On the tangent bundle T SN-1 of the unit sphere SN-1, this paper reduces the coupled Burgers equations to two Neumann systems by using the nonlinearization of the Lax pair, whose Liouville integrability is displayed in the scheme of the r-matrix technique. Based on the Lax matrix of the Neumann systems, the Abel-Jacobi coordinates are appropriately chosen to straighten out the restricted Neumann flows on the complex torus, from which the new finite-gap solutions expressed by Riemann theta functions for the coupled Burgers equations are given in view of the Jacobi inversion.
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,本文摆脱现有代数几何方法中使用Riemann定理的限制,构造mKdV(modifed 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.
