  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.
