多分量可积系统的非局域对称群

This project aims at the multi-component nonlinear integrable systems with physical significance. By using their geometry integrability and Lax integrability, it studies the systems’ non-local symmetries, recursion operators, infinite number of conservation laws, the construction of interaction solution between elliptic periodic waves and solitary waves, as well as the dynamic behavior of the solution. The main research content includes the following three aspects: Firstly, structure more non-local symmetries of 2-component μ-Camassa-Holm equations, generalized 2-component μ-Camassa-Holm equations and 3-component Camassa-Holm-Hunter-Saxton equations based on their geometry integrability, so as to further obtain systems’ the exact solutions, infinite number of conservation laws, the recursion operators and Darboux transformations. Secondly, construct non-local symmetries which reply on the eigenfuction of 2-component μ-Camassa-Holm equations, the generalized 2-component μ-Camassa-Holm equations and 3-component Camassa-Holm-Hunter-Saxton equations, and then we obtain solutions of interaction between elliptic periodic waves and solitary waves, and analyze the dynamic behavior of the solutions. Thirdly, search the relationship between different types of non-local symmetries of the same systems, explore the connections between exact solutions, recursion operators and conservation laws, which are based on different types of non-local symmetries of the same systems. And the initial value problems of the systems are studied. The construction methods of non-local symmetries are summarized and extended to more systems, such as 2-component Degesperis-Procesi equations.

本项目主要针对有物理意义的多分量非线性可积系统，利用其几何可积性和Lax可积性，研究系统的非局域对称、递推算子、无穷多守恒律、椭圆周期波与孤立波相互作用解的构造方法以及解的动力学行为。主要研究内容包括：(1)构造更多的2分量μ-CH方程组、广义2分量μ-CH方程组以及3分量CH-HS方程组基于几何可积性的非局域对称，从而进一步获得精确解、无穷多守恒律、递推算子以及Darboux变换。(2)构造更多依赖本征函数的2分量μ-CH方程组、广义2分量μ-CH方程组、3分量CH-HS方程组的非局域对称，进而获得椭圆周期波与孤立波相互作用解，并分析这种解的动力学行为。(3)寻找同一个系统不同类型的非局域对称之间的联系，探索基于不同类型非局域对称所构造出来的精确解、守恒律、递推算子之间的相互关系，并研究系统的初值问题。将非局域对称的构造方法总结并推广到更多的系统中，例如2分量DP方程组。

本项目主要针对有物理意义的多分量非线性可积系统，利用其几何可积性和Lax可积性，研究系统的非局域对称、递推算子、无穷多守恒律、椭圆周期波与孤立波相互作用解的构造以及解的动力学行为。主要研究内容包括：(1) 构造更多的2分量μ-CH方程组，广义2分量μ-CH方程组，以及3分量CH-HS方程组基于几何可积性的非局域对称，从而进一步获得精确解，无穷多守恒律，递推算子以及Darboux变换。(2) 构造更多依赖本征函数的2分量μ-CH方程组，广义2分量μ-CH方程组，3分量CH-HS方程组的非局域对称，进而获得椭圆周期波与孤立波相互作用解，并分析这种解的动力学行为。(3) 寻找同一个系统不同类型的非局域对称之间的联系，探索基于不同类型的非局域对称所构造出来的精确解，守恒律，递推算子之间相互关系，并研究系统的初值问题。将非局域对称的构造方法总结，推广到更多的系统中，例如2分量DP方程组。