On the problem of dynamical localization in the nonlinear Schrödinger equation with a random potential (Q937090)

The authors consider the problem of a dynamical localization of waves in a nonlinear Schrödinger equation (NLSE) with a random potential term \[ i\partial_t\psi=(-\partial_{xx} +V_\omega)\psi+| \psi| ^2 \psi , \] where \(\psi=\psi(x,t),x \in Z \) and \(\{V_\omega\}_{\omega\in\Omega}\) is a collection of random potentials chosen from the set \(\Omega\) ,with probability measure \(\mu(\omega)\). It is well known that NLSE has stationary solutions \[ E\psi_E=(-\partial_{xx} +V_\omega)\psi_E+| \psi_e| ^2 \psi_E , \] which are exponentially localized for almost all E with a localization length. The authors prove that for times of order \(O(\beta^{-2})\) the solution remains exponentially localized.