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 Schr\"odinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and dispersive wave. 
Masaya Maeda
math.AP