Mathematical methods for nonlinear wave interactions

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Abstract, DMS 0506495, R Goodman, New Jersey Inst of technologyTitle: Mathematical methods for nonlinear wave interactions This project studies interactions between waves in models from optics and mathematical physics, and the relative stability of waves trapped by localized potentials. Wave interactions in many applications have been known for many years to display resonance structures in which wave capture and reflection alternate in a surprising fractal-like manner. Methods from the theory of dynamical systems, including Melnikov methods, matched asymptotics, and transport theory of iterated maps, are to be applied to ordinary differential equations models of such wave interactions, extending recently published work of the PI. In a second thread of research, we look at the nonlinear interaction between trapped modes at potentials engineered into optical fibers, modeled by the nonlinear Schrodinger equation (NLS), or the nonlinear coupled mode equations (NLCME) in the case of fiber Bragg gratings. The NLCME system lacks an energy-minimization principle (all bound states are saddle points of the energy), yet somehow a ground state is chosen. We study this using numerical analysis, derivation of simplified ordinary differential equation models, and ideas from the spectral theory of Hamiltonian systems.Nonlinear wave phenomena are ubiquitous in physics. An important engineering example is in optical communications, where information is sent as pulses of light through glass fibers. It is important to understand how these waves interact with each other, and with local structures in the medium through which they travel, in order to better design devices that exploit their novel properties. The aim of this research is to extend mathematical theory from dynamical systems to explain phenomena seen in wave interaction, such as wave-trapping. It will use sophisticated computational and analytical methods, some developed in the study of Bose-Einstein condensates, to study the behavior of light in novel physical configurations.