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

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.