#### 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

$i{\partial }_{t}u-{\Delta }_{{ℍ}^{1}}u={\left|u\right|}^{2}u,\phantom{\rule{1em}{0ex}}{\Delta }_{{ℍ}^{1}}=\frac{1}{4}\left({\partial }_{x}^{2}+{\partial }_{y}^{2}\right)+\left({x}^{2}+{y}^{2}\right){\partial }_{s}^{2},\phantom{\rule{1em}{0ex}}\left(t,x,y,s\right)\in ℝ×{ℍ}^{1}.$

This equation is a model for totally nondispersive evolution equations. We show existence of ground state traveling waves with speed $\beta \in \left(-1,1\right)$. 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.

##### Keywords
nonlinear Schrödinger equation, traveling wave, orbital stability, Heisenberg group, dispersionless equation, Bergman kernel
##### Mathematical Subject Classification 2010
Primary: 35B35, 35C07, 35Q55, 43A80