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摘要0204714 Schober该项目包括非线性色散波的理论和应用方面的三个相互关联的主题：扰动非线性薛定谔（NLS）方程中的混沌动力学，同宿混沌在产生高振幅（流氓）海浪中的作用，以及非线性波动方程保结构积分器的发展和分析。 结合实验和理论研究的水波最近产生的扰动NLS方程，是通用的，可观察的，物理意义上的混沌演化的表征。 初步分析表明，类似的结果也适用于光纤。 在这个项目中，将进行进一步的实验，数值和理论分析。 后者包括扩展Floquet判别式和相关Melnikov积分的几何解释以获得全相空间解，并获得NLS方程对称破缺扰动存在同宿结构的准则。 将研究这些结果在同宿混乱背景下的含义和流氓波形成的可能性。 该研究还涉及开发高效，稳定的多辛积分方程的水波和光学问题的兴趣。 将进行多辛，能量和动量误差的积分器的整体性能（包括系统的定性特征的准确捕获）的比较。 将寻求局部守恒律近似保持的误差界。 拟议的研究重点是数学和计算问题的核心水波动力学和非线性光学。 流氓波事件可能对海上结构物和船舶产生破坏性影响。 该项目的结果将解决流氓波生成的基本特性，并可能对海上石油钻井平台等结构的设计和分析产生影响。 NLS方程的孤子解被用来模拟高速光通信系统中的光脉冲。 多辛积分器的研究将产生快速，有效的代码，可用于高数据速率通信系统和流氓波事件的数值模拟。

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.