Mathematical Sciences: Nonlinear Localization

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The subject of Professor Faris' research is the relation between two frontier areas in wave propagation: random waves and nonlinear waves. The understanding of these types of waves and their interaction is of importance both for our knowledge of fundamental physical phenomena as well as technological applications in a wide variety of fields; for example, electromagnetic waves in plasma, optical fibers with impurities, and doped superlattices. When the medium through which a wave passes is complicated and irregular, it may be treated as a random medium. The solutions, called random waves, of the equations that describe wave propagation in a highly random medium, are just beginning to be sufficiently well understood to compare with experiment. In some situations a remarkable phenomenon occurs, called localization, in which repeated scattering actually traps a wave. The nature of the transition between localization and the more common case of diffusion is not yet well understood, and mathematical progress here would be a conceptual advance in the study of complicated interactions of waves and matter. In previous work, Professor Faris has found new situations, in dimension higher than one, in which localization takes place despite the fact that this is not the only possible behavior. In the current proposal, he treats localization in the presence of nonlinearities. In this context, the wave interacts with itself and produces solitary waves, or solitons. These waves behave in many ways like classical particles. The goal is to obtain rigorous mathematical results for solitons in random media. This would be a major step in understanding the dynamics of complex random systems.