Nonlinear Fourier Analysis and Partial Differential Equations

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This project will involve research in partial differential equations, geometry, and nonlinear Fourier analysis. Its intent is twofold. On the one hand, it is concerned with the behavior of nonlinear waves and solutions to nonlinear dispersive equations arising in physics, nonlinear optics, and ferromagnetism. On the other, it is focused on wave-packet analysis techniques and the study of multilinear singular operators, in both the non-translation-invariant and nontensorial settings. These are two areas that are intimately related to one another by way of decompositions, frequency interaction analysis, and nonlinear estimates. The first part of the project concentrates on the study of certain nonlinear partial differential equations and systems, including the spin-model known as the hyperbolic Ishimori system, which plays a central role in the theory of ferromagnetism. This system arises naturally from the Landau-Lifshitz equation governing both the static and the dynamic properties of magnetization when coupling to a mean field is taken into account. The global-in-time behavior of solutions with special symmetries and initially carrying small energy will be studied. In particular, one would like to know whether such systems are close to equilibrium as time evolves. The principal investigator will also study soliton solutions of the associated hyperbolic cubic nonlinear Schroedinger equation. Of special interest here is the existence of symbions, which are solutions of symbiotic form to dark and bright solitons. A longer term goal is to understand the blow-up dynamics associated with large energy data. In a slightly different direction, the principal investigator plans to obtain sharpened local well-posedness and almost sure global existence results (i.e., for "generic data") for certain periodic nonlinear equations for which there remains a gap between the local-in-time results and those that could be globally achieved for all solutions. The approach is to construct and to exploit the invariance of the associated Gibbs measure that, just like typical conserved quantities, controls the growth in time of the solutions through its support. The second major component of the project is part of a comprehensive program to develop wave packet analysis and time frequency techniques to study multilinear pseudo-differential operators. Their treatment departs from the classical multilinear theory because the behavior of the associated symbols may be governed by a variety that is allowed to change at each spatial point or curvature assumptions are not necessarily imposed in certain directions.Wave phenomena in physics such as light, sound, and gravity, are mathematically modeled using partial differential equations. Nonlinear wave models arise in quantum mechanics and ferromagnetism, as well as in the study of vibrating systems, semiconductors, and optical fibers. Nonlinear Schroedinger equations are fundamental physical equations, for they govern the motion of quantum particles, such as electrons. Some of the topics that the project will explore are of basic interest in connection to both the theory of vortex filaments in three-dimensional fluids and aerodynamics -- a vortex filament can be visualized as a thin tube in which the flow has vorticity -- and to current work in nonlinear fiber optics that is of fundamental importance in today's telecommunication systems. The hyperbolic nonlinear Schroedinger equation has recently received increased attention by physicists and applied mathematicians studying the evolution of optical pulses in normally dispersive nonlinear array structures. Nonlinear Fourier analysis in general (and adapted wave-packet analysis in particular) consists in decomposing complex structures via modulated waveforms into basic building blocks that are localized and thus relatively easy to understand. These blocks can then be put back together in a straightforward manner. The modulated waveforms capture amplitude, scale, frequency, and position, just like a musical score. The objects to which the technique applies include speech, radar signals, oscillatory expressions arising in optics, wave propagation, and other phenomena of a nonlocal nature. This analysis is thus well adapted to study the nonlinear effects that allow waves to interact and produce new modified propagation patterns.