具有奇特性质非线性波的数学理论研究

项目名称： 具有奇特性质非线性波的数学理论研究

项目编号： No.11271362

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 赵俊霄

作者单位： 中国科学院大学

项目金额： 50万元

中文摘要： 关于具有奇特性质非线性波的研究，一直是数学物理、流体力学等领域的重要研究内容。我们将应用可积系统理论重点研究怪波、共振波和周期波。希望通过具体问题的研究，探索发展非线性波解的孤立子数学理论。我们将应用可积系统理论研究特殊孤立子非线性作用时的极大振幅，探索怪波与孤立波之间的关系，以及其它支持怪波的非线性模型，研究玻色-爱因斯坦凝聚试验中的怪波问题。目前，可积系统孤立子τ-函数一般由行列式或pfaff式表示，相对于行列式而言，pfaff式更具一般性。我们将研究具有pfaff式解可积系统的线孤立子解及其非线性作用，重点研究共振作用和网状结构。关于非线性发展方程的周期波解，Hirota猜想具有N-孤立子解的孤子方程也有N-周期波解。由于目前很难直接证明这一猜想，我们将应用可积数值算法，设计3-周期波解的数值解法，检验Hirota的猜想，探索证明该猜想的方法。

中文关键词： 离散可积系统；波色－爱因斯坦凝聚；怪波；共振；孤立子

英文摘要： The study of nonlinear waves with special or unusual properties is an important aspect of mathematical physics, fluid mechanics and related fields. We will study some exact solutions modelling extreme or rogue waves, resonant waves and periodic wave by using integrable theories.Based on studies of particular examples, we will explore and develop mathematical theories for these special types of wave. One task of our project is to explore whether other nonlinear model equations exhibit rogue waves by analyzing the maximum amplitude of nonlinear interaction of solitons. This will also help us to understand the relationship between rogue waves and solitons. Besides, we will also study rogue waves related to Bose - Einstein condensation problem. Another part of our project is concerned with integrable systems having pfaffian solutions. This type of solution is in contrast with the much more common situation that multi-soliton solutions are expressed in terms of determinants. We conjecture that such systems will exhibit line soliton solutions, which make up the web-structure associated with resonant interaction. The third part of our project is concerned with periodic waves. Hirota conjectured that nonlinear evolution equations possessing N-soliton solutions also have N-periodic solutions. Up to now, the scientifi

英文关键词： discrete integrable system；Bose-Einstein Condensates；rogue wave；resonance；soliton