Nicholas M. Ercolani

Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.

Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation

In this paper the homoclinic geometric structure of the integrable sine-Gordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to N-tori. 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 one-to-one manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Backlund transformations.

Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

Abstract We give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e −n(V(x)+W(y)−2τxy) d x d y 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 non-local Riemann–Hilbert 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 multi-matrix models. We show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of V and W coincide with a semi-infinite generalization of the completely integrable full Kostant–Toda lattice. This connection could be relevant for understanding aspects of scaling limits for the multi-matrix model.

Complete integrability of the reduced Maxwell—Bloch equations with permanent dipole

Abstract We obtain the Lax pair, hierarchy of commuting flows and Backlund transformations for a reduced Maxwell–Bloch (RMB) system. This system is of particular interest for the description of unipolar, nonoscillating electromagnetic solitons (also called “electromagnetic bubbles” ).

The geometry of real sine-Gordon wavetrains

The characterization ofreal, N phase, quasiperiodic solutions of the sine-Gordon 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 by-product of this approach is a naturalalgebraic 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.

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

SummaryIn this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. 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.

Cycle structure of random permutations with cycle weights

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.

Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns

Abstract We show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns. Further, by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing Gaussian curvature (in 3D, all sectional curvatures) excetp at points, we are able to derive explicit expressions which capture the anatomies of point and line (and surface) defects in two and three dimensional patterns, together with their topological characters and energetic constraints.

Painlevé property and geometry

Abstract The Painleve 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 Hamilton-Jacobi 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 Hamilton-Jacobi equation. It is shown how branching can arise naturally in a Painleve system.

Toward a Topological Classification of Integrable PDE’s

We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both of its arguments provides a transparent realization of a Bott function and of the remaining building blocks of the stratification; in this manner, Fomenko’s structure theorems are expressed through the inverse spectral transform. Thus, soliton equations are shown to provide natural representatives of the classification in the context of PDE’s.