A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS

TL;DR

This paper introduces a numerical method based on modulation equations to accurately simulate moving solitons and radiation in the nonlinear Schrödinger equation, allowing for natural boundary passage and separate solution of components.

Contribution

It develops a novel modulation equations approach coupled with a numerical scheme to effectively handle multichannel solutions with moving solitons and radiation.

Findings

Accurately captures moving solitons and radiation separately.

Allows solutions to pass through boundaries naturally.

Provides a stable and precise numerical method for all times.

Abstract

Based on our previous work for solving the nonlinear Schrodinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which can not be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.