Existence and asymptotic stability of quasi-periodic solution of discrete NLS with potential in $mathbb{Z}$

We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schrodinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and dispersive wave.

On weak interaction between a ground state and a trapping potential

We study the interaction of a ground state with a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.