变系数孤子方程的达布变换及行列式解

With the inhomogeneous media, nonuniform boundaries and external potential fields considered, variable-coefficient soliton equations(Nonautonomous system) can describe certain situations in nature, science and engineering problems more realistically than their constant-coefficient counterparts. This subject mainly focuses on the soltion equations with the variable coefficients (include the N-coupled and non-isospectral ones) in fluid dynamics, plasma, Bose-Einstein condensates and optical communications. The main issues and innovations include: (1) The variable-coefficient N-fold Darboux transformation wil be proposed; (2) Two types of determinant representations of the variable-coefficient soliton equations will be investigated, especially the N-Wronkian solution; (3) The dynamical properties of the novel solutions are analyzed (the effects of the variable coefficients on the rogue waves and the nonautonomous soliton control and management). Such study will provide some new ideas for the N-fold Darboux transformations as well as determinant solutions of other variable-coefficient soliton equations, and the valuable information to the study of autonomous soliton in experiments and engineering applications.

考虑到介质的不均匀性、边界条件的不一致性以及外部势场影响的情况下，变系数孤子方程（非自治系统）能够更好地刻画自然界和科学、工程问题中的非线性现象。本课题研究广泛存在于流体力学、等离子体、玻色-爱因斯坦凝聚和光纤通信领域中的变系数孤子方程(包含N耦合非等谱情况)，其主要研究内容和创新点包括：(1) 提出变系数N重达布变换方法；(2) 变系数孤子方程的两类行列式解的计算问题，重点研究利用变系数N重达布变换构造N朗斯基行列式解的技术；(3) 新解的力学性质分析，包括物理变系数对Rogue波的作用规律、非自治孤子的控制及管理。本课题将为研究为其他变系数孤子方程的N重达布变换和行列式解的计算提供新的思路，并为研究非自治孤子在实验和工程领域中的应用提供有价值的信息。

考虑到介质的不均匀性、边界条件的不一致性以及外部势场影响的情况下，变系数孤子方程能够更好地刻画自然界和科学、工程问题中的非线性现象。根据研究计划，我们研究了非均匀等离子体中变系数DNLS方程的广义达布变换及其行列式解，获该模型的非自治高阶怪波解和呼吸子解，实现了高阶怪波的周期激发与控制，并探讨物理变系数对两类波的作用机制。除了上述计划内的研究内容外，我们还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。