Dynamical systems Methods and Geometric Integrators for Nonlinear Wave Equations

本项目主要集中在两个主题上：(1)发展动力系统方法来分析深水中大振幅瞬变(无赖)波的产生；(2)发展和分析非线性波动方程的几何积分器。这将通过综合研究非线性现象的数值、物理和理论研究来实现。本杰明-费尔(BF)不稳定性和非线性聚焦是产生深水无赖波的一种机制。在早期对高阶非线性薛定谔(NLS)方程的数值研究中，我们发现混沌背景极大地增加了无赖波形成的可能性，并且由于混沌产生的最佳相位调制而增强了聚焦。在这个研究项目中，我们研究了以下问题：(1)HONLS方程中大振幅同宿结构的持久性；(2)NLS对不稳定和同宿数据的“接近”概念是否可以用来发展一个稳健的判据来预测流氓波的发生；(3)合并模和流氓波是否可以与高阶相位奇点的存在联系起来；(4)阻尼对流氓波早期发展的影响；(5)它们的实验验证。这些问题将利用NLS方程的Floquit谱理论，并通过将Mel‘nikov理论推广到偏微分方程组和相位奇异性分析来解决。我们研究的另一个重点是多辛格式的定性性质。需要解决的几个主要问题是：(1)多辛积分器的拓扑稳定性；(2)向后误差分析，以获得系统时空域中局部守恒律的近似保持的误差界和此类估计的有效域；(3)多辛有限元方法的发展和分析，并应用于海森堡磁体模型。对流浪波的建模和结构保持算法的研究在许多应用领域都具有很强的跨学科和相关性，例如水波、非线性光学(其中也与大振幅结构相关)、工程和物理。对流氓波的研究将同时影响建模和预测能力，并需要进一步开发相关的数学工具。多辛积分器的研究有望导致结构保持算法的改进，提高对此类系统的长时间行为的分辨率，并减少所需的计算时间。这个项目的主要特点是理论、计算和实验方法的结合使用，以及研究生的参与。学生将接受从理论到建模再到实施和验证的跨学科协同观点的培训。基于拟议工作的研究论文将发表在数学、科学计算、地球物理和流体动力学(海洋学)的期刊上。

This project focuses on two main topics: (1) the development of dynamical systems methods to analyze the generation of large amplitude transient (rogue) waves in deep water, and (2) the development and analysis of geometric integrators for nonlinear wave equations. This will be accomplished by a synthesis of numerical, physical and theoretical studies investigating the nonlinear phenomena. One mechanism for generating rogue waves in deep water is the Benjamin-Feir (BF) instability and nonlinear focusing. In an earlier numerical study of a higher order nonlinear Schrodinger (NLS) equation we found that a chaotic background greatly increases the likelihood of rogue wave formation and that enhanced focusing occurs due to chaotically generated optimal phase modulations. In this research project we investigate the following questions: (1) persistence of large amplitude homoclinic structures in the HONLS equation; (2) whether the notion of "proximity" to instabilities and homoclinic data of the NLS can be used to develop a robust criterium for predicting the occurence of rogue waves; (3) whether coalesced modes and rogue waves can be linked to the presence of higher order phase singularities; (4) the effect of damping on the early developmentof rogue waves; (5) their experimental validation. These issues will be addressed using the Floquet spectral theory of the NLS equation and by extending Mel'nikov theory for PDEs andphase singularity analysis. The other focus of our research is on the qualitative properties of multisymplectic schemes.Several of the main questions to be addressed are: (1) the topological stability of multisymplectic integrators; (2) backward error analysis to obtain error bounds on the approximate preservation of the local conservation laws and validity regions for such estimates in the space-time domain of the system; (3) development and analysis of multisymplectic finite element methodswith application to the Heisenberg magnet model. The research on modeling rogue waves and structure preserving algorithms isstrongly interdisciplinary and relevant in many areas of application, e.g. water waves, nonlinear optics (where large amplitude structures are also releveant), engineering and physics. The proposed work on rogue waves is expected to simultaneously impact modeling and predicting capablilities as well as to require further development of the relevant mathematical tools.The study of multi-symplectic integrators is expected to lead to improved structure preserving algorithms, providing enhanced resolution of the long time behavior of such systems with a reduction in the required computational time. Key features of this project are the combined use of theoretical, computational and experimental methods and the involvement of graduate students. The students will be trained from a synergistic viewpointacross disciplines from theory to modeling to implementation and validation. Research articles based on the proposed work will be published in journals in mathematics, scientific computing, geophysics and fluid dynamics (oceanography).