Dynamical Systems Methods and Geometric Integrators for Nonlinear Wave Equations

NSF奖摘要-DMS-0204714数学科学：非线性波动方程的动力系统方法和几何积分器这个项目包括非线性色散波理论和应用中的三个相互关联的主题：扰动的非线性薛定谔(NLS)方程中的混沌动力学，同宿混沌在高振幅(无赖)海浪产生中的作用，以及非线性波动方程的结构保持积分器的开发和分析。最近，对水波的实验和理论研究相结合，得到了扰动NLS方程中混沌演化的特征，这是普遍的、可观察的和具有物理意义的。初步分析表明，类似的结果也适用于光纤。在本项目中，将进行进一步的实验、数值和理论分析。后者包括推广全相空间中解的Floquet判别式和相应的Melnikov积分的几何解释，以及得到NLS方程对称破缺扰动的同宿结构存在的判据。这些结果在同宿混沌的背景下的含义和流氓波形成的可能性将被检验。这项研究还包括开发高效、稳定的多辛积分器，用于水波和光学问题中的兴趣方程。将对多辛误差、能量误差和动量误差与积分器的整体性能进行比较(包括准确捕获系统的定性特征)。我们将寻求关于局部守恒律的近似保持的误差界。拟议的研究集中在水波动力学和非线性光学的核心数学和计算问题上。无赖海浪事件可能对近海结构物和船舶造成毁灭性影响。该项目的结果将解决无赖波产生的基本特性，并可能对近海石油钻井平台等结构的设计和分析产生影响。利用NLS方程的孤子解对高速光通信系统中的光脉冲进行建模。多辛积分器的研究将产生快速、高效的程序，可用于高数据速率通信系统和流浪波事件的数值模拟。

NSF Award Abstract - DMS-0204714Mathematical Sciences: Dynamical Systems Methods and Geometric Integrators for Nonlinear Wave EquationsAbstract0204714 SchoberThis project consists of three interrelated topics in the theory and application of nonlinear dispersive waves: chaotic dynamics in perturbed nonlinear Schrodinger (NLS) equations, the role of homoclinic chaos in the generation of high amplitude (rogue) ocean waves, and the development and analysis of structure-preserving integrators for nonlinear wave equations. Combined experimental and theoretical studies of water waves recently yielded a characterization of the chaotic evolution in perturbed NLS equations that is generic, observable, and physically significant. Preliminary analysis indicates that similar results hold in fiber optics. In this project, further experimental, numerical, and theoretical analysis will be carried out. The latter includes extending the geometric interpretation of the Floquet discriminant and the associated Melnikov integrals for solutions in the full phase space and obtaining criteria for the existence of homoclinic structures for symmetry-breaking perturbations of the NLS equation. The implication of these results in the context of homoclinic chaos and the likelihood of rogue wave formation will be examined. The research also involves the development of efficient, stable multi-symplectic integrators for equations of interest in the water wave and optics problems. A comparison of the multi-symplectic, energy, and momentum errors to the overall performance of the integrators (including accurate capture of qualitative features of the system) will be carried out. Error bounds on the approximate preservation of the local conservation laws will be sought. The proposed research focuses on mathematical and computational issues central to water wave dynamics and nonlinear optics. Rogue wave events can have a devastating effect on offshore structures and ships. The results of this project will address fundamental properties of rogue wave generation and potentially have impact on the design and analysis of structures such as offshore oil rigs. Soliton solutions of the NLS equation are used to model light pulses in high-speed optical telecommunication systems. The research on multi-symplectic integrators will yield fast, efficient codes that can be used in numerical simulation of high data rate communication systems and rogue wave events.