Dynamical Systems Methods and Geometric Integrators for Nonlinear Wave Equations

点击翻译按钮获取中文摘要

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.