previously by examining the uniform validity of the asymptotic closure govern how close wave turbulence stays to joint Gaussianity. A new result in the case of small separations in that the system behavior is organized by a special point in the (α,β) plane where α and β are the homogeneity parameters for the linear and nonlinear coupling coefficients. Finally, we explore how modifications of the breakdown criteria are necessary because of the strengths of non-local long-wave-short-wave interactions. © 2003 Elsevier B.V. All rights reserved.
• Connaughton, C., Nazarenko, S., & Newell, A. C. (2003). Dimensional analysis and weak turbulence. Physica D: Nonlinear Phenomena, 184(1-4), 86-97.
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  Abstract: In the study of weakly turbulent wave systems possessing incomplete self-similarity, it is possible to use dimensional arguments to derive the scaling exponents of the Kolmogorov-Zakharov spectra, provided the order of the resonant wave interactions responsible for nonlinear energy transfer is known. Furthermore, one can easily derive conditions for the breakdown of the weak turbulence approximation. It is found that for incompletely self-similar systems dominated by three wave interactions, the weak turbulence approximation usually cannot break down at small scales. It follows that such systems cannot exhibit small scale intermittency. For systems dominated by four wave interactions, the incomplete self-similarity property implies that the scaling of the interaction coefficient depends only on the physical dimension of the system. These results are used to build a complete picture of the scaling properties of the surface wave problem where both gravity and surface tension play a role. We argue that, for large values of the energy flux, there should be two weakly turbulent scaling regions matched together via a region of strongly nonlinear turbulence. © 2003 Elsevier B.V. All rights reserved.
• Connaughton, C., Newell, A. C., & Pomeau, Y. (2003). Non-stationary spectra of local wave turbulence. Physica D: Nonlinear Phenomena, 184(1-4), 64-85.
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  Abstract: The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is studied in the limit of strongly local interactions where the usual kinetic equation, describing the time evolution of the spectral wave-action density, can be approximated by a PDE. If the wave action is initially compactly supported in frequency space, it is then redistributed by resonant interactions producing the usual direct and inverse cascades, leading to the formation of the K-Z spectra. The emphasis here is on the direct cascade. The evolution proceeds by the formation of a self-similar front which propagates to the right leaving a quasi-stationary state in its wake. This front is sharp in the sense that the solution remains compactly supported until it reaches infinity. If the energy spectrum has infinite capacity, the front takes infinite time to reach infinite frequency and leaves the K-Z spectrum in its wake. On the other hand, if the energy spectrum has finite capacity, the front reaches infinity within a finite time, t*, and the wake is steeper than the K-Z spectrum. For this case, the K-Z spectrum is set up from the right after the front reaches infinity. The slope of the solution in the wake can be related to the speed of propagation of the front. It is shown that the anomalous slope in the finite capacity case corresponds to the unique front speed which ensures that the front tip contains a finite amount of energy as the connection to infinity is made. We also introduce, for the first time, the notion of entropy production in wave turbulence and show how it evolves as the system approaches the stationary K-Z spectrum. © 2003 Elsevier B.V. All rights reserved.
• Rumpf, B., & Newell, A. C. (2003). Localization and coherence in nonintegrable systems. Physica D: Nonlinear Phenomena, 184(1-4), 162-191.
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  Abstract: We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asymptotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps. In systems where the phase space is not compact, the coherent structures can be collapsed, represented in phase space by a heteroclinic connection of some unstable saddle to infinity. © 2003 Elsevier B.V. All rights reserved.
• Galtier, S., Nazarenko, S. V., Newell, A. C., & Pouquet, A. (2002). Anisotropic turbulence of shear-Alfvén waves. Astrophysical Journal Letters, 564(1 II), L49-L52.
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  Abstract: Weak turbulence of shear-Alfvén waves is considered in the limit of strongly anisotropic pulsations that are elongated along the external magnetic field. The kinetic equation thus derived agrees with the Galtier et al. formulation of the full three-dimensional helical case when taking the proper limit, This new approach allows for significant simplification, and, as a result, the applicability conditions for the weak turbulence theory are now more transparent. It thus provides an attractive theoretical framework for describing anisotropic MHD turbulence in astrophysical contexts where a strong magnetic field is present and for which shear-Alfvén waves are important.
• Kremeyer, K., Nazarenko, S., & Newell, A. C. (2002). Shock bowing and vorticity dynamics during propagation into different transverse density profiles. Physica D: Nonlinear Phenomena, 163(3-4), 150-165.
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  Abstract: A 2D numerical investigation is presented of shock wave propagation into a gas whose density is modulated in the transverse direction across the width of a shock tube. These density modulations represent temperature distributions in which low density corresponds to high temperature gas and high density corresponds to low temperature gas. This work is motivated by recent shock-plasma experiments, and mechanisms to explain the experimentally observed shock "splitting" signatures are investigated. It is found that the shock splitting signatures are more pronounced when the shock wave is more strongly curved or bowed. This occurs as the depth of the initial density profile is increased. The gross features of the shock splitting signatures are relatively insensitive to variations in the shape of the initial density profile (into which the shock propagates). Several interesting features of vorticity production and evolution are also indicated. © 2002 Elsevier Science B.V. All rights reserved.
• Newell, A. C. (2002). Wave turbulence is almost always intermittent at either small or large scales. Studies in Applied Mathematics, 108(1), 39-64.
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  Abstract: The asymptotic expansions for (1) the slow changes in particle number/energy density; namely, the kinetic equation, (2) frequency renormalization; and (3) the nth-order structure functions for wave turbulence systems are almost always nonuniform at either small or large length scales. The manifestation of this nonuniformity is fully nonlinear behavior either in the form of localized structures (coherent structures, shocks) or condensates (nonzero mean over large distances). The result is intermittent behavior dominated by large fluctuation events, anomolous scaling, and far from joint Gaussian statistics. Despite this unexpected surprise, and it is a surprise considering that wave turbulence has been the subject of continuous and intense investigation for several decades, wave turbulence still offers an advantage over systems that are nonlinear over all scales. The advantage is that the nature of the fully nonlinear behavior often can be identified, which gives us reasonable hope that wave turbulent systems may be treated as a two species gas of random wavetrains and randomly occurring coherent structures.
• Biven, L., Nazarenko, S. V., & Newell, A. C. (2001). Breakdown of wave turbulence and the onset of intermittency. Physics Letters, Section A: General, Atomic and Solid State Physics, 280(1-2), 28-32.
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  Abstract: This Letter demonstrates that the kinetic equations for wave turbulence, the long time statistical behavior of a sea of weakly coupled, dispersive waves, will almost always develop solutions for which the theory fails due to strongly nonlinear and intermittent events either at small or large scales. © 2001 Elsevier Science B.V.
• Galtier, S., Nazarenko, S. V., & Newell, A. C. (2001). On wave turbulence in MHD. Nonlinear Processes in Geophysics, 8(3), 141-150.
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  Abstract: We describe the fundamental differences between weak (wave) turbulence in incompressible and weakly compressible MHD at the level of three-wave interactions. The main difference is in the structure of the resonant manifolds and the mechanisms of redistribution of spectral densities along the applied magnetic field Bo. Similar to pure acoustic waves, a three-wave resonance between collinear wavevectors is observed but, in addition, we also have a resonance through tilted planes and spheres. The properties of resonances and their consequences for the asymptotics are also discussed.
• Nazarenko, S. V., Newell, A. C., & Galtier, S. (2001). Non-local MHD turbulence. Physica D: Nonlinear Phenomena, 152-153, 646-652.
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  Abstract: We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfvén waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvénic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker-Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfvén waves seems to be general for three-wave systems. © 2001 Elsevier Science B.V.
• Newell, A. C., Nazarenko, S., & Biven, L. (2001). Wave turbulence and intermittency. Physica D: Nonlinear Phenomena, 152-153, 520-550.
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  Abstract: In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three- and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi-Dirac or Bose-Einstein spectra and admit in addition finite flux (Kolmogorov-Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov-Zakharov (KZ) spectra. © 2001 Elsevier Science B.V.
• Rumpf, B., & Newell, A. C. (2001). Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems. Physical Review Letters, 87(5), 054102/1-054102/4.
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  Abstract: Most dynamical systems exhibit a type of universal behavior in which high peaks of some field emerge from a low amplitude noisy background. In nonintegrable systems, stationary states characterized by robust coherent structures immersed in a sea of radiated wavelike fluctuations occur. This work demonstrates that such states are the most likely solutions for systems with additional integrals corresponding to surviving symmetries.
• Galtier, S., Nazarenko, S. V., Newell, A. C., & Pouquet, A. (2000). Weak turbulence theory for incompressible magnetohydrodynamics. Journal of Plasma Physics, 63(5), 447-488.
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  Abstract: We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0 qq∥. Numerically and analytically, we find energy spectra E± approx. k⊥n±, such that n+ + n- = -4, where E± are the spectra of the Elsasser variables z± = v ± b in the two-dimensional case (k∥ = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.
• Komarova, N. L., & Newell, A. C. (2000). Nonlinear dynamics of sand banks and sand waves. Journal of Fluid Mechanics, 415, 285-321.
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  Abstract: Sand banks and sand waves are two types of sand structures that are commonly observed on an off-shore sea bed. We described the formation of these featues using the equations of the fluid motion coupled with the mass conservation law for the sediment transport. The bottom features are a result of an instability due to tide-bottom interactions. There are at least two mechanisms responsible for the growth of sand banks and sand waves. One is linear instability, and the other is nonlinear coupling between long sand banks and short sand waves. One novel feature of this work is the suggestion that the latter is more important for the generation of sand banks. We derive nonlinear amplitude equations governing the coupled dynamics of sand waves and sand banks. Based on these equations, we estimate characteristic features for sand banks and find that the estimates are consistent with measurements.
• Kremeyer, K., Nazarenko, S., & Newell, A. (2000). The effect of fore-shock heating in the plasma drag-reduction problem. 21st Aerodynamic Measurement Technology and Ground Testing Conference.
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  Abstract: A1 2-D numerical investigation is presented of shock wave propagation into a gas, whose density is modulated in the transverse direction across the width of a shocktube. These density modulations represent temperature distributions, in which low density corresponds to high temperature gas and high density corresponds to low temperature gas. This work is motivated by recent shock-plasma experiments, and mechanisms to explain the experimentally observed shock "splitting" signatures are investigated. It is found that the shock "splitting" signatures are reproduced when the shockwave is strongly curved or bowed. This occurs as the depth of the initial density profile is increased. The gross features of the shock "splitting" signatures are relatively insensitive to variations in the shape of the initial density profile (into which the shock propagates).
• Lvov, Y. V., & Newell, A. C. (2000). Finite Flux Solutions of the Quantum Boltzmann Equation and Semiconductor Lasers. Physical Review Letters, 84(9), 1894-1897.
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  Abstract: We propose and illustrate in the context of the semiconductor laser that, in nonequilibrium fermionic systems with sources and sinks, the family of finite flux stationary solutions of the quantum Boltzmann equation is central and more important then the zero flux Fermi-Dirac spectrum. We present the quantum analog of the finite flux Kolmogorov spectra which are central to understanding nonequilibrium classical systems such as high Reynolds number hydrodynamics and the wave turbulence encountered in water waves, plasmas, and optics. In particular, we show how semiconductor laser efficiency can be improved by maximizing the flux of carriers (electrons and holes) towards the lasing frequencies.
• Komarova, N. L., & Newell, A. C. (1999). Competition between nonlinearity, dispersion and randomness in signal propagation. IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 63(3), 267-286.
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  Abstract: The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion, and furthermore the problem is recast to show that it is isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random potential in which the randomness is manifested spatially. It is shown that, under mild assumptions, the SIT result continues to hold when we replace the uniform medium of inhomogeneously broadened two-level atoms by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem confirms and reveals the reason for the fact that nonlinearity can help improve the transparency of the medium. Also, the small-amplitude, almost monochromatic limit of SIT is taken and results in a complex envelope equation which turns out to be an exactly integrable combination of NLS and a modified SIT equation. Finally, some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.
• Bowman, C., & Newell, A. C. (1998). Natural patterns and wavelets. Reviews of Modern Physics, 70(1), 289-301.
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  Abstract: An introductory review of pattern formation in extended dissipative systems is presented. Examples from many areas of physics are introduced, and the mathematical analysis of the patterns formed by these systems is outlined, for patterns near and far from onset. The wavelet transform is introduced as a useful tool for the extraction of order parameters from patterns.
• Bowman, C., Passot, T., Assenheimer, M., & Newell, A. C. (1998). A wavelet based algorithm for pattern analysis. Physica D: Nonlinear Phenomena, 119(3-4), 250-282.
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  Abstract: Patterns with a local roll structure arise in many diverse physical systems which have little in common at the microscopic level. In this paper we construct an algorithm based on the wavelet transform that can be used as a diagnostic tool to extract from such patterns macroscopic information like the local director field, the local amplitude away from defects, and slowly varying fields, such as mean drift, which may be soft modes of pattern. It allows a precise detection of phase grain boundaries and point defects. Several tests are conducted on numerically generated signals to demonstrate the applicability and precision of the algorithm. Finally, the algorithm is applied to actual experimental convection patterns, allowing us to draw several conclusions about the nature of the wave director field in such patterns. © 1998 Published by Elsevier Science B.V.
• Lvov, Y. V., Binder, R., & Newell, A. C. (1998). Quantum weak turbulence with applications to semiconductor lasers. Physica D: Nonlinear Phenomena, 121(3-4), 317-343.
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  Abstract: Based on a model Hamiltonian appropriate for the description of fermionic systems such as semiconductor lasers, we describe a natural asymptotic closure of the BBGKY hierarchy in complete analogy with that derived for classical weak turbulence. The main features of the interaction Hamiltonian are the inclusion of full Fermi statistics containing Pauli blocking and a simple, phenomenological, uniformly weak two-particle interaction potential equivalent to the static screening approximation. We find a new class of solutions to the quantum kinetic equation which are analogous to the Kolmogorov spectra of hydrodynamics and classical weak turbulence. They involve finite fluxes of particles and energy in momentum space and are particularly relevant for describing the behavior of systems containing sources and sinks. We make a prima facie case that these finite flux solutions can be important in the context of semiconductor lasers and show how they might be used to enhance laser performance. © 1998 Elsevier Science B.V. All rights reserved.
• Feng, Q., Moloney, J. V., Newell, A. C., Wright, E. M., Cook, K., Kennedy, P. K., Hammer, D. X., Rockwell, B. A., & Thompson, C. R. (1997). Theory and simulation on the threshold of water breakdown induced by focused ultrashort laser pulses. IEEE Journal of Quantum Electronics, 33(2), 127-137.
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  Abstract: A comprehensive model is developed for focused pulse propagation in water. The model incorporates self-focusing, group velocity dispersion, and laser-induced breakdown in which an electron plasma is generated via cascade and multiphoton ionization processes. The laser-induced breakdown is studied first without considering self-focusing to give a breakdown threshold of the light intensity, which compares favorably with existing experimental results. The simple study also yields the threshold dependence on pulse duration and input spot size, thus providing a framework to view the results of numerical simulations of the full model. The simulations establish the breakdown threshold in input power and reveal qualitatively different behavior for pico-and femto-second pulses. For longer pulses, the cascade process provides the breakdown mechanism, while for shorter pulses the cooperation between the self-focusing and the multiphoton plasma generation dominates the breakdown threshold.
• Komarova, N. L., Malomed, B. A., Moloney, J. V., & Newell, A. C. (1997). Resonant quasiperiodic patterns in a three-dimensional lasing medium. Physical Review A - Atomic, Molecular, and Optical Physics, 56(1), 803-812.
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  Abstract: Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity in these equations is cubic. For certain sets of four spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. This is a nontrivial example of a nonlinear dissipative system in which the cubic terms are resonant. The equations for the four coupled amplitudes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is π. The latter solution gives rise to a quasiperiodic pattern in the infinite 3D cavity. We also consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperiodic four-mode state rather than suppressing it. Using the Lyapunov function, we find that for the ring-laser configuration the four-mode state is metastable. However, we demonstrate that for a Fabry-Pérot cavity, where diffusion washes out the standing-wave grating, this state is absolutely stable. We also consider a number of more complicated patterns. We demonstrate that adding a pair of resonant vectors, or any number of nonresonant ones, always produces an unstable solution. A set containing several resonant quartets without resonant coupling between them may be stable, but it is less energetically favorable than a single quartet.
• L'vov, V., L'vov, Y., Newell, A. C., & Zakharov, V. (1997). Statistical description of acoustic turbulence. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 56(1 SUPPL. A), 390-405.
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  Abstract: We develop expressions for the nonlinear wave damping and frequency correction of a field of random, spatially homogeneous, acoustic waves. The implications for the nature of the equilibrium spectral energy distribution are discussed.
• Lvov, Y. V., & Newell, A. C. (1997). Semiconductor lasers and Kolmogorov spectra. Physics Letters, Section A: General, Atomic and Solid State Physics, 235(5), 499-503.
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  Abstract: We make a prima facie case that there could be distinct advantages to exploiting a new class of finite flux equilibrium solutions of the quantum Boltzmann equation in semiconductor lasers. © 1997 Published by Elsevier Science B.V.
• Kennedy, P. K., Wright, E. M., Feng, Q., Cook, K., Hammer, D. X., Thompson, C. R., Moloney, J. V., Rockwell, B. A., Druessel, J. J., & Newell, A. C. (1996). Effects of laser-induced breakdown, self-focusing, and plasma shielding on ultrashort-pulse propagation in the eye. Proceedings of SPIE - The International Society for Optical Engineering, 2681, 390-401.
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  Abstract: As part of a research program to understand and model eye damage produced by exposure to subnanosecond laser pulses, an effort is currently being made to model and analyze ultrashort pulse propagation from the cornea to the retina. Both analytical models and numerical simulations are being used to analyze the effects of self-focusing, laser-induced breakdown (LIB), and plasma-pulse interaction. The modeling effort is coupled with experimental measurements of LIB thresholds and plasma shielding for visible, picosecond (psec) and femtosecond (fsec) pulses in water, which serves as a reasonable simulant for the vitreous humor of the eye. Comparison of LIB thresholds to the critical power for self-focusing indicates that self-focusing has little effect on LIB thresholds for long psec pulses. For short psec and fsec pulses, however, numerical simulations show that self-focusing is critical to LIB in water. These results indicate that self-focusing may play a role in fsec pulse ocular damage, by influencing whether LIB and plasma-pulse interaction occur at the retina, in the vitreous, or both. Both the location of the LIB event and the amount of plasma shielding can significantly effect the degree of damage.
• Lega, J., Moloney, J. V., & Newell, A. C. (1995). Universal description of laser dynamics near threshold. Physica D: Nonlinear Phenomena, 83(4), 478-498.
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  Abstract: Complex order parameter descriptions of large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, valid near onset of lasing and for small detunings of the laser from the peak gain, are given in terms of a complex Swift-Hohenberg equation for Class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a Class B laser. The latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems. These universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex Ginzburg-Landau equation description of the laser system valid for finite negative detunings, and traveling wave states, described by coupled Newell-Whitehead-Segel equations valid for finite positive detunings. One of the main conclusions of the present paper is that the usual Eckhaus instability boundary associated with a long wavelength phase instability, and which delineates the region of the stable traveling wave solutions for Class A and C lasers, no longer defines the stability boundary for the mathematically stiff Class B laser. Instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system. This prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser, a Class B system. © 1995.
• Nazarenko, S. V., & Newell, A. C. (1995). Communication with reentry space vehicles via short pulses. Radio Science, 30(6), 1753-1766.
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  Abstract: The resonant absorption of a short pulse of electromagnetic radiation in a plasma slab is considered in order to examine the effectiveness of using short pulses for information transfer through the plasma sheaths surrounding reentry vehicles. The amplitudes of the Raman scattering (RS) on the Langmuir oscillations associated with resonant absorption and the amplitude of the Brillouin scattering (BS) on the nonlinear modifications of the plasma profile are calculated. On-board detection of the RS and BS signals are proposed as parallel and independent channels for reading the message encoded on the electromagnetic pulse sequence. The information transfer capacity of two other nonlinear effects, the generation of the fast electrons in the wave breaking process, and the excitation of the ion-sound waves by the radiative pressure of the Langmuir oscillations are also considered.
• Nazarenko, S. V., Newell, A. C., & Rubenchik, A. M. (1995). Resonant absorption of short pulses. Physics Letters A, 197(2), 159-163.
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  Abstract: The resonant transformation of a Gaussian pulse of electromagnetic radiation into a Langmuir wave in an inhomogeneous plasma is studied. It is shown that if the pulse duration is smaller than the Langmuir wave (plasmon) lifetime, then substantial changes in the Langmuir field profile occur. Namely, the width of this profile becomes greater and its maximum is smaller compared to the correspondent values for the case of long pulses and the same value of the plasmon lifetime. This fact may be useful for experimental measurements of the electron collision frequency. Indirectly, the collision frequency can be obtained by measuring the intensity of the the second harmonic of the postpulse emission, the expression for which in terms of the Langmuir field is given in this Letter. The limits of validity of linear theory due to plasma profile distortion and wavebreaking are discussed. © 1995 Elsevier Science B.V. All rights reserved.
• Newell, A. C., & Pomeau, Y. (1995). Phase diffusion and phase propagation: interesting connections. Physica D: Nonlinear Phenomena, 87(1-4), 216-232.
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  Abstract: We explore parallels between Whitham theory (nonlinear geometrical optics) applied to gradient systems, such as high Prandtl number pattern forming convection layers, and Hamiltonian systems, such as superfluids at zero temperature and thin elastic shells. In particular, we discuss certain universal features such as the canonical nature of the averaged equations and the relation between the conditions for the onset of vortices, the Eckhaus and Landau criteria respectively. We also show that there is an analogue to the zig-zag instability experienced by gradient systems for Hamiltonian systems and discuss how the asymptotic states may relate. We discuss a new approach for obtaining weak and singular solutions (concave and convex disclinations in phase gradient systems and their composites, phase grain boundaries) which takes advantage of a geometrical property of the phase surface, namely that it has zero Gaussian curvature almost everywhere. We exploit similar ideas in the Hamiltonian context. © 1995.
• Feng, Q., Lega, J., Moloney, J. V., & Newell, A. C. (1994). Pattern dynamics in large aspect ratio lasers. IEEE Nonlinear Optics: Materials, Fundamentals and Applications - Conference Proceedings, 66-68.
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  Abstract: A novel out-of-phase oscillating standing wave pattern which appears as a stable output of the laser near threshold is discussed in this paper. When time-averaged, this pattern appears as a stationary square lattice bearing a remarkable similarity to the recent experimental observations in a high Fresnel number CO2 laser. Also, the existence of complex spatio-temporal pattern evolutions are predicted. This, when time integrated, appear stationary and rather regular.
• Feng, Q., Moloney, J. V., & Newell, A. C. (1994). Transverse patterns in lasers. Physical Review A - Atomic, Molecular, and Optical Physics, 50(5), R3601-R3604.
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  Abstract: This paper presents pattern formation of lasers. Traverse patterns in lasers are alternating standing waves (ASW's) in which two standing waves are oriented perpendicular to each other and differ by π/2 in phase. Through Maxwell-Bloch laser equations, it is founded the ASW appears in the presence of appropriate symmetry of the transverse boundary. These boundaries played an important role in selection of wave patterns.
• Jakobsen, P. K., Lega, J., Feng, Q., Staley, M., Moloney, J. V., & Newell, A. C. (1994). Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: I. Analysis and numerical simulation. Physical Review A, 49(5), 4189-4200.
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  Abstract: Transverse pattern evolution is investigated in single-longitudinal-mode two-level and Raman lasers with flat end reflectors, subjected to uniform transverse pumping. The natural nonlinear modes of the laser are identified as spatially homogeneous when the detuning from the gain peak is negative and as "local" plane traveling waves when the detuning is positive. The latter correspond to an off-axis emission of the laser. Stability characteristics of the underlying patterns are predicted to be quite different for one-dimensional and two-dimensional (2D) lasers. As an illustration, we provide direct numerical evidence for weakly turbulent behavior of a 2D Raman laser where Eckhaus and zigzag phase instabilities act in concert to spontaneously nucleate topological defects and ridgelike illuminated regions. Our numerics also confirm that the complicated patterns persist for finite transverse pumping as long as the characteristic width of the pump source contains a sufficient number of selected pattern wavelengths. © 1994 The American Physical Society.
• Lega, J., Jakobsen, P. K., Moloney, J. V., & Newell, A. C. (1994). Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: II. Pattern analysis near and beyond threshold. Physical Review A, 49(5), 4201-4212.
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  Abstract: Complex order-parameter equation descriptions of pattern evolution in large-aspect-ratio two-level and Raman lasers are derived systematically as solvability conditions in a multiple-scales asymptotic expansion of the original Maxwell-Bloch laser equations in powers of a small parameter. These amplitude equations, although strictly valid near threshold for lasing, are shown to capture the essential features of pattern instability and evolution well beyond lasing threshold. A technical difficulty that can arise in the Raman laser, namely, subcriticality of the bifurcation near the critical wave number, is not addressed in the present paper and the order-parameter equations as derived are valid only when this situation does not arise. Analytical expressions for long-wavelength phase instabilities of the underlying traveling-wave pattern, which appears as the natural nonlinear lasing mode when the detuning of the laser from the gain peak is positive, are obtained from the coefficients of a Cross-Newell phase equation. Phase and amplitude instability boundaries, when computed via the original laser equations, the complex order-parameter equations and the phase equation, are shown to be consistent for all cases studied with the exception of the case when a subcritical bifurcation approaches the critical wave number kc. © 1994 The American Physical Society.
• Lega, J., Moloney, J. V., & Newell, A. C. (1994). Swift-Hohenberg equation for lasers. Physical Review Letters, 73(22), 2978-2981.
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  Abstract: Pattern formation in large aspect ratio, single longitudinal mode, two-level lasers with flat end reflectors, operating near peak gain, is shown to be described by a complex Swift-Hohenberg equation for class A and C lasers and by a complex Swift-Hohenberg equation coupled to a mean flow for the case of a class B laser.
• Luther, G. G., Moloney, J. V., Newell, A. C., & Wright, E. M. (1994). Self-focusing threshold in normally dispersive media. Optics Letters, 19(12), 862-864.
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  Abstract: This paper presents the analytical development and numerical simulation of a scaling law for accurately forecasting the self-focusing threshold in normally dispersive media, without any arbitrary factors, which also correlates to the threshold for effects in bulk media, like supercontinuum generation, which are dependent on the intensity. The scaling law also helps understand pulse propagation dynamics such as spectral broadening.
• Luther, G. G., Moloney, J. V., Wright, E. M., & Newell, A. C. (1994). Femtosecond pulse splitting, supercontinuum generation and conical emission in normally dispersive media. IEEE Nonlinear Optics: Materials, Fundamentals and Applications - Conference Proceedings, 400-402.
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  Abstract: In this paper, reported is a self-consistent theory of critical self-focusing in the presence of normal group velocity dispersion. The underlying physical phenomena are all contained in the spatiotemporal dynamics of short laser pulses described by the nonlinear Schrodinger equation (NLS). This work will show that the conical emission predicted by this equation is due to a fundamental four-wave interaction that promotes the transport of energy and modes with finite frequency and energy shifts.
• Luther, G. G., Newell, A. C., & Moloney, J. V. (1994). The effects of normal dispersion on collapse events. Physica D: Nonlinear Phenomena, 74(1-2), 59-73.
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  Abstract: It is shown analytically that positive or normal dispersion arrests at least the initial stage of the critical self-similar collapse events associated with the nonlinear Schrödinger equation. This is done by pertubing a wave packet that undergoes critical self similar collapse in d dimensions by adding weak positive dispersion in one additional direction. Singular perturbation analysis then yields a closed set of simple equations for the parameters of the collapsing mode from which it is clear that in the absence of normal dispersion the self similar collapse will continue while, in its presence, the collapse attractor disappears. Direct numerical integrations are in excellent agreement with the predictions of the simple reduced equations. © 1994.
• Luther, G. G., Newell, A. C., Moloney, J. V., & Wright, E. M. (1994). Conical emission from short-pulse propagation in normally dispersive media. Proceedings of the International Quantum Electronics Conference (IQEC'94), 103-104.
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  Abstract: A uniform asymptotic approximation of Maxwell's equations for propagation of a short pulse in Kerr media yields the well-known nonlinear Schrodinger equation 2ik (∂A÷∂z + k′ ∂A÷∂t) + ▽⊥2 - kk″ ∂2A÷∂t2 + 2k2n2÷n |A|2A = 0, (1) which describes the dynamical evolution of the light pulse subject to diffraction, normal dispersion, and cubic nonlinearity. This equation reproduces many of the features associated with short-pulse propagation that are observed experimentally. In this paper we show that a form of conical emission, due to a four-wave interaction, is captured by Eq. (1). In Fig. 1 the evolution of the intensity and spectrum of a pulse, calculated by integrating Eq. (1), is shown before and after pulse splitting. As the pulse propagates, its bandwidth spreads both in transverse wave number, k⊥, and in frequency, ω-0$/ + Ω. Note that the energy of the pulse is not redistributed uniformly. Instead, energy is preferentially coupled into the band of wave numbers and frequencies indicated by the cross in Fig. 1, on which Ω2 ∝ k⊥2. In Fig. 2 the farfield time-integrated image of the pulse is shown both with and without a filter. Whereas the unfiltered image has no noticeable structure, the filtered image clearly shows off-axis emission at the frequency of the filter. Figure 1 shows that conical emission predicted by Eq. (1) occurs at frequencies of the pulse. The emission angle increases with the magnitude of the shift. As the pulse power is increased, a threshold is reached where the conical signal grows explosively. Here self-focusing plays an important role by reducing the scale length for growth. The same instability that is responsible for conical emission causes an enhancement in the broadening of spectrum of the pulse. An explosive broadening of the spectrum also occurs at this threshold.
• Luther, G. G., Newell, A. C., Moloney, J. V., & Wright, E. M. (1994). Short-pulse conical emission and spectral broadening in normally dispersive media. Optics Letters, 19(11), 789-791.
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  Abstract: This paper describes theoretically a fundamental four-wave interaction which is one of the mechanisms for spectral superbroadening, and can be expressed by the nonlinear Schrodinger equation. The interaction induces conical emission and promotes pulse splitting, and moves energy out along a band of wave numbers in unbounded space, broadening the frequency spectrum in a technique that cannot be modeled by one-dimensional analysis.
• Luther, G. G., Wright, E. M., Moloney, J. V., & Newell, A. C. (1994). Nonlinear propagation of short pulses in the eye. Conference Proceedings - Lasers and Electro-Optics Society Annual Meeting, 8.
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  Abstract: At or above a threshold in peak pulse power that is easily accessible by laser sources available today, a short light pulse can undergo tremendous changes as it propagates from the cornea to the retina. The spot size, the pulse shape, the peak intensity, and the spectral content of the pulse can all undergo changes not predicted with either linear or stationery nonlinear propagation models. Such changes play an important role in understanding the mechanisms by which energy is deposited in ocular tissues in the short-pulse regime. Understanding breakdown, tissue damage or other absorption processes in the eye hinges on understanding the details of the delivery of the light energy. In this paper we develop a simple model which contains the basic short-pulse propagation processes relevant to the optical system of the eye. This model assumes that propagation through the anterior elements of the eye such as the cornea and lens is linear, and that deviations from linear propagation become significant as the pulse propagates through the vitreous humor. Such a model provides information about the irradiance at or near the retina. In the simplest nonlinear media, short pulses propagate under the influence of diffraction, group velocity dispersion, and cubic nonlinearity. The vitreous humor, having optical properties that are very similar to water, is normally dispersive throughout the visible region and becomes anomalous only above 1.1 μm. Normal dispersion tends to inhibit self focusing and broaden the spectrum. As a result, increased input energies are required to cause breakdown. Anomalous dispersion acts in the opposite way, promoting self focusing and localizing the delivery of the pulse energy both in space and time. The Raman response can play an important role in the dynamics even for short pulse propagation. While the Raman effect tends to reduce the time averaged intensity, it causes a temporal modulation that breaks-up the pulse, delocalizing the delivery of energy to the affected tissue. At the same time, the peak intensity can be significantly higher. Further extensions to the basic propagation model and their limitations will also be discussed.
• Nazarenko, S. V., Newell, A. C., & Zakharov, V. E. (1994). Communication through plasma sheaths via Raman (three-wave) scattering process. Physics of Plasmas, 1(9), 2827-2834.
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  Abstract: It was shown in this paper that the process of the Raman scattering of a pump electromagnetic wave on Langmuir oscillations produced by the signal wave can be efficient as a mechanism of information transfer to the vehicle during the reentry blackout of radio communications. An idealized model was utilized based on a simple slab geometry of the plasma sheath and the complexity of various physical chemical processes inside the sheath were all disregarded. An analysis taking account for a real plasma sheath geometry and possible extra ionization because of the pump electromagnetic field is in progress, with the aid of computer simulations.
• Passot, T., & Newell, A. C. (1994). Towards a universal theory for natural patterns. Physica D: Nonlinear Phenomena, 74(3-4), 301-352.
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  Abstract: Our goal is to find a macroscopic description of patterns that both unifies and simplifies classes of externally stressed, dissipative, pattern forming systems, such as convecting fluids, liquid crystals, wideband lasers, that are seemingly unrelated at the microscopic level. We construct an order parameter equation which provides a controlled approximation of the original microscopic field in the limit of large aspect ratios. It is built from, and is a regularization of, the Cross-Newell phase diffusion equation obtained by averaging over the local periodicity of the pattern. Unlike the latter, it is valid for all wavenumbers and can correctly capture the nucleation, shape and nontrivial properties of the far fields of disclinations, dislocations and grain boundaries. It reduces to the Cross-Newell equation away from pattern singularities and to the Newell-Whitehead-Segel equation near onset. As a consequence, it correctly determines all the long wave instability boundaries (zig-zag, Eckhaus-skew-varicose) of the Busse balloon. Far from onset, the order parameter is a real variable but its equation involves a functional corresponding to its local amplitude. The local amplitude and phase, required for the order parameter equation and the reconstruction of the approximation to the original field respectively, are extracted from the order parameter field by wavelet analysis. Numerical comparisons between solutions of the original equation and the regularized equation are carried out. We also explore a new class of singular and weak solutions of the Cross-Newell equation which take account of the energetics of defects as well as their topologies. These solutions correspond to convex and concave disclinations and their composites, including saddles, vortices, targets, dislocations and two new objects, handles and bridges. Finally, we show that phase grain boundaries, lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation. © 1994.
• Taha, T., Ablowitz, M., Bona, J., & Newell, A. (1994). Foreword. Mathematics and Computers in Simulation, 37(4-5), 247-.
• Feng, Q., Moloney, J. V., & Newell, A. C. (1993). Amplitude instabilities of transverse traveling waves in lasers. Physical Review Letters, 71(11), 1705-1708.
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  Abstract: A new class of amplitude instabilities severely restricts the stability domain of the transverse patterns in a laser that takes the form of a continuum of waves traveling transverse to the laser axis. Linear stability analyses and numerical solution of the laser equations reveal that one of the instabilities, which should be manifest in a broad range of contexts for which traveling waves are the preferred pattern, results in the death of the traveling wave and its replacement by a wave traveling in the opposite direction which exhibits novel transient and asymptotic behavior.
• Moloney, J. V., Jakobsen, P. K., Lega, J., Wenden, S. G., & Newell, A. C. (1993). Space-time complexity in nonlinear optics. Physica D: Nonlinear Phenomena, 68(1), 127-134.
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  Abstract: Traveling wave solutions are found to be the natural nonlinear modes of wide aperture two-level and Raman lasers for frequency detunings to the positive side of the gain peak. © 1993.
• Newell, A. C., & Pomeau, Y. (1993). Turbulent crystals in macroscopic systems. Journal of Physics A: Mathematical and General, 26(8), L429-L434.
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  Abstract: The authors demonstrate circumstances in which gradient systems can minimize their free energies with a spatially turbulent planform.
• Newell, A. C., Passot, T., & Lega, J. (1993). Order parameter equations for patterns. Annual Review of Fluid Mechanics, 25(1), 399-453.
• Powell, J. A., Moloney, J. V., Newell, A. C., & Albanese, R. A. (1993). Beam collapse as an explanation for anomalous ocular damage. Applied Optics, 32(29), 1230-1241.
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  Abstract: The topic of ocular damage is discussed. A two-dimensional beam collapse analysis and the nonlinear Schrodinger equation. The effects of Debye relaxation are discussed. A three-dimensional collapse analysis is also presented.
• Dyachenko, S., Newell, A. C., Pushkarev, A., & Zakharov, V. E. (1992). Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 57(1-2), 96-160.
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  Abstract: The nonlinear Schrödinger (NLS) equation iΨt + ∇2Ψ + α{curly logical or}Ψ{curly logical or}sΨ = 0 is a canonical and universal equation which is of major importance in continuum mechanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension and s is the order of nonlinearity, can be understood in terms of a combination of weak turbulence theory and condensate and collapse formation. The results are derived in the broad context of a class of Hamiltonian systems of which NLS is a member, so that the reader can gain a perspective on the ingredients important for the realization of the various equilibrium spectra, thermodynamic, pure Kolmogorov and combinations thereof. We also present time-dependent, self-similar solutions which describe the relaxation of the system towards these equilibrium states. We show that the number of particles lost in an individual collapse event is virtually independent of damping. Our numerical simulation of the full governing equations is the first to show the validity of the weak turbulence approximation. We also present a mechanism for intermittency which should have widespread application. It is caused by strongly nonlinear collapse events which are nucleated by a flow of particles towards the origin in wavenumber space. These highly organized events result in a cascade of particle number towards high wavenumbers and give rise to an intermittency and a behavior which violates many of the usual Kolmogorov assumptions about the loss of statistical information and the statistical independence of large and small scales. We discuss the relevance of these ideas to hydrodynamic turbulence in the conclusion. © 1992.
• Moloney, J. V., Newell, A. C., & Aceves, A. B. (1992). Spatial soliton optical switches: a soliton-based equivalent particle approach. Optical and Quantum Electronics, 24(11), S1269-S1293.
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  Abstract: The spatial soliton plays a fundamental role as a robust, particle-like object which can be manipulated to switch or spatially scan collimated light channels or optical pulses which are incident at an oblique angle to an interface separating two or more selffocusing Kerr slab dielectrics. The underlying equivalent particle/multiparticle theory can also be employed to predict stability properties of nonlinear surface or guided waves, to derive an analytic expression for the nonlinear Goos-Hanchen shift at a nonlinear interface and to quantify the effect of material diffusion, linear and twophoton absorption on the switching behaviour of the light beam. Pulsed spatial switching effects are studied in order to illustrate the general predictive power of the equivalent particle theory. © 1992 Chapman & Hall.
• Newell, A. C. (1992). Inverse cascades in turbulence. Physica D: Nonlinear Phenomena, 61(1-4), 213-216.
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  Abstract: In this article, I report on some joint work with Vladimir E. Zakharov. We suggest that inverse cascades of finite flux motion constants can play a central role in the production of intermittency in both optical and hydrodynamic turbulence and that they may also be responsible for building the dipole component of planetary and solar dynamos. © 1992.
• Newell, A. C., & Passot, T. (1992). Instabilities of dislocations in fluid patterns. Physical Review Letters, 68(12), 1846-1849.
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  Abstract: Dislocations are important for wave-number adjustment in the almost periodic patterns found in convecting fluids and a wide variety of other physical contexts. Using a model for infinite-Prandtl-number fluids, we show that, far from onset, climbing dislocations can undergo local, finite-amplitude instabilities which lead to the formation of disclinations, bridgelike structures, sudden roll disappearance, and gliding.
• Newell, A. C., & Zakharov, V. E. (1992). Rough sea foam. Physical Review Letters, 69(8), 1149-1151.
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  Abstract: A simple theory is developed for rough seas in which the energy cascade towards small scales is too much for a surface tension dominated wrinkling of the surface to handle. We suggest that a new phase, consisting of an air-water foam, is created and becomes the principal medium for energy dissipation. The foam depth and droplet size are calculated and we suggest how the predicted dependence on energy flux and surface tension could be verified experimentally.
• Kuznetsov, E., Newell, A. C., & Zakharov, V. E. (1991). Intermittency and turbulence. Physical Review Letters, 67(23), 3243-3246.
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  Abstract: We suggest a mechanism for the appearance of intermittency in fully developed turbulence consistent with the pictures recently presented by Kraichnan [Phys. Rev. Lett. 65, 575 (1990)] and She [Phys. Rev. Lett. 66, 600 (1991)]. The key features in our model are (i) an inverse cascade associated with the spectral density of an additional finite flux motion invariant, leading to (ii) an attempt to form large-scale structures which (iii) are intrinsically unstable to a broadband spectrum of perturbing modes resulting in a secondary transfer of energy to small dissipative scales in intermittent bursts.
• Powell, J. A., Newell, A. C., & K., C. (1991). Competition between generic and nongeneric fronts in envelope equations. Physical Review A, 44(6), 3636-3652.
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  Abstract: Arguments are presented for understanding the selection of the speed and the nature of the fronts that join stable and unstable states on the supercritical side of first-order phase transitions. It is suggested that from compact support, nonpositive-definite initial conditions, observable front behavior occurs only when the asymptotic spatial structure of a trajectory in the Galilean ordinary differential equation (ODE) corresponds to the most unstable temporal mode in the governing partial differential equation (PDE). This selection criterion distinguishes between a nonlinear front, which has its origin in the first-order nature of the bifurcation, and a linear front. The nonlinear front has special properties as a strongly heteroclinic trajectory in the ODE and as an integrable trajectory in the PDE. Many of the characteristics of the linear front are obtained from a steepest-descent linear analysis originally due to Kolmogorov, Petrovsky, and Piscounov [Bull. Univ. Moscow, Ser. Int., Sec. A 1, 1 (1937)]. Its connection with global stability arguments, and in particular with arguments based on a Lyapunov functional where it exists, is pursued. Finally, the point of view and results are compared and contrasted with those of van Saarloos [Phys. Rev. A 37, 211 (1988); 39, 6367 (1989)]. © 1991 The American Physical Society.
• Moloney, J. V., & Newell, A. C. (1990). Nonlinear optics. Physica D: Nonlinear Phenomena, 44(1-2), 1-37.
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  Abstract: Nonlinear optics provides a wonderful playground for the scientist interested in developing a general understanding of nonlinear behavior and for the innovator for whom the subject holds a wealth of technological promise. The goals of this review are to present a unified overview which emphasizes the universal nature of much of the observed behavior and to bring the rich potential of the field to the attention of the scientific community. © 1990.
• Newell, A. C., Passot, T., & Souli, M. (1990). Convection at finite Rayleigh numbers in large-aspect-ratio containers. Physical Review Letters, 64(20), 2378-2381.
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  Abstract: The phase diffusion and mean drift equations which describe the behavior of a convection pattern are derived, a step which is essential for obtaining quantitative comparisons between theory and experiment. The theory recovers the boundaries of the Busse balloon, agrees closely with the dominant wave numbers observed by Heutmaker and Gollub [Phys. Rev. A 35, 242 (1987)] and Steinberg, Ahlers, and Cannell [Phys. Scr. 32, 534 (1985)] in natural and target patterns, predicts a new instability which is important in facilitating wave-number adjustment in circular target patterns and in initiating time dependence, and predicts Rayleigh numbers at which loss of spatial correlation due to global defect nucleation will occur.
• Newell, A. C., Passot, T., & Souli, M. (1990). Phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers. Journal of Fluid Mechanics, 220, 187-252.
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  Abstract: We derive the phase diffusion and mean drift equations for the Oberbeck-Boussinesq equations in large-aspect-ratio containers. We are able to recover all the long-wave instability boundaries (Eckhaus, zigzag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. Moreover, the development of the skew-varicose instability can be followed and it becomes clear how the mean drift field conspires to enhance the necking of phase contours necessary for the production of dislocation pairs. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker & Gollub at Prandtl number 2.5, and by Steinberg, Ahlers & Cannell at Prandtl number 6.1. We find a new instability, the focus instability, which causes circular target patterns to destabilize and which, at sufficiently large Rayleigh numbers, may play a major role in the onset of time dependence. Further, we predict the values of the Rayleigh number at which the time-dependent but spatially ordered patterns will become spatially disordered. The key difficulty in obtaining these equations is the fact that the phase diffusion equation appears as a solvability condition at order ε (the inverse aspect ratio) whereas the mean drift equation is the solvability condition at order ε2. Therefore, we had to use extremely robust inversion methods to solve the singular equations at order ε and the techniques we use should prove to be invaluable in a wide range of similar situations. Finally, we discuss the introduction of the amplitude as an active order parameter near pattern defects, such as dislocations and foci.
• Varatharajah, P., Newell, A. C., Moloney, J. V., & Aceves, A. B. (1990). Transmission, reflection, and trapping of collimated light beams in diffusive Kerr-like nonlinear media. Physical Review A, 42(3), 1767-1774.
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  Abstract: An earlier equivalent-particle theory, describing the propagation of a self-focused light channel at an oblique angle of incidence to the interface separating two nonlinear dielectric media [A. B. Aceves, J. V. Moloney, and A. C. Newell, J. Opt. Soc. Am. B 5, 559 (1988); Phys. Lett. A 129, 231 (1988); Phys. Rev. A 39, 1809 (1989); 39, 1828 (1989)], is extended to include diffusion of the nonlinear excitation within each medium. The theory replaces the computationally intensive problem of beam propagation by the much simpler and intuitive picture of the motion of an equivalent particle in an equivalent potential. This simpler Newtonian dynamical problem provides quantitative information on the asymptotes of the reflected, transmitted, or trapped channels as well as the stability of the latter as a function of increasing diffusion length. Our main results are that increased diffusion makes light transmission more difficult and tends to wash out the local equilibria of the equivalent potential representing unstable or stable TE nonlinear surface waves. © 1990 The American Physical Society.
• Aceves, A. B., Moloney, J. V., & Newell, A. C. (1989). Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface. Physical Review A, 39(4), 1809-1827.
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  Abstract: A theory is presented that describes the global reflection and transmission characteristics of a self-focused channel propagating at an oblique angle of incidence to an interface separating two or more self-focusing nonlinear dielectric media. The nonlinear wave packet representing the self-focused channel is represented as an equivalent particle moving in an equivalent potential. The dynamics of the particle is described by Newtons equations of motion, with the asymptotic propagation paths of the channel being read off from the associated phase portraits of the equivalent potential. Equilibria of the potential, or equivalently, critical points in the phase plane, represent stationary (stable or unstable) nonlinear surface waves. Stability of the latter follows immediately from a simple inspection of the potential. The shape of the equivalent potential changes with the power in the incident beam. Our theory provides the nonlinear analog of the well-known linear Snells laws of reflection and transmission. Conditions on the validity of the theory are established in parameter space by extensive numerical solution of the nonlinear partial differential equation describing beam propagation. One important conclusion of the paper is that the predictions of the equivalent-particle theory encompass a wide physical parameter space. As an illustration of an application of the theory, we show how to design an all-optical angle or power adjustable spatial scanning element. Contact is made with earlier numerical studies of beam propagation and nonlinear surface-wave stability at a linear-nonlinear interface. © 1989 The American Physical Society.
• Aceves, A. B., Moloney, J. V., & Newell, A. C. (1989). Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions. Physical Review A, 39(4), 1828-1840.
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  Abstract: The theory presented in the preceding paper is extended to account for incident optical beam breakup into multiple self-focused channels and to deal with multiple reflection and transmission at multiple interfaces. Beam breakup is explained by examining the decomposition of the channel in one medium, into its soliton and radiation components after it has crossed into the new medium. A formula is derived which gives the criterion for the number of channels appearing as a consequence of breakup. This formula also provides analytic expressions for the individual self-focused channel powers and the amount of radiation generated. An important observation here is that the amount of radiation generated at the interface shrinks rapidly as a function of increasing channel number N. Each new component generated can be treated as a separate equivalent particle moving in its own equivalent potential. The theory of the preceding paper [Aceves, Moloney, and Newell, Phys. Rev. A 39, 1809 (1989)] can therefore be applied directly to show that low-power channels generated in the breakup will suffer reflection while higher-power channels will undergo transmission. An added ingredient to allow for mutual-channel interaction is the soliton-collision formula. The multiple-interface extension of the single-interface problem results from patching individual single-interface equivalent potentials together. The theory is illustrated with two applications: (i) a nonlinear version of a directional coupler requiring just two interfaces and (ii) trapping of a channel at an interface by a ramped linear refractive index. © 1989 The American Physical Society.
• Aceves, A. B., Moloney, J. V., & Newell, A. C. (1988). Snell's laws at the interface between nonlinear dielectrics. Physics Letters A, 129(4), 231-235.
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  Abstract: An equivalent particle theory is developed which is effective in capturing the global behavior of surface waves at the interface between two nonlinear dielectrics. © 1988.
• Aceves, A., Moloney, J. V., & Newell, A. C. (1988). Mathematical modeling of beam reflection, transmission and trapping at nonlinear optical interfaces. Mathematical and Computer Modelling, 11(C), 101-105.
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  Abstract: The propagation of laser beams as self-focussed channels in nonlinear dielectric media, their mutual interaction and interaction with one or more dielectric interfaces can be described in a unified manner in terms of the dynamics of one or more equivalent particles in equivalent potentials. Beam reflection, transmission and trapping as nonlinear surface waves is described by the phase space trajectories of the appropriate equivalent particle. The equivalent particle theory can, with minor extensions, provide quantitative estimates on beam break-up at the interface into multiple self-focussed channels and accurately estimate the amount of incident beam power converted into scattered radiation. The theoretical predictions are supported by full scale numerical simulations. © 1988.
• Adachihara, H., McLaughlin, D. W., Moloney, J. V., & Newell, A. C. (1988). Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis. Journal of Mathematical Physics, 29(1), 63-85.
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  Abstract: The transverse behavior of a laser beam propagating through a bistable optical cavity is investigated analytically and numerically. Numerical experiments that study the (one-dimensional) transverse structure of the steady state profile are described. Mathematical descriptions of (i) an infinite-dimensional map that models the situation, (ii) the solitary waves that represent the transverse steady state structures, (iii) a projection formalism that reduces the infinite-dimensional map to a finite-dimensional one, and (iv) the theoretical analysis of this reduced map are presented in detail. The accuracy of this theoretical analysis is established by comparing its predictions to numerical observations. © 1987 American Institute of Physics.
• Gibbon, J. D., Newell, A. C., Tabor, M., & Zeng, Y. B. (1988). Lax pairs, Backlund transformations and special solutions for ordinary differential equations. Nonlinearity, 1(3), 481-490.
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  Abstract: The authors investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Backlund transformations for systems of ordinary differential equations. This method can yield both auto-Backlund transformations and, where necessary, Backlund transformations between different equations. In the latter case they investigate the circumstances under which the general Backlund transformations reduce to auto-Backlunds. In addition, special solution families for the second and fourth Painleve transcendents are obtained.
• Newell, A. C., Rand, D. A., & Russell, D. (1988). Turbulent dissipation rates and the random occurrence of coherent events. Physics Letters A, 132(2-3), 112-123.
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  Abstract: In this Letter, we suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events that correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor. Homoclinic excursions are trajectories in the non-compact phase space that are attracted to special orbits which connect saddle points in the finite region of phase space to infinity and represent coherent structures in the flow field. © 1988.
• Newell, A. C., Rand, D. A., & Russell, D. (1988). Turbulent transport and the random occurence of coherent events. Physica D: Nonlinear Phenomena, 33(1-3), 281-303.
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  Abstract: We suggest that the transport properties and dissipation rates of a wide class of turbulent flows are determined by the random occurrence of coherent events which correspond to certain orbits which we call homoclinic excursions in the high dimensional strange attractor. Homoclinic excursions are trajectories in the noncompact phase space that are attracted to special orbits which connect saddle points in the finite region of phase space to infinity and represent organized structures in the flow field. This picture also suggests that one can compute fluxes using a relatively low dimensional description of the flow. A method for extracting the organized structures from a time-series is given and provides a local analogue of the notion of Lyapunov exponents. © 1988.
• Newell, A. C., Tabor, M., & Zeng, Y. B. (1987). A unified approach to Painlevé expansions. Physica D: Nonlinear Phenomena, 29(1-2), 1-68.
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  Abstract: The Painlevé test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Bäcklund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy. © 1987.
• Aceves, A., Adachihara, H., Jones, C., Lerman, J. C., McLaughlin, D. W., Moloney, J. V., & Newell, A. C. (1986). Chaos and coherent structures in partial differential equations. Physica D: Nonlinear Phenomena, 18(1-3), 85-112.
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  Abstract: This paper addresses the possible connections between chaos, the unpredictable behavior of solutions of finite dimensional systems of ordinary differential and difference equations and turbulence, the unpredictable behavior of solutions of partial differential equations. It is dedicated to Martin Kruskal on the occasion of his 60th birthday. © 1986.
• Brand, H. R., Lomdahl, P. S., & Newell, A. C. (1986). Benjamin-Feir turbulence in convective binary fluid mixtures. Physica D: Nonlinear Phenomena, 23(1-3), 345-361.
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  Abstract: Binary mixtures of water and alcohol or 3He and 4He provide an excellent vehicle for studying the onset of a rich variety of dynamical behavior. For different values of the separation ratio, these systems exhibit both steady and oscillatory convection, competition between the two, and in another parameter range, a weakly turbulent state. In this paper we develop a general theory which is valid near onset and which can account for each of these features. A universal equation of Ginzburg-Landau type with complex coefficients obtains and is investigated numerically and analytically in one and two spatial dimensions. The most novel prediction is that some of the a periodic behavior seen in experiment is due to the universal Benjamin-Feir instability and we call this phenomenon Benjamin-Feir turbulence. Although the possibility of this effect has been discussed theoretically in the literature for over ten years, binary mixture convection provides the first vehicle in which the prediction can be quantitatively tested. We also discuss mean drift effects and the consequences of fixed lateral boundaries. © 1986.
• Brand, H. R., Lomdahl, P. S., & Newell, A. C. (1986). Evolution of the order parameter in situations with broken rotational symmetry. Physics Letters A, 118(2), 67-73.
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  Abstract: We describe the evolution of the envelope of a wavetrain which, at a critical value of the stress parameter, breaks the rotational symmetry of the governing equations. The new envelope equation is a generalization to wavelike disturbances of the Newell-Whitehead-Segel equation and the one-dimensional complex Ginzburg-Landau equation. The most novel prediction of the new theory is that spatially uniform envelopes are rarely stable and that the dynamics is dominated by terms which c...onserve phase space volume. In particular, it turns out that the strongest spatial modulation takes place along the wave crests, perpendicular to the direction of the propagation. We also discuss mean drift effects and in particular their consequences near lateral boundaries. © 1986.
• Campbell, D., Newell, A., Schrieffer, B., & Segur, H. (1986). Preface. Physica D: Nonlinear Phenomena, 18(1-3), vii-viii.
• Newell, A. C., & Yunbo, Z. (1986). The Hirota conditions. Journal of Mathematical Physics, 27(8), 2016-2021.
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  Abstract: The condition on the polynomial P for a Hirota equation Pτ·τ=0 to have an N-soliton solution for arbitrary N is examined and simplified. © 1986 American Institute of Physics.
• Knickerbocker, C. J., & Newell, A. C. (1985). Propagation of solitary waves in channels of decreasing depth. Journal of Statistical Physics, 39(5-6), 653-674.
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  Abstract: The changes which occur in a right-going solitary wave as it travels a channel of decreasing depth are discussed. In addition to the changes in the solitary wave, we have found through a judicious use of the conservation laws two secondary structures (a shelf and a reflection). Each of these structures is small with respect to the solitary wave, though the mass flux associated with each is of the same order as that of the solitary wave. Of interest is that the amplitude of the reflected wave does not satisfy Green's law. But rather, the amplitude of the reflected wave is constant along left-going characteristics. This finding allows us to satisfy the mass flux conservation laws to leading order and establishes the perturbed Korteweg-deVries equation as a consistent approximation for the right-going profile. © 1985 Plenum Publishing Corporation.
• Knickerbocker, C. J., & Newell, A. C. (1985). REFLECTIONS FROM SOLITARY WAVES IN CHANNELS OF DECREASING DEPTH.. Journal of Fluid Mechanics, 153, 1-16.
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  Abstract: The paper discusses the finding that the reflected wave that is created by a right-going solitary wave as it travels in a region of slowly changing depth does not satisfy Green's law. The amplitude of the reflected wave is constant along left-going characteristics rather than proportional to the negative fourth root of depth. This new finding allows the mass-flux conservation laws to be satisfied to leading order and establishes that the perturbed Korteweg-de Vries equation is a consistent approximation for the right-going profile.
• McLaughlin, D. W., Moloney, J. V., & Newell, A. C. (1985). New class of instabilities in passive optical cavities. Physical Review Letters, 54(7), 681-684.
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  Abstract: In this Letter we show that the fixed points of the Ikeda map are more unstable to perturbations with a short-scale transverse structure than to plane-wave perturbations. We correctly predict the most unstable wavelength, the critical intensity, and the growth ratés of these disturbances. Our result establishes that, for a large class of nonlinear waves, spatial structure is inevitable and drastically alters the route to chaos. In an optical cavity the consequence is that the period-doubling cascade is an unlikely scenario for transition to optical chaos. © 1985 The American Physical Society.
• Meiron, D., & Newell, A. C. (1985). The shape of stationary dislocations. Physics Letters A, 113(5), 289-292.
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  Abstract: It is shown that the structure of the stationary dislocations which occur in a wide variety of patterns in nonequilibrium systems is given by a self-similar solution of the universal Cross-Newell equation. © 1985.
• Cross, M. C., & Newell, A. C. (1984). Convection patterns in large aspect ratio systems. Physica D: Nonlinear Phenomena, 10(3), 299-328.
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  Abstract: A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-Bénard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are discussed. The effects of adding the local mean drift important in low Prandtl number situations are investigated. Our theory includes the notion of the Busse stability balloon, reduces near critical values of the stress parameter to the Newell-Whitehead-Segel equations, and contains the Pomeau-Manneville phase equation. It also gives a detailed description of the way in which the field amplitude, which is slaved to the phase gradient away from threshold, becomes an independent order parameter near the critical point. © 1984.
• Briggs, W. L., Newell, A. C., & Sarie, T. (1983). Focusing: A mechanism for instability of nonlinear finite difference equations. Journal of Computational Physics, 51(1), 83-106.
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  Abstract: A mechanism for the destabilization of numerical algorithms for partial differential equations is suggested. The novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations. © 1983.
• Laughlin, D., Moloney, J. V., & Newell, A. C. (1983). Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity. Physical Review Letters, 51(2), 75-78.
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  Abstract: Phase-locked solitary waves are shown to be the stable fixed points of an infinite-dimensional map obtained from a bistable optical ring cavity. © 1983 The American Physical Society.
• Newell, A. C. (1983). Two Dimensional Convection Patterns in Large Aspect Ratio Systems. North-Holland Mathematics Studies, 81(C), 205-231.
• Eilbeck, J. C., Lomdahl, P. S., & Newell, A. C. (1981). Chaos in the inhomogeneously driven sine-Gordon equation. Physics Letters A, 87(1-2), 1-4.
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  Abstract: Some numerical studies are reported of solutions of the sine-Gordon equation with damping and a mixture of two types of driving force: a constant (dc) force and a spatially inhomogeneous force varying harmonically (ac) in time. For some ranges of parameters the excitation of kink-antikink pairs is chaotic. © 1981.
• Knickerbocker, C., & Newell, A. C. (1981). Propagation of surface and internal solitary waves in nonuniform media. Applied Scientific Research, 37(1-2), 111-126.
• Knickerbocker, C. J., & Newell, A. C. (1980). HOW INTERNAL WAVES TRAVELLING ON A THERMOCLINE CHANGE POLARITY NEAR CRITICAL POINTS.. Rundfunktechnische Mitteilungen, 10, 581-603.
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  Abstract: The paper examines the effects of a slowly varying depth on a solitry wave propagating on the surface and on a thermocline near the shoreline. The mechanism by which an internal solitary wave reverses its polarity is reported.
• Knickerbocker, C. J., & Newell, A. C. (1980). Internal solitary waves near a turning point. Physics Letters A, 75(5), 326-330.
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  Abstract: We report an unexpected result and illustrate the mechanism by which a solitary wave propagating on a thermocline reverses its polarity as it passes through a turning point near the shoreline. © 1980.
• Knickerbocker, C. J., & Newell, A. C. (1980). SHELVES AND THE KORTEWEG-DE VRIES EQUATION.. Journal of Fluid Mechanics, 98(pt 4), 803-818.
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  Abstract: An extension of the analytical results of D. J. Kaup & A. C. Newell concerning the effect of a perturbation on a solitary wave of the Korteweg-de Vries equation is given and numerical studies are conducted to verify the conclusions. In all cases, the numerical results agree with the results predicted by the theory. The most striking feature of the perturbed flow is the presence of a shelf in the lee of the solitary wave whose role is to absorb (provide) the extra mass which is created (depleted) by the perturbation. Refs.
• Kaup, D. J., & Newell, A. C. (1979). Evolution equations, singular dispersion relations, and moving eigenvalues. Advances in Mathematics, 31(1), 67-100.
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  Abstract: A new approach for finding the class of integrable evolution equations associated with a given eigenvalue problem is developed. The key point to note is that the squares of the eigenfunctions form a natural basis in which to expand the solutions of the evolution equation. Once this step is taken, the class of integrable equations may usually be read off by inspection. Of particular interest are those equations for which the bound state eigenvalues are not invariant but move in a way prescribed by the coefficients of the evolution equation. The corresponding solitons have the property that they retain their identity on collision with other solution components, but this identity is no longer a constant one. The Hamiltonian structure and the causality properties of these systems are also explored. © 1979.
• Newell, A. C. (1979). GENERAL STRUCTURE OF INTEGRABLE EVOLUTION EQUATIONS.. Proc R Soc London Ser A, 365(1722), 283-311.
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  Abstract: This paper presents some new results in connection with the structure of integrable evolution equations. The idea in solving the initial value problem for partial differential equation was to transform it to a separable system of ordinary differential equations.
• Kaup, D. J., & Newell, A. C. (1978). Theory of nonlinear oscillating dipolar excitations in one-dimensional condensates. Physical Review B, 18(10), 5162-5167.
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  Abstract: We find that for a one-dimensional condensate in the presence of an applied ac field, a significant absorption peak should occur for values of the external frequency just below the pinning frequency ωF. This power absorption is due to those nonlinear normal modes of the condensate known as "breathers," which are shown to phase lock with the applied field. © 1978 The American Physical Society.
• Newell, A. C. (1978). The inverse scattering transform, nonlinear waves, singular perturbations and synchronized solitons. Rocky Mountain Journal of Mathematics, 8(1-2), 25-52.
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  Abstract: It is not the intention of this article to cover, in detail, all of the material presented in my lecture notes at the Conference. Rather, it is the plan to attempt to point out (a) the universal nature and origin of the partial differential equations which the inverse scattering transform enables one to solve and (b) the rich potential for examining partial differential equations which are close, in some perturbation sense, to integrable ones. Several examples are investigated. We discuss the synchronous response of a nonlinear Schrödinger soliton to an applied field and also examine the effects of density gradients, damping and diffusion. We also consider the behavior of a kink of the sine-Gordon equation in the presence of an impurity and finally develop 4π pulse solutions of the double sine- Gordon equation which are formed by a pair of synchronized 2π pulses. Copyright © 1978 Rocky Mountain Mathematics Consortium.
• Kaup, D. J., & Newell, A. C. (1977). An exact solution for a derivative nonlinear Schrödinger equation. Journal of Mathematical Physics, 19(4), 798-801.
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  Abstract: A method of solution for the "derivative nonlinear Schrödinger equation" iqt = -qxx±i(q*q 2)x is presented. The appropriate inverse scattering problem is solved, and the one-soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess "algebraic solitons." © 1978 American Institute of Physics.
• Kaup, D. J., & Newell, A. C. (1977). On the coleman correspondence and the solution of the massive thirring model. Lettere Al Nuovo Cimento Series 2, 20(9), 325-331.
• Newell, A. C. (1977). Nonlinear tunnelling. Journal of Mathematical Physics, 19(5), 1126-1133.
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  Abstract: The propagation of a soliton pulse towards a finite potential barrier is examined, and it is found that in certain circumstances, depending on the ratio of soliton amplitude to barrier height, the soliton can tunnel through the barrier in a lossless manner. It is suggested that the phenomenon of barrier penetration by nonlinear pulses may have widespread application. © 1978 American Institute of Physics.
• Newell, A. C., & Redekopp, L. G. (1977). Breakdown of Zakharov-Shabat theory and soliton creation. Physical Review Letters, 38(8), 377-380.
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  Abstract: The Zakharov-Shabat theory of integrable systems with more than one spatial dimension breaks down when certain resonances between solitons occur. Remarkably, the resonance condition is precisely the same as that for the strong interaction of three weakly coupled linear dispersive waves. This provides a mechanism for the creation of new particles by collision. © 1977 The American Physical Society.
• Newell, A. C. (1976). Synchronized solitons. Journal of Mathematical Physics, 18(5), 922-926.
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  Abstract: We find that, under certain conditions, two solitons of an integrable system can form a synchronized bound state when the system is perturbed. Copyright © 1977 American Institute of Physics.
• Ablowitz, M. J., Kaup, D. J., Newell, A. C., & Segur, H. (1974). INVERSE SCATTERING TRANSFORM-FOURIER ANALYSIS FOR NONLINEAR PROBLEMS.. Studies in Applied Mathematics, 53(4), 249-315.
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  Abstract: A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.
• Lange, C. G., & Newell, A. C. (1974). STABILITY CRITERION FOR ENVELOPE EQUATIONS.. SIAM Journal on Applied Mathematics, 27(3), 441-456.
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  Abstract: The statistical initial value problem for the envelope equation dw/dt minus gamma d**2w/dx**2 equals chi w- beta w**2w*; gamma , beta complex, is discussed. (The derivatives are both partial derivatives. ) It is shown that there exists an instability (stability) criterion which determines whether the system underlying the above equation achieves or does not achieve a monochromatic state.
• Ablowitz, M. J., & Newell, A. C. (1973). The decay of the continuous spectrum for solutions of the Korteweg-deVries equation. Journal of Mathematical Physics, 14(9), 1277-1284.
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  Abstract: The asymptotic behavior of the solution u(x, t) of the Korteweg-deVries equation ut + uux + uxxx = 0 is investigated for the class of problems where the initial data does not give rise to an associated discrete spectrum. It is shown that the behavior is different in the three regions (i) x ≫ t1/3, (ii) x = 0(t1/3), (iii) x ≪ - t1/3. Asymptotic solutions in each of these regions are found which match at their respective boundaries. One of the Berezin-Karpman similarity solutions is the asymptotic state in (ii) and u(x, t) decays like 1/t2/3 in this region. For region (i) there is exponential decay, whereas in region (iii) the structure of u(x, t) is highly oscillatory and the amplitudes of the oscillations decay as 1/t1/2 when - x/t is of order unity. For x/t large and negative these amplitudes decay at least as 1/(- x/t)1/4t1/2. Copyright © 1973 by the American Institute of Physics.
• Ablowitz, M. J., Kaup, D. J., & Newell, A. C. (1973). Coherent pulse propagation, a dispersive, irreversible phenomenon. Journal of Mathematical Physics, 15(11), 1852-1858.
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  Abstract: The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect "this problem" has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state. Copyright © 1974 American Institute of Physics.
• Ablowitz, M. J., Kaup, D. J., Newell, A. C., & Segur, H. (1973). Method for solving the sine-gordon equation. Physical Review Letters, 30(25), 1262-1264.
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  Abstract: The initial value problem for the sine-Gordon equation is solved by the inverse-scattering method. © 1973 The American Physical Society.
• Ablowitz, M. J., Kaup, D. J., Newell, A. C., & Segur, H. (1973). Nonlinear-evolution equations of physical significance. Physical Review Letters, 31(2), 125-127.
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  Abstract: We present the inverse scattering method which provides a means of solution of the initial-value problem for a broad class of nonlinear evolution equations. Special cases include the sine-Gordon equation, the sinh-Gordon equation, the Benney-Newell equation, the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and generalizations. © 1973 The American Physical Society.
• Lange, C. G., & Newell, A. C. (1973). SPHERICAL SHELLS LIKE HEXAGONS. CYLINDERS PREFER DIAMONDS - 1.. ASME Pap.
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  Abstract: Describes the initial postbuckling behavior of cylindrical shells under axial compression. The thesis is that, out of a single infinity of possible buckling configurations which all correspond to diamond-shaped patterns, the square diamond pattern dominates. It is not claimed that this pattern will be seen under all circumstances; if no substantial bias is present in either the initial imperfections or initial conditions, a natural selection mechanism exists which favors the square diamond configuration. A multiple scale technique is used to describe both the dynamic interaction and evolution of competing diamond patterns and the propagation of spatial inhomogeneities. Stability analysis on the resulting differential equation system lends support to the stated thesis. In addition, some initial numerical studies are presented. Part 2 will be devoted to more extensive numerical experiments.
• Lange, C. G., & Newell, A. C. (1973). SPHERICAL SHELLS LIKE HEXAGONS: CYLINDERS PREFER DIAMONDS - 1.. Journal of Applied Mechanics, Transactions ASME, 40 Ser E(2), 575-581.
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  Abstract: Paper No. 73-APM-7.
• Lange, C. G., & Newell, A. C. (1973). The role of resonant interactions in the dynamic stability of elastic systems. Journal of Elasticity, 3(4), 271-276.
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  Abstract: A resonance phenomenon which can occur in elastic systems supporting wave motion is discussed. An analytical and numerical study of the dynamic stability of a cylindrical shell under axial compression illustrates the potential importance of this resonance phenomenon for imperfection-sensitive structures. © 1973 Noordhoff International Publishing.
• BENNEY, D. J., & NEWELL, A. C. (1969). RANDOM WAVE CLOSURES. Studies in Applied Mathematics, 48(1), 29-53.
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  Abstract: Detailed study is made of the way in which weak nonlinearities affect the statistical properties of a system of dispersive waves. Given that at some initial instant the spectral cumulants are sufficiently smooth it is shown that they will remain smooth to a zeroth order, save in one dimension where a discrete spectrum may eventually be generated. Of prime interest is the fact that on considering the long time behavior of the system, one is led to a sequence of closures for the zeroth order spectral functions. Apparent difficulties associated with the irretraceability of the solution are discussed. The structure of the closure equations depends on the asymptotic behavior of a class of singular integrals.
• NEWELL, A. C. (1969). ROSSBY WAVE PACKET INTERACTIONS. Journal of Fluid Mechanics, 35(pt 2), 255-271.
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  Abstract: Mechanism is proposed whereby planetary zonal flows can be generated by the resonant interaction of Rossby wave packets whose amplitudes are slowly varying functions of both space and time. Equations are derived describing the longtime behavior of a resonantly interacting triad. At the first closure certain properties analogous to those already known for discrete waves are deduced. At the second closure, in the particular case when one of the members of the triad is a zonal flow, it is shown that the sideband resonance mechanism can cause energy to be gained or lost by this zonal flow. It is also shown that a single Rossby wave packet can exchange energy with a zonal flow with weak shear.
• NEWELL, A. C., & WHITEHEAD, J. A. (1969). FINITE BANDWIDTH, FINITE AMPLITUDE CONVECTION. Journal of Fluid Mechanics, 38(pt 2), 279-303.
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  Abstract: It is shown how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Benard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space maye 180//0 out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.
• Newell, A. C. (1968). An alternative proof of the Poincaré-Bertrand formula for real integrals and its generalization to n dimensions. Journal of Mathematical Analysis and Applications, 24(1), 149-155.
• Benney, D. J., & Newell, A. C. (1967). Statistical properties of the sea. Physics of Fluids, 10(9), S281-S290.
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  Abstract: The long-time behavior of weakly interacting random waves on the ocean surface is shown to be a solvable theoretical problem. Asymptotic closures are obtained for both the free and forced cases.