Perturbation theory for nonlinear Schrodinger equations

TL;DR

This paper develops a perturbation theory for the nonlinear Schrödinger equation, proving convergence of the power series solution when the nonlinear term's strength is below a certain threshold.

Contribution

It introduces a convergent Rayleigh-Schrodinger perturbation series for the nonlinear Schrödinger equation with a nonlinear term treated as a perturbation.

Findings

Power series converges for nonlinear strength below a threshold.

Provides stationary solutions to the nonlinear Schrödinger equation.

Establishes conditions for convergence of the perturbation series.

Abstract

Treating the nonlinear term of the Gross-Pitaevskii nonlinear Schrodinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh-Schrodinger power series. This power series is proved to be convergent when the parameter representing the intensity of the nonlinear term is less in absolute value than a threshold value, and it gives a stationary solution to the nonlinear Schrodinger equation.