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

Abstract

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger 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 Schrödinger 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${}_r$\,($r\ge 3$) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.

34 pages, 3 figures. Extended version of arXiv:1110.3108[nlin.SI], to appear in J. Phys. A: Math. Theor

References in corpus (5)

• Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates
• Exact soliton solutions of coupled nonlinear Schrödinger equations: Shape changing collisions, logic gates and partially coherent solitons
• Bright-dark solitons and their collisions in mixed N-coupled nonlinear Schrödinger equations
• Multicomponent integrable wave equations I. Darboux-Dressing Transformation
• Periodic energy switching of bright solitons in mixed coupled nonlinear Schr{ö}dinger equations with linear self and cross coupling terms

Cited by in corpus (3)

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• Dark-dark soliton breathing patterns in multi-component Bose-Einstein condensates