The nonlinear Schr\"odinger equation with forcing involving products of eigenfunctions

TL;DR

This paper introduces a novel method to construct integrable forced nonlinear Schrödinger equations involving eigenfunction products, enabling efficient solutions and modeling of physical phenomena.

Contribution

It develops a new methodology to generate integrable forced equations from known integrable systems using eigenfunction products, with explicit solution techniques.

Findings

Forced equations can be solved via Riemann-Hilbert problems with explicit data

Some forced equations model important physical phenomena

Method applies to nonlinear Schrödinger equation and potentially others

Abstract

We elaborate on a new methodology, which starting with an integrable evolution equation in one spatial dimension, constructs an integrable forced version of this equation. The forcing consists of terms involving quadratic products of certain eigenfunctions of the associated Lax pair. Remarkably, some of these forced equations arise in the modelling of important physical phenomena. The initial value problem of these equations can be formulated as a Riemann-Hilbert problem, where the "jump matrix" has explicit x and t dependence and can be computed in terms of the initial data. Thus, these equations can be solved as efficiently as the nonlinear integrable equations from which they are generated. Details are given for the forced versions of the nonlinear Schrodinger.