Mathematical Methods for Nonlinear Wave Equations

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Proposal ID: 0604546PI: Bernard DeconinckInstitution: University of WashingtonTitle: Mathematical Methods for Nonlinear Wave EquationsAbstractI will develop two general methods for the study of nonlinear wave equations. These methods pertain to the explicit computation of (quasi-)periodic solutions of integrable equations, and the numerical stability analysis of these and other solutions of integrable or near-integrable equations. The first method uses the computational calculus on Riemann surfaces I have developed over the last years. The second method extends and rigorously examines Hill's method, which we introduced in its full generality in 2006.The applications of the methods will allow us to compute and analyse solutions of equations describing such diverse phenomena as tsunamis, laserlight, and new states of matter occurring at extreme temperatures. The proposed research will involve different graduate and undergraduate students, and put them on their way to making their own important contributions to applied mathematical research in the United States.