Equivariant Solutions to a System of Nonlinear Wave Equations with Ginzburg-Landau Type Potential

TL;DR

This paper constructs solutions to a coupled nonlinear wave system with an interface, where one component forms a time-like surface and the other exhibits exponential decay, revealing complex geometric and phase interactions.

Contribution

It introduces a novel class of solutions with interfaces for a two-component nonlinear wave system, linking interface geometry to phase dynamics in Minkowski space.

Findings

Existence of solutions with interface and phase coupling.

Profiles determined by winding number density.

Interface evolution coupled to phase in Minkowski space.

Abstract

It is known that there exist solutions with interfaces to various scalar nonlinear wave equations. In this paper, we look for solutions of a two-component system of nonlinear wave equations where one of the components has an interface and and where the second component is exponentially small except near the interface of the first component. A formal asymptotic expansion suggests that there exist solutions to this system with these characteristics whose profiles are determined by the winding number density of the second component and where the interface of the first component is a time-like surface in Minkowski space whose geometric evolution is coupled in a highly nonlinear way to the phase of the second component. We verify this heuristic when $n=2$ and for equivariant maps.