From the solution of the Tsarev system to the solution of the Whitham equations

Details From the solution of the Tsarev system to the solution of the Whitham equations From the solution of the Tsarev system to the solution of the Whitham equations

2000-07-12

arxiv.org/abs/nlin/0007016v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Latex2e 41 pages, 5 figures

Scientific paper

We study the Cauchy problem for the Whitham modulation equations for monotone 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 transform 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. Next we characterize initial data such that the solution of the Whitham equations has genus $g\leq N$, $N>0$, for all real $t\geq 0$ and $x$.

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