耦合可积非线性薛定谔系统的高阶Peregrine呼吸子以及湍流中怪波现象研究

The rogue wave phenomenon appears in various backgrounds such as ocean, fiber, atmosphere, plasma and so on. It is a hot topic in the fields of mathematical physics all over the world. In this project, we will use symbolic computation and numerical simulation to discuss higher order Peregrine breathers and rogue waves in turbulence for integrable coupled nonlinear Schrodinger-type equations. In detail, there are two points: 1) we will study firstly higher order Peregrine breathers in coupled nonlinear Schrodinger equations, matrix nonlinear Schrodinger equation, three-wave interaction equation, long wave–short wave resonance model, Massive-Thirring model, Schrodinger-Boussinesq equation, Davey-Stewartson equation and 2+1 dimensional long wave–short wave resonance equation, etc. Furthermore, we will study their multi-rogue waves to establish some interesting geometric patterns superimposed among them, such as trigon, circularity, circularity-trigon, pentagram. 2) Considering random waves of fixed statistical properties as initial conditions in a class of integrable coupled nonlinear Schrodinger-type equations, we will use numerical simulation and statistical methods to investigate the formation of rogue waves in integrable turbulence.

怪波现象普遍存在于海洋、光纤、等离子体等环境中，是当前国内外数学物理领域中研究的热点。本项目将以孤立子理论为支撑，以符号计算、数值模拟为辅助研究工具，进一步探讨非线性耦合可积薛定谔系统中怪波的Peregrine呼吸子解以及湍流中怪波现象，着重从两个方面探讨：1）建立和完善耦合非线性薛定谔方程、矩阵非线性薛定谔方程、三波共振方程、长短波共振方程、Massive-thirring模型、Schrodinger-Boussinesq方程、Davey-Stewartson方程以及2+1维长短波共振方程等耦合可积模型的高阶Peregrine呼吸子解，在此基础上进一步利用多怪波解开发出三角形、环形、环形–三角形组合、五角星形等有趣的怪波叠加动力学模式；2）考虑一些耦合可积非线性薛定谔系统的随机初始值问题，通过数值模拟和统计方法来研究这些系统中湍流产生的怪波现象。

怪波现象普遍存在于海洋、光纤、等离子体等环境中，是当前国内外数学物理领域中研究的热点。本项目以孤立子理论为支撑，以符号计算、数值模拟为辅助研究工具，进一步探讨非线性耦合可积薛定谔系统中怪波的Peregrine呼吸子解以及湍流中怪波现象，着重从两个方面探讨：1）建立和完善耦合非线性薛定谔方程、矩阵非线性薛定谔方程、三波共振方程、长短波共振方程、Massive-thirring模型、Schrodinger-Boussinesq方程、Davey-Stewartson方程以及2+1维长短波共振方程等耦合可积模型的高阶Peregrine呼吸子解，在此基础上进一步利用多怪波解开发出三角形、环形、环形–三角形组合、五角星形等有趣的怪波叠加动力学模式；2）考虑一些耦合可积非线性薛定谔系统的随机初始值问题，通过数值模拟和统计方法来研究这些系统中湍流产生的怪波现象。本项目研究成果将促进学科交叉发展，对海洋、大气、光学、金融等复杂系统怪波行为的研究提供理论基础和有力工具。