负阶孤立子方程及其有限带解

The present proposal is going to be getting explicit solutions for 1+1 dimensional negative order soliton equations through reverse finite dimensional Neumann type systems; to realize this target, we do research mainly about the scientific problems: (1) the integrable characteristics associated with negative order soliton equations, such as the bi-Hamiltonian structures, the infinite conservation laws and the Lax compatibility; (2) the Neumann type integrable reduction of negative order soliton equations on the symplectic submanifold separating the temporal and spatial variables; (3) specifying the finite dimensional invariant subspace of 1+1 dimensional negative order flows, and establishing the direct relationship between the finite-gap potential and the reverse Neumann potential; (4) constructing the Abel-Jacobi variable to straighten out 1+1 dimensional negative order flows on the Jacobian of Riemann surface that signifies the integrable structure in the sense of Liouville-Arnold; (5) and finally, based on the theory of algebraic curves and complex analysis, writing down the Riemann theta function representation of finite-gap potential by means of the Jacobi inversion to display the quasi-periodic behavior of negative order soliton equations. Summing up, the proposed investigation would enrich the content of finite dimensional integrable systems together with its mathematical theory, and then further develop the theory of finite dimensional integralbe reduction for the infinite dimensional integrable system,as well as facilitate the application of reverse Neumann type system for getting solutions to 1+1 dimensional negative order soliton equations.

该项目拟将以对1+1维负阶孤立子方程显式求解为主线，致力于研究：1)负阶孤立子方程的双Hamiltonian结构，无穷守恒律，Lax相容性等可积特征；2)负阶孤立子方程在辛子流形上的Neumann型有限维可积约化，实现其变量分离；3)探明1+1维负阶流作用的有限维不变子空间，并建立有限带势与反向Neumann势之间的直接联系；4)制作Abel-Jacobi变量，在Riemann面Jacobi簇上拉直相关1+1维负阶流，明确其在Liouville-Arnold意义下的可积结构；5)最终应用代数曲线理论和复分析，经Jacobi反演显式写出负阶孤立子方程有限带势的Riemann theta函数表示，揭示其拟周期机制。据此，研究内容将丰富有限维可积系统的研究对象及其数学理论，从而进一步发展无穷维可积系统的有限维可积约化理论及反向Neumann型系统在1+1维负阶孤立子方程解析求解中的应用。

根据原研究计划，我们圆满地完成了各项预订任务。由Lenard算子K的核，导出了包括负阶Jaulent-Miodek系统，负阶耦合Harry-Dym系统(当α=-1/4时，即2元Camassa-Holm方程)，负阶Kaup-Newell系统等一批新非局部可积非线性发展方程，并识别其在Lax相容性和双Hamiltonian结构意义下的可积特征。基于Lax对非线性化，在球丛或椭球丛上将一个负阶可积系统约化为两个反向Neumann型系统，并证明所得反向Neumann型流的Liouville可积性，实现其便捷求解。借助于Lax对非线性化中的对称约束，我们提出负阶Novikov方程，它在负阶流的作用下从无穷维函数空间截出一个有限维不变子空间，并明确反向Neumann型系统的对合解经Neumann映射直接生成负阶可积系统的有限参数解，及其有限带势，为进一步解析求解非局部可积非线性发展方程提供理论依据。从反向Neumann型系统的椭圆变量出发，制作1+1维负阶流的Abel-Jacobi变量，实现1+1维负阶流在紧Riemann面Jacobian簇上的有限带积分，探明其Liouville-Arnold可积结构。进而将谱位势表为椭圆变量的对称函数，结合Riemann定理和迹公式，探讨了1+1维负阶流Abel-Jacobi解的Riemann-Jacobi反演，从而显式写出负阶可积系统有限带势的Riemann Theta函数表示。综上，该项目建立了反向Neumann型系统与负阶可积系统之间的直接联系，提供一个系统、有效的方式寻求非局部可积非线性发展方程的拟周期解，进一步发展了反向有限维可积系统的应用，以及负阶可积系统的有限带积分理论。