Interacting Solitary Waves in Nonlinear Wave Equations

Introduction

Various models encountered in mathematical physics possess special solutions called solitary waves, which preserve their shape as time passes. In the case of dispersive models, small perturbations of the field tend to spread, so that their amplitude decays. The Soliton Resolution Conjecture predicts that, generically, a solution of a nonlinear dispersive partial differential equation decomposes into a superposition of solitary waves and a perturbation of small amplitude called radiation.

Focus of Study

Our study will focus on topological solitons appearing in models motivated by Quantum Field Theory:

1. Kinks in the phi4 theory
2. Rational maps in the O(3) sigma model

We expect that the developed techniques will have applications in the study of other topological solitons like vortices, monopoles, Skyrmions, and instantons.

Objectives

Our general ultimate objective goes beyond the Soliton Resolution and consists in obtaining an asymptotic description in infinite time, in both time directions, of solutions of the considered model. Such a description should be correct at least at the main order and reflect interesting features of the problem, which are the soliton-soliton interactions and soliton-radiation interactions.

Concrete Situations

We pursue this general goal in various concrete situations, namely:

1. The problem of unique continuation after blow-up for the equivariant wave maps equation
2. The collision problem for the phi4 equation
3. The study of pure multi-solitons in the regime of strong interaction
4. The multi-soliton uniqueness and stability problem

Their solution requires a mixture of non-perturbative and perturbative techniques. While the former rely heavily on the concrete model, the latter will be applicable to any dispersive equation having solitary waves.