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

Abstract

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

$i{\partial }_{t}u-{\mathrm{\Delta }}_{{ℍ}^1}u={\left|u\right|}^{2}u,{\mathrm{\Delta }}_{{ℍ}^1}=\frac{1}{4}\left({\partial }_x^2+{\partial }_y^2\right)+\left(x^2+y^2\right){\partial }_s^2,\left(t,x,y,s\right)\in ℝ\times {ℍ}^1.$

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.