Quasi-integrability in the modified defocusing non-linear Schr\"odinger model and dark solitons

TL;DR

This paper investigates quasi-integrability in a modified defocusing nonlinear Schrödinger model with dark solitons, revealing an infinite sequence of conserved and asymptotically conserved charges through analytical and numerical methods, with potential applications in nonlinear science.

Contribution

It demonstrates quasi-integrability in the modified defocusing NLS with dark solitons, including exact conservation in special cases and analysis of charge renormalization.

Findings

Quasi-integrability holds for the modified defocusing NLS with dark solitons.

Special two-soliton solutions exhibit exact charge conservation.

Numerical simulations confirm elastic scattering in the saturable NLS model.

Abstract

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schr\"odinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential $V =\eta I^2 - \frac{\epsilon}{6} I^3 $ and the saturable type potential satisfying $V'[I] =2 \eta I - \frac{\epsilon I^q}{1+ I^q},\,q \in \IZ_{+}$, with a deformation parameter $\epsilon \in \IR$ and $I=|\psi|^2$. The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of $\{\eta, \epsilon, n\}$. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.