# Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

## TL;DR

This paper studies traveling wave solutions for a class of nonlocal 1D Gross-Pitaevskii equations with nonzero boundary conditions, establishing existence, stability, and conditions for global solutions.

## Contribution

It introduces new conditions on nonlocal interactions that guarantee traveling wave solutions and their orbital stability, extending known results for contact interactions.

## Key findings

- Existence of traveling wave solutions under specific nonlocal interaction conditions
- Orbital stability of these traveling waves
- A simple criterion for global-in-time solutions to the Cauchy problem

## Abstract

We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.