Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy

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

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 submo…

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

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger 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$-$da…

Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation

A Kac-Moody algebra construction for the integrable hierarchy containing the Gardner equation is proposed. Solutions are systematically constructed employing the dressing method and deformed vertex operators which takes into account the nonvanishing boundary value problem for the mKdV hierarchy. Explicit examples are given and besides usual KdV like solitons, our solutions contemplate the large amplitude table-top solitons, kinks, dark solitons, breathers and wobbles.

Nonvanishing boundary conditions and dark solitons in the NLS model

We consider non-vanishing boundary conditions (NVBC) for the NLS model [6,7,27] in the context of the hybrid dressing transformation and $τ$-function approach. In order to write the NLS model in a suitable form to deal with non-vanishing boundary conditions it is introduced a new spectral parameter in such a way that the usual NLS parameter will depend on the affine parameter through the so-called Zukowsky function. In the context of the dressing transformation the introduction of the affine parameter avoids the construction of certain Riemann sheets for the usual NLS spectral parameter. In th…

广义黎曼-希尔伯特-伯克霍夫分解与一类新的高次可积层级

Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies

该研究提出广义黎曼-希尔伯特-伯克霍夫分解，通过引入三个参数扩展可积层级形式体系，统一构造并分析新的高次可积层级，覆盖已知模型并展现广泛适用性。