A uniqueness result for the two-vortex traveling wave in the nonlinear Schrödinger equation

Chiron, David; Pacherie, Eliot

doi:10.2140/apde.2023.16.2173

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

A uniqueness result for the two-vortex traveling wave in the nonlinear Schrödinger equation

David Chiron and Eliot Pacherie

Vol. 16 (2023), No. 9, 2173–2224

DOI: 10.2140/apde.2023.16.2173

Abstract

For the nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel, Gravejat and Saut (2009) (see also work of Chiron and Mariş (2017)). This minimizer is a traveling wave for the nonlinear Schrödinger equation. For large momenta, the propagation speed is small and the minimizer behaves like two well-separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this traveling wave. This work is a follow up to two previous papers, where we constructed and studied a particular traveling wave of the equation. We show a uniqueness result on this traveling wave in a class of functions that contains in particular all possible minimizers of the energy.

Keywords

Gross–Pitaevskii, uniqueness, traveling waves, vortices

Mathematical Subject Classification

Primary: 35A02, 35A15, 35B35, 35C07, 35Q56

Milestones

Received: 16 September 2021

Revised: 26 February 2022

Accepted: 9 April 2022

Published: 11 November 2023

Authors

David Chiron
Université Côte d’Azur, CNRS, LJAD Nice France

Eliot Pacherie
NYUAD Research Institute New York University Abu Dhabi United Arab Emirates

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 16, No. 9, 2023