On the orbital stability of a family of traveling waves for the cubic Schr{\"o}dinger equation on the Heisenberg group

We consider the focusing energy-critical Schr{"o}dinger equation on the Heisenberg group in the radial case$$i{\mathrm{\partial }}_{t}u-{\mathrm{\Delta }}_{{\mathbb{H}}^1}u=\mathrm{\mid }u{\mathrm{\mid }}^{2}u,{\mathrm{\Delta }}_{{\mathbb{H}}^1}=\frac{1}{4}({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)+(x^2+y^2){\mathrm{\partial }}_s^2,(t,x,y,s)\in \mathbb{R}\times {\mathbb{H}}^1,$$which is a model for non-dispersive evolution equations. For this equation, existence of smooth global solutions and uniqueness of weak solutions in the energy space are open problems. We are interested in a family of ground state traveling waves parametrized by their speed $\beta \in (-1,1)$. We show that the traveling waves of speed close to $1$ present some orbital stability in the following sense. If the initial data is radial and close enough to one traveling wave, then there exists a global weak solution which stays close to the orbit of this traveling wave for all times. A similar result is proven for the limiting system associated to this equation.