Self-Similar Solutions of the Non-Strictly Hyperbolic Whitham Equations
  We study the Whitham equations for the fifth order KdV equation. The 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. We classify the step like initial data into eight different types. We construct self-similar solutions for each type. 
V. U. Pierce and Fei-Ran Tian