Perturbation theory for the Nonlinear Schroedinger Equation with a random potential

TL;DR

This paper develops a perturbation theory for the 1D nonlinear Schrödinger equation with a random potential, analyzing its asymptotic behavior and comparing theoretical predictions with numerical results.

Contribution

It introduces a probabilistic approach to handle small denominators and demonstrates the asymptotic nature of the perturbation series for the NLSE.

Findings

Series grows exponentially with order

Perturbation series is asymptotic

Theoretical results align with numerical simulations

Abstract

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.