无色散和非交换系统的可积耦合推广及其求解

Dispersionless systems and noncommutative systems have wide applications in the physics and they become two hot topics in the soliton theory and integrable systems. However, the study on the coupled generalization of the above two systems is very little. So the coupled generalization of dispersionless systems and of noncommutative systems will be explored in our project. This project consists of three parts: (1) The coupled generalization of dispersionless BKP, Toda and Harry-Dym hierarchies is considered within dispersionless Sato's framework. A new method is proposed to solve the dispersionless integrable systems and the second type of dispersionless equations with self-consistent sources. B?cklund transformation between two coupled dispersionless systems is explored. In addition, the algebraic and geometric structures of the new systems are also studied. (2) Within noncommutative Sato's framework, the coupled noncommutative mKP amd Toda hierarchies are constructed by introducing squared eigenfunctions in the Moyal-deformed Lax equations. Multi-soliton solutions to the coupled noncommutative systems are given, and the effects of source terms on the dynamical behaviors of Multi-soliton solutions are analyzed as well. Furthermore, B?cklund transformation between two coupled noncommutative systems is also investigated. The applications of the coupled noncommutative systems in physics are finally studied. (3) Noticing the fact that the second types of the coupled BKP and CKP hierarchies are difficult to solve owing to the complexity of equations itself, dressing method is used to solve these two systems. Moreover, the dynamical behaviors for the multi-soliton solutions to the second type of BKP and CKP equations with self-consistent sources are analyzed.

无色散系统以及非交换系统在物理中均有广泛应用, 是孤立子和可积系统的两个研究热点。但目前对于这两个系统的可积耦合推广研究还很少。本项目拟对此展开研究，内容包括：(1)在无色散Sato理论框架下, 构造无色散耦合BKP、Toda及Harry-Dym方程族；提出新的方法求解无色散可积方程族以及第二型带源的无色散方程；研究无色散耦合系统间的B?cklund变换及其代数几何结构。(2)在非交换Sato理论框架下，引入Moyal形变下的平方特征函数，研究非交换耦合mKP和Toda方程族的构造及其约化；给出非交换耦合系统的多孤子解，分析源项的引入对非交换系统的孤子解的动力学行为的影响；研究非交换耦合系统间的B?cklund变换；研究非交换耦合系统在物理上的应用。(3)考察耦合BKP、CKP方程族的dressing解法；求解第二型带源BKP、CKP方程，并分析其孤子解的动力学行为。

一、无色散可积系统在流体力学，拓扑量子场论，光学，超弦理论，复变函数，共形映射等领域中有着广泛的应用；而非交换可积系统是经典可积系统的一种重要推广，它在弦理论、非交换场的规范理论以及量子理论中亦有重要的应用。这两个可积系统的研究是孤立子和可积系统的两个研究热点。本项目主要研究了这两个系统的可积耦合推广及其求解问题。具体研究成果如下：.（1）给出了无色散BKP，无色散Harry-Dym方程的对称约束；.（2）构造了新的耦合的无色散KP，BKP，Harry-Dym方程族及其相应的守恒方程；.（3）利用约化的方法并结合速端变换求解了第一型带源的无色散BKP、Harry-Dym方程;.（4）考察了耦合的无色散mKP方程族和耦合的无色散Harry-Dym方程族之间的Bäcklund 变换；.（5）考察了q-形变的耦合KdV方程族的Darboux-Bäcklund变换，并求出了该方程族的孤子解；.（6）探讨了q-形变的耦合KdV方程与q-形变的耦合mKdV方程的Bäcklund 变换；.（7）考察了耦合的KdV方程的爆破解的动力学性质，分析了源项的引入对爆破点轨迹的影响；.（8）推导出了不带色散项的modified Camassa-Holm方程的双线性方程，并给出了该方程的参数形式表示的soliton解；.（9）利用标准的Crank-Nicolson及Dufor-Frankel 有限差分法来别求解二维的阻尼、非阻尼的sine-Gordon方程，并探讨了它们的稳定性和收敛性；最后举例加以验证。.本项目的研究成果是孤立子和可积系统的重要完善和补充。.二、反应-扩散过程不仅仅存在于自然界中，同时也存在生物体体内。其主要原因是由于分子的运动总是从浓度高向浓度低的方向进行。而反应-扩散方程通常被用来描述这一过程。本项目在反应-扩散方程方面做了如下工作：.（1）探讨了一般反应-扩散方程的Bogdanov-Takens奇异性问题；.（2）全面分析了带有两个时滞的基因表示的反应-扩散系统在Neumann边界条件下的解的相关问题。