非线性波的时空复杂性研究

In resent years, rogue wave and soliton have been observed both experimentally and numerically. It is demonstrated that nonlinear waves have some internal complex spatial-temporal profiles and structures. Previously, we have obtained some results for the diversity of nonlinear waves dynamic. Subsequently, we plane to study the complexity of nonlinear wave for some equations such as Schodinger-Boussinesq equation, (2+1) D KdV equation, KP equation, Boussinesq equation and DS equation. These complexities mainly include: the diversity of nonlinear waves profiles, the complexity of the waves structures, the change of waves profiles in the course of wave motion, the diversity of homoclinic and heteroclinic orbit and the long time behavior of periodic waves in the presence of initial perturbation. It is shown that the complexity of the structures of multi-wave and the diversity in the course of multi-wave interaction. We try to obtain some advance about diversity and complexity of multi-wave with the help of many theories, methods, and numeric computations. Besides, we will explore the spatial-temporal variation as well as chaos of many kinds of multi-wave for some nonlinear evolution equation. It is expected that we will get some meaningful results including novel methods, the structures of the solution of nonlinear problems and numeric simulation.

近几年来，怪波和多类多孤波的发现，数值模拟和实验都证明了非线性波有着非常复杂的时空形态和结构。本课题在对非线性波动力学多样性研究已积累的成果的基础上，深入研究Schodinger-Boussinesq方程、1+2 维KdV方程、KP方程、Boussinesq方程、DS方程等非线性波的时空复杂性,包括波的形态的时空多样性和结构复杂性、波传播过程中的形态变异的复杂性、同宿和异宿的局域形态多样性、种子解的变化和参数小扰动下多类孤波和同宿解的时空变化以及周期类多孤波长时间形态，揭示多波复杂结构以及它们在相互作用过程中的时空多样性，力图在理论、方法、数值计算和图像展示相结合的平台上研究不同类多孤波的形态多样性和结构复杂性方面取得进展，探索非线性发展方程解特别是多类多孤波解的时空演变乃至时空混沌。在研究方法创新、非线性问题解的结构、解的数值模拟等方面都获得新的有意义的成果。

国家自然科学基金项目《非线性波的时空复杂性研究》（批准号11361048）主要研究了KdV方程、Kundu方程、Davey-Stewartson方程、Schrödinger方程、kadomtsev-Petviashvili方程、Yu-Toda-Sasa-Fukuyama方程、coupled long-wave-short-wave方程、Ito方程等非线性系统数学模型的时空复杂现象，利用本课题组提出的三波法、广义辅助方程法、同宿呼吸极限法、同宿测试法、三孤子极限法等新方法，成功地发现了多孤子解、孤立波解、同宿波解、类周期解、同宿呼吸波解、怪波解、呼吸扭结波和周期孤子等新的时空现象，并得到了Schrödinger-Kirchhoff型方程解的存在性和Emden-Fowler方程若干新的振动准则。四年来已在国内外专业期刊发表学术论文24篇，其中18篇论文被SCI检索，1篇论文发表在中文期刊《应用数学学报》上。所有这些成果不仅揭示了高维系统丰富的动力学形态，揭示了孤波类形态的多样性、复杂性和多波相互作用导致的复杂结果，而且对进一步探讨时空混沌的复杂结构提供了一定的基础，对由非线性发展方程所定义的时空动力学研究具有重要的学术意义和价值，也必将大大丰富这一领域的研究方法和内容。