On the orbital stability of a family of traveling waves for the cubic Schr{\"o}dinger equation on the Heisenberg group

TL;DR

This paper investigates the orbital stability of a family of traveling wave solutions for the focusing energy-critical Schr{"o}dinger equation on the Heisenberg group, demonstrating stability near speed 1 for radial initial data.

Contribution

It establishes the orbital stability of traveling waves close to speed 1 for the Schr{"o}dinger equation on the Heisenberg group, a novel result in this non-dispersive setting.

Findings

Traveling waves with speed close to 1 are orbitally stable.

Radial initial data near a traveling wave leads to solutions remaining close to the wave's orbit.

Stability results also apply to the associated limiting system.

Abstract

We consider the focusing energy-critical Schr{\"o}dinger equation on the Heisenberg group in the radial case\[i\partial_t u-\Delta_{\mathbb{H}^1} u=|u|^2u,\quad\Delta_{\mathbb{H}^1}=\frac{1}{4}(\partial_x^2+\partial_y^2)+(x^2+y^2)\partial_s^2,\quad(t,x,y,s)\in \mathbb{R}\times\mathbb{H}^1,\]which is a model for non-dispersive evolution equations. For this equation, existence of smooth global solutions and uniqueness of weak solutions in the energy space are open problems. We are interested in a family of ground state traveling waves parametrized by their speed $\beta \in (-1,1)$. We show that the traveling waves of speed close to $1$ present some orbital stability in the following sense. If the initial data is radial and close enough to one traveling wave, then there exists a global weak solution which stays close to the orbit of this traveling wave for all times. A similar result is proven for the limiting system associated to this equation.