超椭圆曲线理论在2+1维孤立子方程拟周期解研究中的应用

The problem of constructing the quasiperiodic solutions (QPS) for soliton equations is one of the most challenging problems of the theory of integrable systems. The algebro-geometric technique developed in the late 1970s are powerful solution generating methods. Significantly, entering the 1990s, Gesztesy and Holden proposed a systemic to study QPS of soliton equations. However, now their method are only applied to 1+1 dimensional equations and (modified) Kadomtsev-Petviashvili hierarchies. Based on the theory of hyperelliptic curves and the constraint of high-dimensional system, this project concentrate on constructing QPS for several 2+1 dimensional soliton equations by Gesztesy-Holden's method.

构造孤立子方程的拟周期解是可积系统领域最具挑战性的问题之一。上个世纪70年代后期发展起来的代数几何方法是生成拟周期解强有力的工具。值得注意的是，进入90年代，Gesztesy和Holden的发展了一种系统的方法用于计算孤立子方程拟周期解。然而，目前他们的方法仅应用于1+1维方程及（修正）Kadomtsev-Petviashvili族。本项目拟以超椭圆曲线理论和高维系统的约束为基础，运用Gesztesy和Holden的方法展开对若干2+1维孤立子方程拟周期解的研究。