非线性光学等领域中带有高阶非线性项的薛定谔模型的修正Darboux变换及解析解

Nonlinear Schrodinger equations (NLSEs)can be used to describe the optical pulse propagation in optical fibers, the plasma wave packets and the wave propagation of protein, etc. It has important practical significance to study the analytical solutions and their interactions of NLSEs. In some cases, some higher-order nonlinear terms are required to be incorporated into the NLSEs to describe certain physical phenomena. With higher order nonlinear terms, especially with variable coefficients, or coupled NLS type model, due to the complexity of the structure correlation algorithm, the study becomes so difficult that the research results are far less than the ordinary NLSEs. This subject mainly focuses on a class of NLSEs with higher-order nonlinear terms in nonlinear optics by modified Darboux transformation (DT) and Mathematics, including the higher order NLSEs with variable coefficients and coupled types. The main issues and innovations include: (1) The modified DTs of the NLSEs with higher order nonlinear terms will be investigated to get the analytical solutions, including rogue waves, breathers and solitons; (2) Study the interactions and conversions among different types of nonlinear waves and the manipulation; (3) Dynamical properties and physical applications of the solutions will be analyzed. Those results might be of some value for the ultrashort optical pulse propagation in experiments and engineering applications.

非线性薛定谔模型,可用于描述光纤中光脉冲的传输、等离子体中的波包及蛋白质中波的传播等。研究这类模型的解析解及其相互作用具有重要的现实意义。在某些情况下，薛定谔方程中需要加入高阶非线性项才能描述某类物理现象。但是对于带有高阶非线性项，尤其是带有变系数或是耦合项的非线性薛定谔类型模型，因为构造相关算法的复杂性，往往使得研究变得十分困难，目前对这类模型的研究成果远少于普通薛定谔类型模型。本项目拟结合Mathematica算法程序，利用修正的达布变换等方法研究来源于非线性光学等领域中的一类带有高阶非线性项的薛定谔模型，包括变系数及耦合变系数类型：(1) 得到此类型模型的修正的达布变换及解析解，包括：怪波、呼吸子及孤子等；(2) 分析各类解析解的相互作用、相互转化与控制；(3)研究解析解的各类动力学性质及物理应用。项目的研究结果可以为与光纤通信等领域相关的实验和工程实践提供理论指导。

非线性模型可以描述现实世界中的很多自然现象,研究非线性模型的解析解及其相互作用具有重要的现实意义。本项目结合Mathematica及 Maple算法程序，利用修正的达布变换,双线性等方法研究来源于非线性光学等领域中的一类带有高阶非线性项的薛定谔模型及其他类型非线性模型，包括变系数及耦合变系数类型：(1) 得到此类型模型的解析解，包括：怪波、呼吸子及孤子等；(2) 分析各类解析解的相互作用、相互转化与控制；(3)研究解析解的各类动力学性质及物理应用。项目的研究结果可以为与光纤通信等领域相关的实验和工程实践提供理论指导。