Nicholas M Ercolani
 Professor, Mathematics
 Professor, Applied Mathematics  GIDP
Contact
 (520) 6214343
 Environment and Natural Res. 2, Rm. S415
 Tucson, AZ 85719
 ercolani@math.arizona.edu
Degrees
 Ph.D. Mathematics
 University of California at Berkeley, Berkeley, California, United States
Interests
No activities entered.
Courses
202021 Courses

Dynamical Systems+Chaos
MATH 557A (Fall 2020)
201920 Courses

Complex Analysis
MATH 520B (Spring 2020) 
Dissertation
MATH 920 (Spring 2020) 
Topics Modern Analysis
MATH 529 (Spring 2020) 
Complex Analysis
MATH 520A (Fall 2019) 
Dissertation
MATH 920 (Fall 2019)
201819 Courses

Dissertation
MATH 920 (Spring 2019) 
Complex Analysis
MATH 520A (Fall 2018) 
Dissertation
MATH 920 (Fall 2018)
201718 Courses

Dissertation
MATH 920 (Spring 2018) 
Dissertation
MATH 920 (Fall 2017)
201617 Courses

Complex Analysis
MATH 520B (Spring 2017) 
Dissertation
MATH 920 (Spring 2017) 
Independent Study
MATH 599 (Spring 2017) 
Topics In Applied Math
MATH 577 (Spring 2017) 
Complex Analysis
MATH 520A (Fall 2016) 
Dissertation
MATH 920 (Fall 2016) 
Independent Study
MATH 599 (Fall 2016)
201516 Courses

Dissertation
MATH 920 (Spring 2016) 
Independent Study
MATH 599 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Ercolani, N. M., Jansen, S., & Ueltschi, D. (2019). Singularity Analysis for Heavy Tailed Rendom Variables. Journal of Theoretical Probability, 32(1), 1  46. doi:https://doi.org/10.1007/s1095901808322
 Lega, J. C., Kamburov, N., & Ercolani, N. M. (2018). The Phase Structure of Grain Boundaries. Philosophical Transactions of the Royal Society A.
 Chouchkov, D., Ercolani, N. M., Rayan, S., & Sigal, I. M. (2017). GinzburgLandau equations on Riemann surfaces of higher genus. TBD.More infoWe study the GinzburgLandau equations on Riemann surfaces of arbitrarygenus. In particular: we explicitly construct the (local moduli space ofgaugeequivalent) solutions in a neighbourhood of a constant curvature branchof solutions; in linearizing the problem, we find a relation with de Rhamcohomology groups of the surface; we classify holomorphic structures on linebundles arising as solutions to the equations in terms of the degree, theAbelJacobi map, and symmetric products of the surface; we construct explicitlythe automorphy factors and the equivariant connection on the trivial bundleover the Poincar\'e upper complex half plane.[Journal_ref: ]
 Ercolani, N. M., Jansen, S., & Ueltschi, D. (2014). Random Partitions in Statistical Mechanics. Electronic Journal of Probability, 19(82), 1  37. doi:10.1214/EJP.v193244
 Ercolani, N. M. (2014). Conservation Laws of Random Matrix Theory. MSRI Publications, 68.More infoThe volume is in press (galleys were finalized).
 Ercolani, N. M., & Ueltschi, D. (2014). Cycle structure of random permutations with cycle weights. Random Structures and Algorithms, 44(1), 109133.More infoAbstract: We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables. © 2012 Wiley Periodicals, Inc.
 Ercolani, N. M., Jansen, S., & Ueltschi, D. (2014). Random Partitions in Statistical Mechanics. Electronic Journal of Probability, 19(82), 1  37.
 Ercolani, N. M., & Pierce, V. U. (2012). The continuum limit of Toda lattices for random matrices with odd weights. Communications in Mathematical Sciences, 10(1), 267305.More infoAbstract: This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitian random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a sequence of prior works whose ultimate aim is to reveal and understand, in a rigorous way, the deep connections between correlation functions for eigenvalues of these random matrix ensembles on the one hand and the enumerative interpretations of their matrix moments in terms of map combinatorics (a branch of graph theory) on the other. In doing this we make essential use of the link between the asymptotics of the random matrix partition function and orthogonal polynomials with exponential weight equal to the random matrix potential. Along the way we develop and analyze the continuum limits of both the Toda lattice equations and the difference string equations associated to these orthogonal polynomials. The former are found to have the structure of a hierarchy of nearconservation laws; the latter are a novel semiclassical extension of the traditional string equations. One has these equations for each class of regular maps of a given valence. Our methods apply to regular maps of both even and odd valence, however we focus on the latter since that is the relevant case for this paper. These methods enable us to rigorously determine closed form expressions for the generating functions that enumerate trivalent maps, in general implicitly, but also explicitly in a number of cases. © 2012 International Press.
 Ercolani, N. M. (2011). Caustics, counting maps and semiclassical asymptotics. Nonlinearity, 24(2), 481526.More infoAbstract: This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion (and its derivatives), are generating functions for avariety of graphical enumeration problems. The main results are to prove that these generating functions are,in fact, specific rational functions of a distinguished irrational (algebraic) function, z 0(t). This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of weight of the parameter t ). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions. As an intriguing application, one gains new insights into the relation between certain derivatives of the genus expansion, in a doublescaling limit, and the asymptotic expansion of the first Painlevé transcendent. This provides a precise expression of the Painlevé asymptotic coefficients directly in terms of the coefficients of the partial fractions expansionof the rational form of the generating functions established in this paper. Moreover, these insights point towards a more general program relating the first Painlevé hierarchy to the higher order structure of the doublescaling limit through the specific rational structure of generating functions in the genus expansion. The paper closes with a discussion of the relation of this work to recent developments in understanding the asymptotics of graphical enumeration. As a byproduct, these results also yield new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights, the calculation of correlation functions for certain tied. © 2011 IOP Publishing Ltd and London Mathematical Society.
 Ercolani, N. M., & Venkataramani, S. C. (2009). A variational theory for point defects in patterns. Journal of Nonlinear Science, 19(3), 267300.More infoAbstract: We derive a rigorous scaling law for minimizers in a natural version of the regularized CrossNewell model for pattern formation far from threshold. These energyminimizing solutions support defects having the same character as what is seen in experimental studies of the corresponding physical systems and in numerical simulations of the microscopic equations that describe these systems. © 2008 Springer Science+Business Media, LLC.
 Ercolani, N. M., McLaughlin, K. D., & Pierce, V. U. (2008). Random matrices, graphical enumeration and the continuum limit of toda lattices. Communications in Mathematical Physics, 278(1), 3181.More infoAbstract: In this paper we derive analytic characterizations for and explicit evaluations of the coefficients of the matrix integral genus expansion. The expansion itself arises from the large N asymptotic expansion of the logarithm of the partition function of N × N Hermitian random matrices. Its g th coefficient is a generating function for graphical enumeration on Riemann surfaces of genus g. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. Our results are based on a hierarchy of recursively solvable differential equations, derived through a novel continuum limit, whose solutions are the coefficients we want to characterize. These equations are interesting in their own right in that their form is related to partitions of 2g + 1 and joint probability distributions for conditioned random walks. © 2007 SpringerVerlag.
 Ercolani, N. M., & Lozano, G. I. (2006). A biHamiltonian structure for the integrable, discrete nonlinear Schrödinger system. Physica D: Nonlinear Phenomena, 218(2), 105121.More infoAbstract: This paper shows that the AL (AblowitzLadik) hierarchy of (integrable) equations can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J, and a new nonlocal, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R = K J 1. In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering biHamiltonian structures for other families of discrete, integrable equations. © 2006 Elsevier Ltd. All rights reserved.
 Glasgow, S. A., Agrotis, M. A., & Ercolani, N. M. (2005). An integrable reduction of inhomogeneously broadened optical equations. Physica D: Nonlinear Phenomena, 212(12), 8299.More infoAbstract: We introduce a set of reduced MaxwellBloch equations that incorporates the effects of a permanent dipole and inhomogeneous broadening. We demonstrate the integrability of this reduction by providing a Lax pair, which is found using the pseudopotential technique. An appropriate Bäcklund transformation is employed to produce a family of solitonic solutions. Published by Elsevier B.V.
 Ercolani, N., & Taylor, M. (2004). The dirichlettoneumann map, viscosity solutions to eikonal equations, and the selfdual equations of pattern formation. Physica D: Nonlinear Phenomena, 196(34), 205223.More infoAbstract: We study the limiting behavior as ε ↘ 0 of solutions u ε to the Dirichlet problem ε2 δu ε  uε = 0 on Ω, u∂Ω = e θ/ε. where ̄Ω is a bounded domain and θ a given smooth function on its boundary ∂Ω. We provide a natural criterion on θ in order to obtain an estimate εvu ε(x)/uε(x) ≤C≤∞, x ∈ ∂ Ω, independent of ε as ε 0, where ∂vu ε denotes the normal derivative of uε. The results of this paper serve to significantly strengthen the analysis of asymptotic minimizers for a GinzburgLandau variational problem for irrotational vector fields (gradient vector fields) known as the regularized CrossNewell variational problem in the pattern formation literature. In particular, this yields estimates on the asymptotic energy of these minimizers for general Dirichlet viscosity boundary conditions. The class of boundary conditions for this variational problem to which our methods apply is quite general (even including ' domains which are general Riemannian manifolds with boundary). This, for instance, provides a first step for extending the GinzburgLandau type model we consider to the larger class of vector fields that are locally gradient (often called director fields). © 2004 Elsevier B.V. All rights reserved.
 Ercolani, N. M., & McLaughlin, K. D. (2003). Asymptotics of the partition function for random matrices via RiemannHilbert techniques and applications to graphical enumeration. International Mathematics Research Notices.
 Ercolani, N. M., Jin, S., Levermore, C. D., & MacEvoy Jr., W. D. (2003). The Zerodispersion Limit for the Odd Flows in the Focusing Zakharovshabat Hierarchy. International Mathematics Research Notices, 25292564.
 Ercolani, N. M., & McLaughlin, K. T. (2001). Asymptotics and integrable structures for biorthogonal polynomials associated to a random twomatrix model. Physica D: Nonlinear Phenomena, 152153, 232268.More infoAbstract: We give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form en(V(x)+W(y)2τxy)dx dy for polynomial V and W. We are further able to show that the zeroes of these polynomials are all real and distinct. A complex analytical construction of the biorthonormal polynomials is given in terms of a nonlocal RiemannHilbert problem which, given our prior result, provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large. The biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multimatrix models. We show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of V and W coincide with a semiinfinite generalization of the completely integrable full KostantToda lattice. This connection could be relevant for understanding aspects of scaling limits for the multimatrix model. © 2001 Elsevier Science B.V.
 Ercolani, N. M., Kuznetsov, E. A., & Levermore, C. D. (2001). Physica D: Nonlinear Phenomena: Preface. Physica D: Nonlinear Phenomena, 152153, viiix.
 Agrotis, M., Ercolani, N. M., Glasgow, S. A., & Moloney, J. V. (2000). Complete integrability of the reduced MaxwellBloch equations with permanent dipole. Physica D: Nonlinear Phenomena, 138(12), 134162.More infoAbstract: We obtain the Lax pair, hierarchy of commuting flows and Bäcklund transformations for a reduced MaxwellBloch (RMB) system. This system is of particular interest for the description of unipolar, nonoscillating electromagnetic solitons (also called "electromagnetic bubbles"). © 2000 Elsevier Science B.V. All rights reserved.
 Ercolani, N. M., Levermore, C. D., & Zhang, T. (1997). The behavior of the Weyl function in the zerodispersion KdV limit. Communications in Mathematical Physics, 183(1), 119143.More infoAbstract: The moment formulas that globally characterize the zerodispersion limit of the KortewegdeVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zerodispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem.
 Caflisch, R. E., Ercolani, N., & Steele, G. (1996). Geometry of singularities for the steady boussinesq equations. Selecta Mathematica, New Series, 2(3), 369414.More infoAbstract: Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for twodimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the CauchyKowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates. © 1996 Birkhäuser Verlag,.
 Calini, A., Ercolani, N. M., McLaughlin, D. W., & Schober, C. M. (1996). Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena, 89(34), 227260.More infoAbstract: Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrödinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the NLS. Using heuristic Mel'nikov methods, the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments. In a neighborhood of the persistent homoclinic orbits, the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study. © 1996 Elsevier Science B.V. All rights reserved.
 Miller, P. D., Ercolani, N. M., & Levermore, C. D. (1996). Modulation of multiphase waves in the presence of resonance. Physica D: Nonlinear Phenomena, 92(12), 127.More infoAbstract: The phenomenon of spatiotemporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the AblowitzLadik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments. © 1996 Elsevier Science B.V. All rights reserved.
 Senouf, D., Caflisch, R., & Ercolani, N. (1996). Pole dynamics and oscillations for the complex Burgers equation in the smalldispersion limit. Nonlinearity, 9(6), 16711702.More infoAbstract: A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the ColeHopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinitedimensional Calogerotype dynamical system. The initial data are provided by highorder asymptotic approximations of the poles at the critical time ts for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solution as the dispersion parameter ε → 0+. A uniform asymptotic expansion as ε → 0+ of the dispersive solution is derived in terms of the Pearcey integral in a neighbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representation of the solution to the inviscid Burgers equation. © 1996 IOP Publishing Ltd and LMS Publishing Ltd.
 Ercolani, N. M., Forest, M. G., McLaughlin, D. W., & Sinha, A. (1993). Strongly nonlinear modal equations for nearly integrable PDEs. Journal of Nonlinear Science, 3(1), 393426.More infoAbstract: The purpose of this paper is the derivation of reduced, finitedimensional dynamical systems that govern the nearintegrable modulations of Nphase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any Nphase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosophy with the damped, driven sineGordon system and summarize our present progress toward application of the modulation equations to this numerical study. © 1993 SpringerVerlag New York Inc.
 Ercolani, N. M., McLaughlin, D. W., & Roitner, H. (1993). Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis. Journal of Nonlinear Science, 3(1), 477539.More infoAbstract: In this paper we rigorously show the existence and smoothness in ε of traveling wave solutions to a periodic KortewegdeVries equation with a KuramotoSivashinskytype perturbation for sufficiently small values of the perturbation parameter ε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions. © 1993 SpringerVerlag New York Inc.
 Ercolani, N., & Montgomery, R. (1993). On the fluid approximation to a nonlinear Schrödinger equation. Physics Letters A, 180(6), 402408.More infoAbstract: We present a heuristic proof that the nonlinear Schrödinger equation (NLS)  iθ{symbol}Ψ θ{symbol}t= 1 2ΔΨ+ 1 2(1Ψ2)Ψ in 2 + 1 dimensions has a family of solutions which can be well approximated by a collection of point vortices for a planar incompressible fluid. The novelty of our approach is that we begin with a representation of the NLS as a compressible perturbation of Euler's equations for hydrodynamics. © 1993.
 Ercolani, N., & Sinha, A. (1991). On the construction of a hyperKähler metric on the moduli space of monopoles. Physics Letters A, 153(23), 8189.More infoAbstract: This note describes the explicit construction of the 4(k1)dimensional moduli space of kmonopoles in centerofmass coordinates. This space is known to have a natural hyperKähler structure. We show how the construction of this hyperKähler metric can be reduced to solving a gauge orthogonality condition. © 1991.
 Ercolani, N., & McKean, H. P. (1990). Geometry of KDV (4): Abel sums, Jacobi variety, and theta function in the scattering case. Inventiones Mathematicae, 99(1), 483544.
 Ercolani, N., Forest, M. G., & McLaughlin, D. W. (1990). Geometry of the modulational instability. III. Homoclinic orbits for the periodic sineGordon equation. Physica D: Nonlinear Phenomena, 43(23), 349384.More infoAbstract: In this paper the homoclinic geometric structure of the integrable sineGordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to Ntori. Simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given. A labeling is provided which identifies and catalogues all such orbits. These orbits are related in a onetoone manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Bäcklund transformations. © 1990.
 Ercolani, N., & Siggia, E. D. (1989). Painlevé property and geometry. Physica D: Nonlinear Phenomena, 34(3), 303346.More infoAbstract: The Painlevé property for discrete Hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times. The augmented manifold is constructed by expanding the HamiltonJacobi equation. A complete classification of the types of poles allowed in complex time is given for Hamiltonians which separate into the direct product of hyperelliptic curves. For such systems, bounds on the degrees of the (polynomial) separating variable change, and the other integrals in involution can be found from the pole series and the HamiltonJacobi equation. It is shown how branching can arise naturally in a Painlevé system. © 1989.
 Ercolani, N., & Sinha, A. (1989). Monopoles and Baker functions. Communications in Mathematical Physics, 125(3), 385416.More infoAbstract: The work in this paper pertains to the solutions of Nahm's equations, which arise in the AtiyahDrinfieldHitchinManinNahm construction of solutions to the Bogomol'nyi equations for static monopoles. This paper provides an explicit construction of the solution of Nahm's equations which satisfy regularity and reality conditions. The Lax form of Nahm's equations is reduced to a standard eigenvalue problem by a special gauge transformation. These equations may then be solved by the method of BakerKrichever. This leads to a compact representation of the solutions of Nahm's equations. The regularity condition is shown to be related to the monodromy of the gauge reduced linear operator. Hitchin showed that the solutions of Nahm's equations can be characterized by an algebraic curve and some data on that curve. Here, this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential. Donaldson coordinatized the moduli space of kmonopoles by a class of rational maps from the Riemann sphere to itself. The data of a Baker function is equivalent to this map. This method gives an "apriori" construction of the (known) two monopole solutions. We also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge. These solutions correspond to reducible curves with elliptic components and the associated Donaldson rational function has a simple partial fraction expansion. © 1989 SpringerVerlag.
 Ercolani, N., & Siggia, E. D. (1986). Painlevé property and integrability. Physics Letters A, 119(3), 112116.More infoAbstract: For an n degree of freedom hyperelliptic separable hamiltonian, the pole series with n+1 free constants, through the HamiltonJacobi equation, bounds the degrees of the npolynomials in involution. When all the pole series have no fewer than 2n constants, the phase space is conjectured to be just the direct product of 2n complex lines cut out by (2n1) integrals. © 1986.
 Ercolani, N., Forest, M. G., & McLaughlin, D. W. (1986). The origin and saturation of modulational instabilities. Physica D: Nonlinear Phenomena, 18(13), 472474.
 Ercolani, N. M., & Forest, M. (1985). The geometry of real sineGordon wavetrains. Communications in Mathematical Physics, 99(1), 149.More infoAbstract: The characterization of real, N phase, quasiperiodic solutions of the sineGordon equation has been an open problem. In this paper we achieve this result, employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature. A significant byproduct of this approach is a natural algebraic representation of the full complex isospectral manifolds, and an understanding of how the real isospectral manifolds are embedded. By placing the problem in this general context, these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions. © 1985 SpringerVerlag.
 Ercolani, N., Forest, M., & McLaughlin, D. W. (1984). MODULATIONAL STABILITY OF TWOPHASE SINEGORDON WAVETRAINS.. Studies in Applied Mathematics, 71(2), 91101.More infoAbstract: A modulational stability analysis is presented for real, twophase sineGordon wavetrains. Using recent results on the geometry of these real solutions, an invariant representation in terms of Abelian differentials is derived for the sineGordon modulation equations. The theory thus attains the same integrable features of the previously completed KdV and sineGordon modulations. The twophase results are as follows: kinkkink trains are stable, while the breather trains and kinkradiation trains are unstable, to modulations.
Presentations
 Ercolani, N. M. (2019, April). INTEGRABLE MAPPINGS FROM A UNIFIED PERSPECTIVE. IMACS 2019 Conference on Nonlinear Waves. Athens, Georgia: University of Georgia.
 Ercolani, N. M. (2019, November). Holomorphization in Field Theories on Riemann Surfaces. AMS Sectional Meeting; Special Session on Effective Equations of Quantum Physics. University of Florida Gainesville, FL: American Mathematical Society.
 Ercolani, N. M. (2019, November). Random Matrix Theory:A Bridge Between Classical Dynamics and Quantum Field Theory. Fields Institute Applied Mathematics Colloquium. Toronto, Canada: Fields Institute/University of Toronto.
 Ercolani, N. M. (2019, November). The Geometry of MongeAmpere Equations. Mathematics Colloquium. University of Central Florida: Department of Mathematics.
 Ercolani, N. M. (2017, December). Patterns, Defects and Phase Singularities. Applied Mathematics Colloquium (UC Boulder). Boulder, Coloradp: University of Colorado at Boulder.
 Ercolani, N. M. (2017, March). Ginzburg Landau Equations on Riemann Surfaces of Higher Genus. AMS Southeastern Sectional Meeting. Charleston, South Carolina: American Mathematical Society.
 Ercolani, N. M. (2017, March). PDE Models of GinzburgLandau Type for Defect Formation in PatternForming Systems. International Conference on Nonlinear Evolutin Equaitons and Wave Phenomena: Computation and Theory. IMACS Center, Athens, Georgia.
 Ercolani, N. M. (2017, November). Random Matrices and Random Surfaces. Talk to Undergraduate Majors Math Club (UNCO). Greely, Colorado: University of Northern Colorado.
 Ercolani, N. M. (2017, November/December). Statistical Mechanics of Anharmonic Nonlinear Lattices. Lecture Series (2 Talks) CSU. Fort Collins, Colorado: Colorado State University.
 Ercolani, N. M. (2017, October). Quantum Gravity and Quantum Groups. Analysis Seminar (U. Toronto). Toronto, Ontario, Canada: University of Toronto.
 Ercolani, N. M. (2017, Ovtober/November). Applications of Random Matrix Theory to Mathematical Physics. Lecture Series (3 Talks) CSU. Fort Collins, Colorado: Colorado State University.
 Ercolani, N. M. (2017, September). Discrete Surfaces and Quantum Gravity. AMS Western Sectional Meeting. Denton, Texas: American Mathematical Society.
 Ercolani, N. M. (2017, September). Statistical Mechanics of Integrable Nonlinear Lattices. AMS Westrn Sectional Meeting. Denton, Texas: American Mathematical Society.
 Ercolani, N. M. (2017, September). Statistical Mechanics of Integrable Nonlinear Lattices. Applied Mathematics Colloquium (SMU). Dallas, Texas: Southern Methodist University.
 Ercolani, N. M. (2017, September/October). Pattern Formation. Lecture Series (3 Talks) CSU. Fort Collins, Colorado: Colorado State University.
 Ercolani, N. M. (2016, April). Backlund Transformations, Old and New. Mathematics Colloquium. Temple University: Mathematics Department.
 Ercolani, N. M., & Brown, T. (2016, September). Combinatorics, Dynamics and Integrability. Mathematics Colloquium. University of Illinois at UrbanaChampaign: Mathematics Department.
 Ercolani, N. M., Ercolani, N. M., Brown, T., & Brown, T. (2016, November). Combinatorics, Dynamics and Integrability. Mathematics Colloquium. University of Toronto: Department of Mathematics.
 Ercolani, N. M., Ercolani, N. M., Brown, T., & Brown, T. (2016, October). Combinatorics, Dynamics and Integrability. AMS Western Sectional Meeting. Denver, CO: American Mathematical Society.
 Ercolani, N. M. (2015, June). The Propagation of Oscillations: Some New/Old Perspectiives. International Conference on Mathematics of Nonlinearity in Neural and Physical Sciences. Shanghai China: NYU Shanghai.
 Ercolani, N. M. (2015, Spring). Quantum Gravity and Quantum Groups. The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory. Athens Georgia.
 Ercolani, N. M., Kamburov, N. A., & Lega, J. C. (2015, December). How Defects are Born. SIAM conference on Analysis of Partial Differential Equations. Scotssdale, AZ.More infoPatternforming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripeforming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a longterm project whose goal is to follow such a defectforming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the SwiftHohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the CrossNewell equation, develops large derivatives in a region of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
 Ercolani, N. M., Kamburov, N. A., & Lega, J. C. (2015, December). How Defects are Born. SIAM conference on Analysis of Partial Differential Equations. Scottsdale, AZ.More infoPatternforming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripeforming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a longterm project whose goal is to follow such a defectforming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the SwiftHohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the CrossNewell equation, develops large derivatives in a region of diminishing size.This is joint work with N. Ercolani and N. Kamburov.
 Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2015, April). How Defects are Born. The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory.More infoPatternforming systems typically exhibit defects, whose nature is associated with the symmetries of the pattern in which they appear; examples include dislocations of stripe patterns in systems invariant under translations, disclinations in stripeforming systems invariant under rotations, and spiral defects of oscillatory patterns in systems invariant under time translations. Numerical simulations suggest that pairs of defects are created when the phase of the pattern ceases to be slaved to its amplitude. Such an event is typically mediated by the build up of large, localized, phase gradients.This talk will describe recent advances on a longterm project whose goal is to follow such a defectforming mechanism in a system that is amenable to analysis. Specifically, we focus on the appearance of pairs of dislocations at the core of a grain boundary of the SwiftHohenberg equation. Taking advantage of the variational nature of this system, we show that as the angle between the two stripe patterns on each side of the grain boundary is reduced, the phase of each pattern, as described by the CrossNewell equation, develops large derivatives in a region of diminishing size.
 Ercolani, N. M. (2014, April). Random Partitions in Statistical Mechanics. Special Session on Nonlinear Waves and Singularities. University of New Mexico, Albuquerque, New Mexico: American Mathematical Society (Western Sectional Meeting).
 Ercolani, N. M. (2014, February). Lindelof Integrals for Combinatorics I & II. Analysis & its Applications Seminar. University of Arizona: Applied Mathematics Graduate Program.
 Ercolani, N. M. (2014, February). Random Partitions in Statistical Mechanics I & II. Mathemtical Physics & Probability Seminar. University of Arizona: Department of Mathematics.
 Ercolani, N. M. (2014, November). Dyson Brownian Motion and Interlacing Processes. Mathemtical Physics & Probability Seminar. University of Arizona: Department of Mathematics.
 Ercolani, N. M. (2014, September). Nonlinear PDE & Random Matrices I & II. Analysis & its Applications Seminar. University of Arizona: Applied Mathematics Graduate Program.
 Lega, J. C., Ercolani, N. M., & Kamburov, N. A. (2014, July). Grain boundaries of the SwiftHohenberg and regularized CrossNewell equations. Special session on Traveling Waves and Patterns, 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications. Madrid, Spain.More infoGrain boundaries in extended twodimensional pattern forming systems are curves separating regions of slanted rolls. When the angle between the rolls in each of the two regions exceeds a certain threshold, it is known [1,2] that the core of the grain boundary transforms into a chain of convexconcave disclinations. Even though the regularized CrossNewell (RCN) phase diffusion equation cannot describe this transition all the way to the appearance of defects, it can nevertheless be used to address the question of whether the transition results from an instability of the grain boundary core, and if so, to describe this instability. To this end, we will take full advantage of the existence of an exact grainboundary solution of RCN and of the variational nature of this equation. I will also show numerical simulations and connect our results to those of Haragus and Scheel [3] on grain boundaries of the SwiftHohenberg equation.References[1] N.M. Ercolani, R. Indik, A.C. Newell, and T. Passot, J. Nonlinear Sci. 10, 223274 (2000).[2] N.M. Ercolani and S.C. Venkataramani, J. Nonlinear Sci. 19, 267300 (2009).[3] M. Haragus and A. Scheel, European Journal of Applied Mathematics 23, 737759 (2012).
 Ercolani, N. M., & Brown, T. (2016, November). Combinatorics, Dynamics and Integrability. Mathematics Colloquium. University of Toronto: Department of Mathematics.
 Ercolani, N. M., & Brown, T. (2016, October). Combinatorics, Dynamics and Integrability. AMS Western Sectional Meeting. Denver, CO: American Mathematical Society.