几类扰动可积系统的Abel积分零点个数和分支图及其应用

本项目拟利用微分方程与动力系统中的经典方法与最新方法，借助于相关的多项式代数符号计算和几何分析结合新技巧，通过研究Abel积分零点个数问题研究若干可积系统小扰动的极限环分支问题。一是研究可积非Hamiltonian系统对应的Abel积分在环域边界的渐进展开式，给出展开式相关系数结构，提出分支定理； 二是利用已有理论，如Chebyshev方法结合新思路新技巧，结合经典方法，研究若干可积二次系统，三次系统的Abel积分零点个数较小上界建立分支图，探究可积非Hamiltonian系统判定Abel积分至多一个零点（即Abel比率具有单调性）的新方法；三是研究高次Hamiltonian系统的一般扰动(特别是三项情况)的分支图，重点针对高余维的横截系统；四是把研究结果、理论和方法应用到含有小参数耗散项的非线性波方程的孤立子和周期波的存在性和共存性问题。

In this project, we will study the zeros of Abelian integrals associated with some perturbed integrable systems. The main mathematical tools are chosen from classical theories and new methods in differential equations and dynamical systems with the aid of symbolic computation and geometric analysis. Firstly, we will study the asymptotic expansions of the Abelian integral near the endpoints of the periodic annulus for some non-Hamiltonian systems, and investigate its structure of coefficients and build the bifurcation theorems; Secondly, we use the algebraic method combined with the new techniques to study the least upper bound on the number of zeros of the Abelian integrals for some quadratic systems, and obtain bifurcation graphs; Thirdly, we will study the bifurcation graphs of three-parameter perturbation of higher order Hamiltonian systems, especially for the truncated systems with co-dimension 5 bifurcation; Finally, we will apply the results, methods and theories to study the nonlinear wave equations with weak dissipation effects, and establish the existence and coexistence of periodic, solitary and Kink waves.