Dynamical systems Methods and Geometric Integrators for Nonlinear Wave Equations

该项目主要集中在两个主题：（1）开发动力系统方法来分析深水中大振幅瞬态（流氓）波的生成，以及（2）开发和分析非线性波动方程的几何积分器。这将是一个综合的数值，物理和理论研究调查的非线性现象。在深水中产生流氓波的一种机制是Benjamin-Feir（BF）不稳定性和非线性聚焦。在较早的高阶非线性薛定谔（NLS）方程的数值研究中，我们发现，一个混乱的背景大大增加了流氓波形成的可能性和增强的聚焦发生由于混沌产生的最佳相位调制。本文研究了以下几个问题：（1）HONLS方程中大振幅同宿结构的持续性;（2）与不稳定性和NLS同宿数据的"接近"概念是否可以用来发展一个预测异常波发生的可靠判据;（3）合并模和异常波是否可以与高阶相位奇异性的存在联系起来;（4）阻尼对粗波早期发展的影响;（5）实验验证。 这些问题将使用Floquet谱理论的NLS方程和扩展梅尔尼科夫理论的偏微分方程和相位奇异性分析。我们的另一个研究重点是多辛格式的定性性质，主要研究的问题有：（1）多辛积分的拓扑稳定性，（2）向后误差分析，得到系统在时空域上近似保持局部守恒律的误差界和有效域，（3）多辛积分的拓扑稳定性，（4）多辛积分的拓扑稳定性，（5）多辛积分的拓扑稳定性，（6）多辛积分的拓扑稳定性，（7）多辛积分的拓扑稳定性。（3）多辛有限元方法的发展和分析及其在Heisenberg磁体模型中的应用。对异常波建模和结构保持算法的研究是一个跨学科的研究，涉及到许多应用领域，如水波、非线性光学（其中大振幅结构也是异常的）、工程和物理。对流氓波的拟议工作，预计将同时影响建模和预测能力，以及需要进一步发展相关的数学工具。多辛积分的研究，预计将导致改进的结构保持算法，提供增强的分辨率的长时间的行为，这样的系统，减少所需的计算时间。该项目的主要特点是理论，计算和实验方法的结合使用和研究生的参与。学生将从理论到建模到实施和验证的跨学科协同观点进行培训。基于拟议工作的研究文章将发表在数学、科学计算、海洋物理学和流体动力学（海洋学）等期刊上。

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).