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

TL;DR

This paper develops a method to solve the Whitham modulation equations for monotone initial data by leveraging solutions of the Tsarev system, enabling characterization of solutions with bounded genus over time.

Contribution

It constructs unique solutions to the Tsarev system for all genera, linking solutions across different genera and characterizing initial data for bounded genus solutions.

Findings

Constructed solutions for the Tsarev system for all g>0

Matched solutions across genus boundaries

Characterized initial data for bounded genus solutions

Abstract

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 $x$.