Generalized AKNS System, Non-vanishing Boundary Conditions and N-Dark-Dark Solitons

TL;DR

This paper explores soliton solutions in the generalized AKNS system under various boundary conditions, deriving new dark-dark solitons and analyzing bound states, with implications for coupled nonlinear Schrödinger equations in physics.

Contribution

It introduces a modified dressing transformation to derive generalized N-dark-dark solitons and analyzes bound state properties in the AKNS system.

Findings

Existence of two-dark-dark-soliton bound states in AKNS_2.

Higher-dark-dark-soliton bound states do not exist.

Derived properties of focusing, defocusing, and mixed CNLS models.

Abstract

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS). Using the dressing transformation (DT) method and the related tau functions we study the AKNS$_{r}$ system for the vanishing, (constant) non-vanishing and the mixed boundary conditions, and their associated bright, dark and bright-dark N-soliton solutions, respectively. Moreover, we introduce a modified DT related to the dressing group in order to consider the free field boundary condition and derive generalized N-dark-dark solitons. We have shown that two$-$dark$-$dark$-$soliton bound states exist in the AKNS$_2$ system, and three$-$ and higher$-$dark$-$dark$-$soliton bound states can not exist. As a reduced submodel of the AKNS$_r$ system we study the properties of the focusing, defocusing and mixed focusing-defocusing versions of the so-called coupled non-linear Schr\"{o}dinger equation ($r-$CNLS), which has recently been considered in many physical applications. The properties and calculations of some matrix elements using level one vertex operators are outlined.