FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory

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Abstract DMS-0354373 FRG: Collaborative Research in Semiclassical Asymptotic Questions in Integrable Nonlinear Wave Theory Peter Miller, Kenneth McLaughlin and Jared BronskiThe goal of applied mathematics is the study of equations of scientific,engineering or industrial interest in a mathematically rigorous way.Understanding such equations often requires consideringa limit in which certain parameters approach zero. Whilesome of these problems behave in a predictable manner in thelimit, other interesting and important problems involve unstablebehavior that becomes less and less predictable the smaller theparameter of interest becomes. One of the best illustrations of thisconcept is the behavior of fluid flows when the viscous drag is small,and the fluid behaves in a turbulent and chaotic way.The primary model under study in this project is the nonlinearSchrodinger equation, which is a fundamental model for the study ofpulses in optical fibers. The small parameter limit for thismodel corresponds to the limit of ultra-short pulse propagation, whichis expected to find many applications to high-speed telecommunications.This limit is known as the semiclassical limit.This project constitutes an in-depth study of initial-value problemsfor several partial differential equations (PDEs) in the semiclassicallimit. Existing formal theories fail because they lead to model problems that areill-posed and thus make no prediction at all for reasonable initialconditions. The goal of this project is to develop asymptotictheories that are not based on any particular ansatz and do notrequire unphysical conditions on the initial data. Among the problemsunder attack is the rigorous semiclassical analysis of the focusingnonlinear Schroedinger equation for general data, a problem that isgenerally considered to be one of the most important open problems inthe field of integrable systems. The specific aims of the projectinclude the development of new ansatz-free methods of asymptoticanalysis --- for spectral theory and for Riemann-Hilbert problems ofinverse-scattering theory --- that are insensitive to analyticityproperties of the initial data, and have a "nonlinearRiemann-Lebesgue" character, directly exploiting cancellation due tooscillations where analytic deformations are impossible. Thesetechniques, once developed, will also have important repercussions infields that are only tangentially related, for example, the theory oforthogonal polynomials of large degree and the statistical analysis oflarge random matrices.The main goal of the proposed work is to develop predictive tools thatapply to extremely unstable systems when the initial conditionsare rough or noisy. An example of such a system is the one governing thepropagation of ultrashort data pulses in certain optical fibers, andtherefor any new insights into such systems will have repercussionsin the field of telecommunications. Some of the problems we propose tostudy admit a detailed analysis because they are idealized and can in somesense be solved exactly; however our work is also expected to lead to generalmethods that are applicable to less idealized systems.These techniques will also have important implications in mathematicalfields that are only tangentially related, for example, the theory oforthogonal polynomials of large degree and the statistical analysis oflarge random matrices.This project also has an important educationalaspect. We plan to involve both graduate students and postdoctoral researchers inthis work, and the advanced training of the next generation of researchersis an important component of this proposal. We also plan aninterdisciplinary workshop to further disseminate the results of this work beyond theboundaries of the mathematical community.