Fine structure of soliton bound states in the parametrically driven, damped nonlinear Schr\"odinger equation

TL;DR

This paper analytically and numerically investigates static soliton bound states in a parametrically driven, damped nonlinear Schrödinger equation, revealing their structure and analytical description using a Schrödinger-like framework.

Contribution

It introduces a novel analytical approach to describe bound soliton states in a non-integrable, driven-damped nonlinear Schrödinger system, connecting it to reflectionless potentials.

Findings

Symmetric two-hump soliton solutions are well described by a three-soliton formula.

The equations can be transformed into Schrödinger-like eigenvalue problems.

The soliton parameters depend on parametric pumping and dissipation.

Abstract

Static soliton bound states in nonlinear systems are investigated analytically and numerically in the framework of the parametrically driven, damped nonlinear Schr\"odinger equation. We find that the ordinary differential equations, which determine bound soliton solutions, can be transformed into the form resembling the Schr\"odinger-like equations for eigenfunctions with the fixed eigenvalues. We assume that a nonlinear part of the equations is close to the reflectionless potential well occurring in the scattering problem, associated with the integrable equations. We show that symmetric two-hump soliton solution is quite well described analytically by the three-soliton formula with the fixed soliton parameters, depending on the strength of parametric pumping and the dissipation constant.