New derivation of soliton solutions to the AKNS$_2$ system via dressing transformation methods

TL;DR

This paper derives new soliton solutions for the AKNS$_2$ system using dressing transformation methods, exploring various boundary conditions and their physical implications, including bound states and extensions to higher systems.

Contribution

It introduces a modified dressing transformation to derive generalized N-dark-dark solitons and analyzes bound state properties in the AKNS$_2$ system, extending the method to AKNS$_r$ systems.

Findings

Existence of two-dark-dark-soliton bound states in AKNS$_2$

Non-existence of three or higher-dark-dark-soliton bound states

Derivation of generalized N-dark-dark solitons for various boundary conditions

Abstract

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS$_r$)\,($r=1,2$). 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. 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. 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. The AKNS$_r$\,($r\geq 3$) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.