Inverse Scattering and Partial Differential Equations

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The Principle Investigator will continue his research in three areas of geometric analysis: (1) Completely integrable, nonlinear dispersive equations in two space and one time dimensions, (2) resonances in chaotic scattering, and (3) inverse scattering for Schrödinger and Dirac-type equations on the line with singular potentials. Completely integrable dispersive equations in 2 + 1 dimensions describe nonlinear surface waves, exhibit "lump" and line-soliton behavior, and provide examples of Schrödinger maps with Kähler targets. The Principal Investigator and collaborators will develop the techniques of inverse scattering and gauge transformations into rigorous analytical methods for the study of completely integrable dispersive equations in 2+1 dimensions, using the tools of functional, harmonic, and global analysis. They seek a complete picture of the orbit structure and stability of these dynamical systems; existence, classification, and stability of soliton solutions; and Hamiltonian structure of the flows. Initially the PI will concentrate on the Davey-Stewartson, Novikov-Veselov, Kadomtsev-Petviashvili and Ishimori equations as test cases whose dynamical behavior can be studied in depth. Asymptotically hyperbolic (AH) and complex hyperbolic (CH) manifolds are variable-curvature manifolds with simple geometry at infinity and chaotic geodesic flows characterized by a compact, fractal trapped set. As such they are natural targets for the investigation of chaotic scattering, to study the relationship between the trapped set of geodesics and the distribution of resonances. Developing the tools of approximate (AH) and exact (CH) trace formulae, the Principal Investigator and collaborators will obtain estimates on the distribution of resonances in terms of dynamical data. Continuing his work on inverse scattering for singular potentials on the line, the Principal Investigator will study wave operators, m-functions, and inverse scattering maps for Schrödinger equations and Dirac systems. The goal of this work will be to develop analogues of Simon's A-function and to study qualitative behavior of solutions to the NLS and KdV equations with singular initial data. Completely integrable and chaotic dynamical systems are important "extremal" cases of infinite-dimensional dynamical systems which occur in many different areas of applied science. The completely integrable method in one dimension (formulated as the solution of a Riemann-Hilbert problem determined by scattering data) gives remarkably precise asymptotics for solutions of integrable partial differential equations, random matrix ensembles, orthogonal polynomials on the circle and the line, and combinatorial problems. The PI seeks to develop analogous asymptotic methods for the oscillatory d-bar-problems that determine solutions of completely integrable partial differential equations in two dimensions, asymptotics of orthogonal polynomials in the plane, and asymptotics of normal matrix distributions. This analysis will require new techniques and results in harmonic analysis. At the other extreme, the quantization of chaotic dynamical systems is an area of intensive current research interest: in the proposed research we will study the relationship between classical trapping and quantum chaos in a geometrical setting where the dynamics and scattering are amenable to a detailed analysis.