From the Solution of the Tsarev System to the Solution of the Whitham Equations |
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Authors: | Grava Tamara |
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Institution: | (1) Department of Mathematics, University of Maryland, College Park, 20742-4015, U.S.A.;(2) Department of Mathematics, Imperial College, London, SW7 2BZ, U.K. |
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Abstract: | We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2, ... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so-called hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g>0, we construct the unique solution of the Tsarev system, which matches the genus g+1 and g–1 solutions on the transition boundaries. |
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Keywords: | Whitham equations hyperelliptic Riemann surfaces linear overdetermined systems of Euler– Poisson Darboux type |
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