On the construction of orbits homoclinic to plane waves in integrable coupled nonlinear Schrödinger systems

Abstract

Plane waves of coupled NLS PDEs are known to possess self-focusing and cross-coupling instabilities. In the integrable limit of coupled NLS systems, these instabilities are accompanied by homoclinic orbits which asymptotically (in time) approach the unstable plane wave. Here we use the integrable method of Bäcklund transformations to construct a two-parameter family of exact spatially periodic CNLS solutions, which consist of a plane wave connected to a nonlinear superposition of self-focusing and cross-coupling homoclinic orbits.