Coercivity for travelling waves in the Gross-Pitaevskii equation in $\mathbb{R}^2$ for small speed

TL;DR

This paper analyzes the stability and uniqueness of travelling wave solutions in the 2D Gross-Pitaevskii equation, establishing coercivity, kernel characterization, and spectral stability for small wave speeds.

Contribution

It provides the first coercivity results and kernel characterization for these travelling waves, advancing their classification and multi-wave construction.

Findings

Kernel of the linearized operator identified

Spectral stability established

Non-degeneracy of travelling waves proven

Abstract

In the previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In particular, our result proves the non degeneracy of these travelling waves, which is a key step in the classification of these waves and for the construction of multi-travelling waves.