若干非线性物理模型的多元朗斯基解及向量半有理畸形波的研究

Many nonlinear phenomena in fluid dynamics, plasma physics, fibre-optical communication and Bose-Einstein condensates can be described by the nonlinear physical models. Analytic study and calculation on such equations are one of the key problems in nonlinear physics. This subject mainly focuses on such nonlinear physical models as the multi-component, vector and nonautonomous models in the areas mentioned above. The topics and innovative points include: (1) The Wronskian method will be improved and the relation between the N-fold Darboux transformation and the multi-component Wronskian solutions will be revealed; (2) Calculation and analysis on the vector semirational rogue wave solutions; (3) The dynamics and applications of the vector semirational rogue waves for different models will be analyzed, especially the effects of the variable coefficients on the rogue waves. Such study will provide certain new ideas for solving the multi-component Wronskian and vector semirational rogue wave solutions of other nonlinear physical models and the theoretical basis and references for the experiments and applications in fluid dynamics, plasma physics, fibre-optical communication and Bose-Einstein condensates.

流体力学、等离子体物理、光纤通信和波色-爱因斯坦凝聚中的很多非线性现象都可用非线性物理模型来描述。对这类模型的解析计算与分析，是当前非线性物理学研究的关键问题之一。本课题主要研究广泛存在于上述领域中的多元向量非自治非线性物理模型，主要研究内容和创新点包括：(1) 改进和发展Wronski行列式技术，揭示N波Darboux变换与计算非线性物理模型多元Wronski行列式解的内在联系；(2) 向量半有理畸形波解的计算与分析；(3) 探讨向量半有理畸形波的形成、运动机制以及物理变系数对畸形波的影响与控制问题。本课题为研究其他非线性物理模型的多元Wronski行列式解和向量半有理畸形波解的计算与分析提供新思路，并为在流体力学、等离子体物理、光纤通信和波色-爱因斯坦凝聚中的实验设计及工程应用提供理论基础和参考信息。

流体力学、等离子体物理、光纤通信和波色-爱因斯坦凝聚中的很多非线性现象都可用非线性物理模型来描述。对这类模型的解析计算与分析，是当前非线性物理学研究的关键问题之一。根据研究计划，我们研究了非线性光纤通信中变系数二元NLS方程的向量半有理解的计算、模拟及动力学性质，实现了向量半有理畸形波的控制及管理。研究表明物理变系数影响向量半有理畸形波的激发频率和振幅。除了上述计划内的研究内容外，我们还研究了流体力学和光纤通信中的变系数AB系统，获得了高阶畸形波的各类结构，实现了高阶畸形波的周期激发，分析了物理变系数的作用；研究了描述浅水波运动的变系数Boussinesq方程的N波达布变换及双朗斯基行列式解，获得了该模型的孤子形状保持性质和孤子复合体结构。