Department of Mathematics, Beijing Jiao Tong University, Beijing 100044, China and State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao Tong University, Beijing 100044, China.
Chaos. 2013 Sep;23(3):033137. doi: 10.1063/1.4821132.
A spectral problem, the x-derivative part of which is a simple generalization of the standard Ablowitz-Kaup-Newell-Segur and Kaup-Newell spectral problems, is presented with its associated generalized mixed nonlinear Schrödinger (GMNLS) model. The N-fold Darboux transformation with multi-parameters for the spectral problem is constructed with the help of gauge transformation. According to the Darboux transformation, the solution of the GMNLS model is reduced to solving a linear algebraic system and two first-order ordinary differential equations. As an example of application, we list the modulus formulae of the envelope one- and two-soliton solutions. Note that our model is a generalized one with the inclusion of four coefficients (a, b, c, and d), which involves abundant NLS-type models such as the standard cubic NLS equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Liu equation, the Kaup-Newell equation, and the mixed NLS of Wadati and/or Kundu, among others.
提出了一个谱问题，其 x 导数部分是标准的 Ablowitz-Kaup-Newell-Segur 和 Kaup-Newell 谱问题的简单推广，并给出了与之相关的广义混合非线性 Schrödinger（GMNLS）模型。借助规范变换构造了谱问题的 N 重 Darboux 变换及其多参数。根据 Darboux 变换，GMNLS 模型的解可以归结为求解一个线性代数系统和两个一阶常微分方程。作为应用实例，我们列出了包络单孤子和双孤子解的模公式。请注意，我们的模型是一个广义模型，包含四个系数（a、b、c 和 d），它涉及到丰富的 NLS 类型模型，如标准立方 NLS 方程、Gerdjikov-Ivanov 方程、Chen-Lee-Liu 方程、Kaup-Newell 方程以及 Wadati 和/或 Kundu 的混合 NLS 等。
