Some Distributional Solutions of the CH, DP and CH2 Equations and the Lax Pair Formalism

by Mohajer, Keivan

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.

We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.