2000/07/12 by Тамара Грава, T. Grava, Grava, T.
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0007016
Latex2e 41 pages, 5 figures
arxiv created 2000/07/12 · arxiv updated 2009/11/30
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≤ N, N>0, for all real t≥ 0 and x.