On the radially symmetric traveling waves for the Schr{ö}dinger equation on the Heisenberg group

Abstract

We consider radial solutions to the cubic Schr{ö}dinger equation on the Heisenberg group$$i{\partial }_{t}u-{\Delta }_{{\mathbb{H}}^1}u=|u{|}^{2}u,{\Delta }_{{\mathbb{H}}^1}=\frac{1}{4}({\partial }_x^2+{\partial }_y^2)+(x^2+y^2){\partial }_s^2,(t,x,y,s)\in \mathbb{R}\times {\mathbb{H}}^1.$$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.