Self-Similar Solutions of the Non-Strictly Hyperbolic Whitham Equations for the KdV Hierarchy

TL;DR

This paper investigates the self-similar solutions of the non-strictly hyperbolic Whitham equations associated with the entire KdV hierarchy, focusing on step function initial conditions.

Contribution

It provides a detailed analysis of self-similar solutions for the non-strictly hyperbolic Whitham equations across all higher order KdV equations, a novel extension beyond previous strictly hyperbolic cases.

Findings

Characterization of self-similar solutions for all higher order KdV Whitham equations

Analysis of solutions with step function initial data

Insights into the behavior of non-strictly hyperbolic systems

Abstract

We study the Whitham equations for all the higher order KdV equations. The Whitham equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solution of the Whitham equations when the initial values are given by a step function.