具有N-peakon的新可积模型与孤子方程的代数几何解

在孤立子理论中, 寻找新的可积系统是最基础而重要的内容之一. 而如何有效的求得一类孤子方程的精确解, 并研究该精确解的性质, 一直是一个基本而又富有挑战性的课题. 本文便是从这两个方面展开, 一方面构造了两个具有N-peakon 的新可积系统, 为目前并不丰富的具有尖孤子解的可积非线性家族提供了极为重要的可积动力模型; 另一方面, 基于超椭圆代数曲线理论, 本文对Lax 对的有限展开法进行了改进, 并将其拓广到求解相联系的孤子方程可积形变后的代数几何解, 给出了著名的KdV(Korteweg de Vries) 6 方程的解. 进一步, 通过研究与孤子方程族相应的亚纯函数、Baker-Akhiezer 函数和超椭圆曲线的渐近性质和代数几何特征, 本文摆脱了现有代数几何方法中使用Riemann 定理的限制, 构造了mKdV (modified Korteweg de Vries) 型方程和混合AKNS (Ablowitz Kaup Newell Segur)方程等孤子方程的代数几何解. 为构造高阶矩阵谱问题所对应的孤子方程族的代数几何解提供了有力的工具.

New integrable models with N-peakons and algebro-geometric solutions of soliton equations

In the soliton theory, it is a basic and challenging problem to search for integrable dynamical models with peaked soliton solutions and obtain the explicit solutions to soliton equations. This paper can be mainly divided into two parts. First, the derivation of two integrable dynamical systems with N-peakon, which provides new models for the family of nonlinear evolution equations with N-peakon; on the other hand, based on the knowledge of hyperelliptic curve, the method of nite-order expansion of Lax pairs is improved to obtain the solution of KdV6 equations. Moreover, to avoid the limit of using the Riemann theorem, the meromorphic function and the Baker-Akhiezer vector are introduced by which quasi-periodic solutions to the mKdV type equations and mixed AKNS equations are constructed according to their asymptotic properties and algebro-geometric characters.