Mathematical Sciences: Spectral Problems and Inverse Spectral Problems

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This project will focus on two fundamental areas of nonlinear mathematical analysis involving integrable systems of differential equations. Particular emphasis is placed on the relationship between eigenvalue algorithms and Hamiltonian mechanics. The first concerns the asymptotics of the oscillatory Riemann-Hilbert problems of the kind that arise in the theory of integrable nonlinear wave equations. Work will be done exploiting a newly developed steepest descent method to study the zero dispersion limit of the Korteweg de Vries equation, integrable statistical models such as the transverse Ising chain at critical magnetic field and spatially discrete problems, such as the Toda lattice. Work will also continue on the eigenvalue algorithms in the context of integrable systems. It was shown earlier how the basic diagonalization of finite dimensional matrices can be realized as flows on manifolds. A framework has been developed in which many well known systems of physical and mathematical interest can be embedded. One focus of this investigation will be to place the recent theory of Moser and Veselov describing a class of variational problems that can be solved by a generalized QR factorization into the framework of Hamiltonian flows. The principal object of this research is the analysis of systems of differential equations arising from models of the physical world and the application of that analysis to studies of finite dimensional matrix theory; in particular the development of new, computationally effective means for diagonalization of matrices. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions.