The solution set of most nonlinear differential equations has little or no structure. In contrast, the class of systems said to be integrable enjoy interesting properties. For example, they admit special symmetries and a class of solutions known as solitons. Integrable systems appear in several fields of physics such as optics, elasticity, and fluid mechanics. Our session is going to cover both theoretical and applied aspects of the subject.
Speakers are named first.
8:45-9:05 Anjan Biswas, Tennessee State University
Quasi-stationary optical solitons with dual-power law nonlinearity
The multiple-scale perturbation analysis is used to study the perturbed nonlinear Schrödinger equation, with dual-power law non-linearity, that governs the propagation of solitons through an optical fiber. The perturbations due to the nonlinear damping and saturable amplifiers are considered. The WKB type ansatz of the perturbed soliton is introduced that captures the corrections ot the pulse where the standard soliton perturbations theory fails. The numerical results support the analysis.
9:10-9:30 Mitch Rothstein, University of Georgia
The Krichever Construction for ruled surfaces over hyperelliptic curves
It is well-known that the algebro-geometric solutions to the K-dV equation can be understood as the linear flow of a family of embeddings of the ring of functions on an affine curve into the ring of differential operators in one variable, parametrized by a dense open subset of the jacobian of the completed curve. A similar situation occurs when the curve is replaced by a variety X of higher dimension, now with matrix-valued differential operators in dim(X)-many variables. When X is a ruled surface over a curve C, the construction can be described in terms of objects living entirely on C. I will indicate how this goes when C is a hyperelliptic curve.
9:35-9:55 Plamen Iliev, Georgia Tech
The heat kernel and the Korteweg-de Vries hierarchy
The fundamental solution of the heat equation on the real line has been linked with soliton theory from early days, by providing a tool for obtaining the integrals of the motion of the KdV equation. In this talk I will explain how one can go in the opposite direction and compute Hadamard's coefficients using the τ-function of the KdV hierarchy. Basic properties of these coefficients can be easily derived from this approach.
As another application of the explicit formula, we prove that the heat expansion is finite if and only if the potential is a rational solution of the KdV hierarchy decaying at infinity studied by Airault-McKean-Moser and Adler-Moser. Equivalently, one can characterize the corresponding operators as the rank one bispectral family in the paper by Duistermaat and Grünbaum.
Late change: 10:00-10:15 Russell L. Herman, University of North Carolina-Wilmington
Revisiting Quasistationary Perturbation Theory for Equations in 1+1 Dimensions
We revisit quasistationary perturbation theory for integrable systems and compare the solutions to those obtained through eigenfunction expansion methods. We will focus our comparisons to the perturbed nonlinear equations such as the Korteweg de Vries, the nonlinear Schrodinger and the sine-Gordon equations.
2:00-2:20 Dmitry Pelinovsky, McMaster University
Evans function for Lax operators with algebraically decaying potentials
We study structural instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop analysis of the Evans function that define eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singularities in the essential spectrum, which may include embedded eigenvalues with algebraically decaying eigenfunctions. We construct a renormalized Evans function and study bifurcations of embedded eigenvalues, when an algebraically decaying potential is perturbed by a generic potential with the same or faster decay at infinity. We show that the bifurcation problem for embedded eigenvalues can be reduced to cubic or quadratic equations, depending on whether the algebraic potential decays to zero or approaches a nonzero constant. Roots of the bifurcation equations define eigenvalues which correspond to nonlinear waves that are formed from structurally unstable algebraic solitons.
2:25-2:45 Alexander Tovbis, University of Central Florida
On semiclassical limit solutions to the focusing nonlinear Schrödinger
equation: Riemann-Hilbert problem approach
We will discuss our recent results (joint work with Venakides and Zhou) on semiclassical (zero dispersion) limit of solutions to a focusing NLS for a special one-parameter family of initial data, which contain both pure radiation and radiation with solitons cases. We present explicit expressions for leading order asymptotic terms and give error estimates. We also discuss different asymptotic regimes and transitions between them (breaks). Our analysis is based on the method of Riemann-Hilbert problem.
2:50-3:10 Stephanos Venakides, Duke University
Rigorous steepest descent methods for nonlinear systems in the semiclassical limit
We will review and outline the extension of the original Deift-Zhou steepest descent method to the fully nonlinear oscillatory regime of semiclassical systems (joint work with Deift and Zhou), that has led to breakthroughs not only in integrable systems but also in the theory of orthogonal polynomials and random matrices. We will then discuss recent asymptotic results in the focusing NLS equation (joint work with Tovbis and Zhou).
3:15-3:35 Roberto Camassa, UNC Chapel Hill
On a Completely Integrable Numerical Scheme for a Nonlinear
Shallow-Water Wave Equation
An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N2) to O(N), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, if time permits, results on the study of the nonlinear equation posed in the quarter (space-time) plane will be presented.