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。

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.