Dynamical systems Methods and Geometric Integrators for Nonlinear Wave Equations

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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).