形变可积系统的怪波解及几何结构

项目名称： 形变可积系统的怪波解及几何结构

项目编号： No.11301179

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 黄晔辉

作者单位： 华北电力大学

项目金额： 22万元

中文摘要： 可积系统的怪波解以及可积系统与微分几何的联系是可积系统理论的重要研究内容，具有重大意义。本项目将研究：（1）采用推广的Darboux变换方法、dressing方法，对带自相容源的非线性薛定谔方程、Hirota方程、Davey-Stewartson方程等带自相容源孤立子方程的怪波解进行研究，得到带自相容源孤立子方程的各阶怪波解的一般表达式。研究带自相容源孤立子方程的不同类型的解之间的相互作用。（2）构造高维可积系统的推广的Kupershmidt形变。研究推广的Kupershmidt形变方程的求解问题，得到推广的Kupershmidt形变可积系统的怪波解及其它类型解，并研究其动力学行为。（3）研究推广的Kupershmidt形变可积系统的几何结构。建立曲线运动与推广的Kupershmidt形变可积系统之间的联系，给形变可积系统几何上的解释。

中文关键词： 可积形变；怪波解；变系数可积系统；达布变换；玻色-爱因斯坦凝聚

英文摘要： The rogue wave and the geometric structure of the integrable system are two important fields in integrable system theory. Our project will study: (1) By using the generalized Darboux transformation and the generalized dressing method, we will study the rogue wave of the NLS equation with self-consistent sources, the Hirota equation with self-consistent sources and the Davey-Stewartson equation with self-consistent sources. We would obtain the general formula for the rogue wave. We would also discuss the interaction between different types of solutions. (2) We will construct the generalized Kupershmidt deformation for the Higher dimensional integrable system. We will obtain different kinds of solutions of the generalized Kupershmidt deformation system, such as the rogue wave, soliton and breathers and study their dynamical property. (3) Study on the geometric structure of the integrable system with generalized Kupershmidt deformation. We will find the connection between the motion of curves and the integrable system with Kupershmidt deformation. We will explain the geometric structure of the deformed integrable system.

英文关键词： Integrable deformation；rogue wave；integrable system with variable coefficients；Darboux transformation；Bose-Einstein condensate