Interacting Solitary Waves in Nonlinear Wave Equations

Various models encountered in mathematical physics possess special solutions called solitary waves, which preserve their shape astime passes. In the case of dispersive models, small perturbations of the field tend to spread, so that their amplitude decays.

TheSoliton Resolution Conjecture predicts that, generically, a solution of a nonlinear dispersive partial differential equation decomposesinto a superposition of solitary waves and a perturbation of small amplitude called radiation.Our study will focus on topological solitons appearing in models motivated by Quantum Field Theory: kinks in the phi4 theory andrational maps in the O(3) sigma model.

We expect that the developed techniques will have applications in the study of othertopological solitons like vortices, monopoles, Skyrmions and instantons.Our general ultimate objective goes beyond the Soliton Resolution, and consists in obtaining an asymptotic description in infinitetime, in both time directions, of solutions of the considered model.

Such a description should be correct at least at main order, andreflect interesting features of the problem, which are the soliton-soliton interactions and soliton-radiation interactions.We pursue this general goal in various concrete situations, namely: the problem of unique continuation after blow-up for theequivariant wave maps equation, the collision problem for the phi4 equation, the study of pure multi-solitons in the regime of stronginteraction, and the multi-soliton uniqueness and stability problem.

Their solution requires a mixture of non-perturbative and perturbativetechniques.

While the former rely heavily on the concrete model, the latter will be applicable to any dispersive equation havingsolitary waves.