Vol. 2, No. 4, 2020

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Radially symmetric traveling waves for the Schrödinger equation on the Heisenberg group

Louise Gassot

Vol. 2 (2020), No. 4, 739–794
Abstract

We consider radial solutions to the cubic Schrödinger equation on the Heisenberg group

itu Δ1u = u2u,Δ1 = 1 4(x2 + y2) + (x2 + y2) s2,(t,x,y,s) × 1.

This equation is a model for totally nondispersive evolution equations. We show existence of ground state traveling waves with speed β (1,1). When the speed β is sufficiently close to 1, we prove their uniqueness up to symmetries and their smoothness along the parameter β. The main ingredient is the emergence of a limiting system as β tends to the limit 1, for which we establish linear stability of the ground state traveling wave.

Keywords
nonlinear Schrödinger equation, traveling wave, orbital stability, Heisenberg group, dispersionless equation, Bergman kernel
Mathematical Subject Classification 2010
Primary: 35B35, 35C07, 35Q55, 43A80
Milestones
Received: 26 April 2019
Revised: 8 April 2020
Accepted: 28 September 2020
Published: 25 February 2021
Authors
Louise Gassot
Université Paris-Saclay, CNRS
Laboratoire de mathématiques d’Orsay
France