Long time asymptotics of Soliton equations

Completely integrable wave equations, also known as soliton equations, are an important topic in physics used to explain many physical effects. More than 200 years after their discovery by John Scott Russell, solitons are still a highly active area in both physics and mathematics producing new and fascinating phenomena. The aim of this project is to study the stability of soliton equations under small perturbations of the initial data and to derive the long time behaviour of the solution. The solution eventually decomposes into two components: a soliton train (corresponding to the eigenvalues of the underlying Lax operator) and a "background radiation" component (corresponding to the continuous spectrum of the Lax operator). In particular, the solitons constitue the stable part of the solution. The goal of this project is to give a rigorous mathematical description of the complete picture and in particular, to understand and explain what happens for "steplike" initial data when the perturbation is different on the two sides. I plan to investigate this situation for the following two models: the Toda lattice system, which is one of the most prominent Soliton equations; and the Ablowitz-Ladik system, a discretized version of the nonlinear Schrödinger equation. The nonlinear Schrödinger equation describes the evolution of small amplitude, slowly varying wave packets in nonlinear media and has been derived in such diverse fields as deep water waves, plasma physics, and nonlinear optical fibers. The methods and ideas developed for the model systems will be applicable to all soliton equations in one space dimension.

Completely integrable wave equations, or soliton equations, are a highly active area in both physics and mathematics explaining new and fascinating phenomena. The goal of this project was to study the stability of soliton equations under small perturbations of the initial data and to understand the behavior of the solution for large times. Our main results concern the most prominent (discrete-space, continuous-time) soliton equation, the Toda equation, which describes the dynamics of an infinite particle chain with nonlinear nearest neighbor interactions. We wanted to understand what happens when the chain is subjected to shock or rarefaction type initial conditions, that is, for steplike constant initial data. In this case, the continuous spectrum of the underlying Lax operator consists of two intervals which might overlap and their mutual location produces essentially different types of asymptotic solutions. These wave phenomena, also known as Toda shock and Toda rarefaction problem, were first discovered numerically, a rigorous investigation of the limiting behavior as time goes to infinity had been carried out so far only for special initial values. For the Toda shock problem we showed that the half-plane of spaceime variables splits into five main regions divided by critical values which we computed explicitly: the two soli- ton regions far outside where the solution is close to the unperturbed background solution plus a number of solitons (corresponding to the eigenvalues of the Lax operator); the middle region, where the solution can be asymptotically described by a two band Toda solution (corresponding to the two intervals of the Lax operator), and two regions separating them, where the solution is asymptotically given by a slowly modulated two band Toda solution (corresponding to a gradual shortening of one of the intervals plus the full second interval). In particular, the form of this solution in the separating regions verifies a conjecture by Venakides, Deift, and Oba from 1991. For the Toda rarefaction problem we showed that the half-plane splits into four main regions: two soliton regions far outside as before and two regions where the asymptotic solution is given by a slope (corresponding to the gradual shortening of one interval). We give a rigorous mathematical description of all regions involved for both shock and rarefaction type initial conditions. The Toda equation appears in such diverse fields as nonlinear optical fibers and plasma physics. The methods and ideas developed will be adaptable to other soliton equations in one space dimension, in particular, we applied them to the Korteweg-de Vries equation. Funded researchers: Johanna Michor (Uni Wien), Iryna Egorova (ILT, Kharkov)