Spectral Problems and Inverse Spectral Problems

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The Principal Investigator will consider a variety of problems concerning the asymptotic behavior of finite and infinite dimensional systems containing external parameters such as space and time, as the parameters become large. In particular, the Principal Investigator will consider the long-time behavior of solutions of the perturbed focusing Nonlinear Schrodinger equation, with initial data that is close to a solution. In addition, the Principal Investigator will consider statistical problems arising in the theory of permutations of N numbers as N becomes large, as well as questions concerning the rational approximation of special numbers such as z (5) where z is the Riemann zeta function. The Principal Investigator will also consider the computation of various physical constants arising in integrable models. In all the above problems, a key analytical role will be played by the steepest descent method for Riemann-Hilbert problems introduced by Xin Zhou and the Principal Investigator in 1993. Much of the work proposed by the Principal Investigator has a strong interdisciplinary flavor. For example, the statistical problems mentioned above for permutations of N numbers, are intimately related to a model for the condensation of a supersaturated liquid on a substrate and also to a version of solitaire ("patience sorting"), and also to the problem of re-ordering a library in which books have been improperly shelved. In addition, a key objective of the Principle Investigator will be to prove "universality" for a variety of physical systems. For example, motivated by earlier work with Xin Zhou, the Principal Investigator plans to show that solutions of the perturbed focusing Non-Linear Schrodinger equation behave just like solutions of the unperturbed focusing Non-linear Schrodinger equation, once the "scales" of the problem are properly adjusted. This is a key step in the development of models for a wide variety of physical phenomena, in particular for phenomena arising in the transmission of signals along optical fibers.