Traveling waves for nonlinear Schr\"odinger equations with nonzero conditions at infinity

TL;DR

This paper proves the existence of finite energy traveling wave solutions for a broad class of nonlinear Schrödinger equations with nonzero conditions at infinity, valid in multiple dimensions and for various nonlinearities.

Contribution

It establishes the existence of traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity across different dimensions and nonlinearities, including Gross-Pitaevskii and cubic-quintic cases.

Findings

Existence of traveling waves for speeds less than sound velocity

Results valid in dimensions N ≥ 3

Applicable to Gross-Pitaevskii and cubic-quintic nonlinearities

Abstract

For a large class of nonlinear Schr\"odinger equations with nonzero conditions at infinity and for any speed $c$ less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed $c$ in any space dimension $N\geq 3$. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity.