若干非线性薛定谔系统的可积湍流现象及畸形波产生机制研究

Rogue wave is first observed in the ocean, and it poses a great threat to ships and drilling platforms at sea. The latter investigations show that rogue wave also exists in the optical fiber, plasma physics and Bose-Einstein condensation. In recent years, the researches on the rogue wave mainly includes the basic characteristics, numerical simulations and physical simulations. Since the rogue wave appears randomly, there are relatively few studies on the mechanism of the rogue wave. Integrable turbulence can be used to study the amount of solitons and breathers in chaotic wave fields of integrable systems with different initial conditions, and then generation mechanism of the rogue wave will be investigated. This project focuses on the integrable turbulence of three nonlinear Schrödinger systems, i.e., the coupled nonlinear Schrödinger system, derivative nonlinear Schrödinger equation and coupled higher-order nonlinear Schrödinger systems. The results of this project can enrich the studies of turbulence phenomena of the integrable systems, and investigation of the higher-order system can help to study the influences of higher-order terms on integrable turbulence. In addition, the above researches can further deepen the understanding of the formation mechanism of the rogue wave, provide inform for the studies of turbulence phenomena of other integrable systems, and offer the theoretical basis for experimental design or engineering application in optical fiber communication, plasma and Bose-Einstein condensation.

畸形波最早于海洋中被观察到，其对海上的轮船和钻井平台等具有很大威胁。后来的研究表明，畸形波也存在于光纤、等离子体物理以及玻色-爱因斯坦凝聚等领域。近几年来，关于畸形波的研究主要包括畸形波的基本特征、数值模拟以及物理模拟等。因畸形波的出现具有随机性，对于畸形波产生机制的研究相对较少。可积湍流可以用于研究在不同初始条件下可积系统的混沌波场中孤子与呼吸子的数量变化，进而研究畸形波的产生机制。本项目主要研究三类非线性薛定谔系统的可积湍流现象，即耦合非线性薛定谔系统、导数非线性薛定谔方程和高阶的耦合非线性薛定谔系统。本课题的结论可以丰富关于可积系统的湍流现象的研究，而对高阶系统的研究还可以探究高阶项对可积湍流的影响。此外，上述研究可以进一步加深对畸形波的形成机制的认识，为研究其他可积系统的湍流现象提供参考，为在光纤通信、等离子体以及玻色-爱因斯坦凝聚中的实验设计或工程应用提供理论基础。

畸形波广泛存在于海洋、光纤、等离子体物理以及玻色-爱因斯坦凝聚等领域。近几年来，关于畸形波的研究主要包括畸形波的基本特征、数值模拟以及物理模拟等,而关于畸形波的产生机制研究较少。可积湍流可以用于研究在不同初始条件下可积系统的混沌波场中孤子与呼吸子的数量变化，进而研究畸形波的产生机制。本项目主要研究了耦合非线性薛定谔系统的可积湍流现象，揭示了耦合非线性薛定谔系统中畸形波的特性。此外，为了深入理解非线性系统中畸形波等非线性现象，本项目还研究了高阶非线性薛定谔系统的暗孤子解和有理解、耦合Hirota方程的半有理解、耦合AB系统的明暗孤子解和半有理解以及NLS-MB系统的退化孤子解和局域解。本课题的结论不仅可以丰富关于可积系统的湍流现象的研究，而且可以完善关于畸形波和孤子的动力学性质研究。