Schrodinger方程类方程的怪波及其相关问题研究

As the phenomenon on rogue wave in ocean,fiber optic,atmosphere and financial area is discovered, a study on rogue waves and some problems concerning them is urgently needed.The intention of this project is to find out the rogue wave solutions,to explore the formation mechanism of rogue waves and to show their dynamical behaviors; consequently to reveal some new nonlinear phenomena in nonlinear science and to provide a way of mathematical thinking for the studies of rogue waves in marine science, optics, aerodynamics, financial area, etc. and make best use of rogue wave theoretically and practically when it occurs. The techniques of the numerical simulation and finding exact solutions of the nonlinear equation are applied to investigate the Schr?dinger equation type, such as the derivative Schr?dinger equation, the Schr?dinger equation of basing on the Schr?dinger map equation, the multi-dimensional Schr?dinger equation, the nonlinear coupled Schr?dinger-KdV equations and nonlinear coupled Schr?dinger-Bousinesq equations and so on.

怪波现象在海洋、光纤、大气、金融等领域的发现，迫切需要对怪波及相关问题进行深入研究。本课题旨在利用数值模拟和求解非线性发展方程精确解的方法，对Schr?dinger方程类方程，如导数Schr?dinger方程、源自Schr?dinger映射方程的Schr?dinger方程，多维Schr?dinger方程、非线性耦合Schr?dinger-KdV方程组，非线性耦合Schr?diner-Bousinesq方程组等进行研究，探索怪波的动力学性态和怪波的形成机制，进一步揭示非线性科学中新的非线性现象，为海洋科学、光学、空气动力学，乃至金融等领域中的怪波现象的研究提供新的数学思路，为在不利的方面抑制怪波的产生，在有利的方面发挥怪波的效用提供理论和实践基础。

该项目通过对非线性耦合Schrödinger-Bousinesq方程组、(2+1)-维Davey-Stewartson 方程、变系数的非线性Schrödinger方程等十余个应用广泛且涉及Schrodinger类的非线性模型波的研究，获得丰富的非线性波结构(包括怪波)和显式表示，较深刻地刻画了怪波的形成及性态。本项目研究结果表明，怪波可以不同形式呈现，而且存在于不同系统中，从而揭示了“怪波”存在于非线性系统的普遍性，有助于解释非线性系统中出现的部分奇异现象。对怪波的动力学分析，研究方法和技术，以及在每个模型中获得的新结果，是对数学及物理学科领域的新贡献，对深入研究非线性系统复杂性具有较高学术价值。