High order perturbation theory for nonlinear Anderson model

The perturbation expansion for the nonlinear Schrödinger equation with a random potential that was developed in earlier works by some of us is extended to higher orders. As the order is increased a solution that is valid for longer time can be found. In particular it is found that Anderson localization persists in the fifth and sixth orders for times when perturbation theory is valid. The perturbation expansion is asymptotic and for the value of the nonlinearity parameter used, the fifth order is the optimal order of the perturbation theory. There are indications that for the sixth order perturbation theory may not be valid. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014