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

TL;DR

This paper establishes the existence and stability of traveling wave solutions for a broad class of nonlinear Schrödinger equations with nonzero boundary conditions at infinity, including the Gross-Pitaevskii and cubic-quintic models.

Contribution

It introduces new variational methods for constructing stable traveling waves and compares different minimization approaches, also proving nonexistence for low-energy waves.

Findings

Existence of nontrivial finite energy traveling waves in dimensions N ≥ 2.

Stable traveling waves obtained via energy minimization at fixed momentum.

Nonexistence of small-energy traveling waves.

Abstract

We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schr\"odinger equations with nonzero conditions at infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic" equations) in space dimension $ N \geq 2$. We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy.