Solitary pulse solutions of a coupled nonlinear Schr\"{o}dinger system arising in optics

TL;DR

This paper proves the existence of traveling-wave solutions in a coupled nonlinear Schrödinger system modeling second-harmonic generation in nonlinear optics, using variational methods and concentration compactness.

Contribution

It introduces a rigorous proof of traveling-wave solutions for a coupled Schrödinger system relevant to nonlinear optics, employing the concentration compactness method.

Findings

Existence of traveling-wave solutions established.

Application of concentration compactness method.

Relevance to second-harmonic generation in optics.

Abstract

We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schr\"{o}dinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $\chi^{(2)}$ nonlinearity. To prove the existence of travelling waves, we employ the method of concentration compactness to prove the relative compactness of minimizing sequences of the associated variational problem.