New derivation of soliton solutions to the AKNS₂system via dressing transformation methods

Abstract

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger equation (AKNSr) (r = 1, 2). Using the dressing transformation (DT) method and the related tau functions we study the AKNSr 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 AKNSr 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 AKNS2 system, and three− and higher−dark−dark−soliton bound states can not exist. The AKNSr (r ≥ 3) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.Dedicated to the memory of S. S. Costa.

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

“…In the first reference a hybrid of the dressing transformation and tau function methods has been used, whereas in the second one they have been derived by the KP-hierarchy reduction method. The ansatz (5.2) for the general N-soliton solution in terms of the tau functions G and F follows the construction presented in [25].…”

Darboux transformation and multi-dark soliton forN-component nonlinear Schrödinger equations

“…Recently, there are some another method to derive the dark soliton for multi-component NLS equations. For instance, the algebraic-geometry reduction method and dressing-Hirota method [28,29].…”

Spatial shifts of colliding dark solitons in deformed non-linear Schrödinger models

“…The 2-dark soliton solution is given by [27] ψ(x, t) = |ψ 0 |e iwt 1 + y 1 e Γ1(x,t) + y 2 e Γ2(x,t) + r y 1 y 2 e Γ1(x,t) e Γ2(x,t) 1 + e Γ1(x,t) + e Γ2(x,t) + r e Γ1(x,t) e Γ2(x,t) (A.5)…”

“…The system (A.1) is hyperbolic for small oscillations around |ψ| = |ψ 0 | and it has an associated sound speed given by v s = √ β|ψ 0 |[17]. This velocity plays an important role in the existence of solitary waves in the NLS model.The 1-dark soliton solution is given by[27] ψ(x, t)= 1 y exp [2P (x−vt−x0)] 1 + exp [2P (x−vt−x0)] , y = e 2iη , η = arctan (−P/v) iv + P tanh [P (x − vt − x 0 )] (A.4)This solution possesses three arbitrary real parameters, v, P and β. The intensity function |ψ| moves at the velocity v, which is the velocity of the dark soliton.…”