We consider radial solutions to the cubic Schrödinger equation on the Heisenberg
group
$i{\partial}_{t}u{\Delta}_{{\mathbb{H}}^{1}}u={\leftu\right}^{2}u,\phantom{\rule{1em}{0ex}}{\Delta}_{{\mathbb{H}}^{1}}=\frac{1}{4}\left({\partial}_{x}^{2}+{\partial}_{y}^{2}\right)+\left({x}^{2}+{y}^{2}\right){\partial}_{s}^{2},\phantom{\rule{1em}{0ex}}\left(t,x,y,s\right)\in \mathbb{R}\times {\mathbb{H}}^{1}.$
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.
