Perturbation theory for the Nonlinear Schroedinger Equation with a random potential

Shmuel Fishman, Yevgeny Krivolapov, Avy Soffer

Published 2009-01-30, updated 2009-09-07Version 2

A perturbation theory for the Nonlinear Schroedinger Equation (NLSE) in 1D on a lattice was developed. The small parameter is the strength of the nonlinearity. For this purpose secular terms were removed and a probabilistic bound on small denominators was developed. It was shown that the number of terms grows exponentially with the order. The results of the perturbation theory are compared with numerical calculations. An estimate on the remainder is obtained and it is demonstrated that the series is asymptotic.