# On the radially symmetric traveling waves for the Schr{\"o}dinger equation on the Heisenberg group

## TL;DR

This paper studies radial solutions to a cubic Schr{"o}dinger equation on the Heisenberg group, demonstrating the existence, uniqueness, and stability of traveling wave solutions with speeds close to the maximum value.

## Contribution

It establishes the existence, uniqueness, and smoothness of ground state traveling waves for the Schr{"o}dinger equation on the Heisenberg group, including their stability analysis as speed approaches a critical limit.

## Key findings

- Existence of ground state traveling waves for speeds in (-1,1)
- Uniqueness of these waves near speed 1 up to symmetries
- Linear stability of the limiting wave as speed approaches 1

## Abstract

We consider radial solutions to the cubic Schr{\"o}dinger equation on the Heisenberg group$$i\partial_t u - \Delta_{\mathbb{H}^1} u = |u|^2u, \quad\Delta_{\mathbb{H}^1} = \frac{1}{4}(\partial_x^2+\partial_y^2) + (x^2+y^2)\partial_s^2, \quad(t,x,y,s) \in \mathbb{R}\times\mathbb{H}^1.$$This equation is a model for totally non-dispersive evolution equations. We show existence of ground state traveling waves with speed $\beta \in (-1,1)$. When the speed $\beta$ is sufficiently close to $1$, we prove their uniqueness up to symmetries and their smoothness along the parameter $\beta$. The main ingredient is the emergence of a limiting system as $\beta$ tends to the limit $1$, for which we establish linear stability of the ground state traveling wave.