Some properties of small perturbations against a stationary solution of the nonlinear Schrodinger equation

TL;DR

This paper analyzes small perturbations around stationary solutions of the nonlinear Schrödinger equation, emphasizing the need for nonlinear equations of motion to correctly preserve conserved quantities and demonstrating additivity of integrals of motion for different modes.

Contribution

It introduces the necessity of nonlinear equations of motion for perturbations to accurately maintain conserved quantities in the nonlinear Schrödinger equation.

Findings

Nonlinear equations of motion are required for correct conserved quantities.

Additivity of integrals of motion holds for different modes up to second order.

Perturbations can be nonlinear yet preserve certain additive properties.

Abstract

In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter).