N ov 2 00 5 Analysis of singular solutions for two nonlinear wave equations

3 Distance defined by optimal transportation problem 33 3.1 A distance functional in the spatially periodic case. Introduction This thesis deals with two strongly nonlinear evolution Partial Differential Equation (in the following named P.D.E.) arising from mathematical physics. The first one was introduced first by Fokas and Fuchssteiner [28] as a bi-Hamiltonian equation, and then was rediscovered by R. Camassa and D.D. Holm [13] as an higher order level of approximation of the unidirectional shallow water wave equation than the Korteweg-de Vries equation [38]. It can be written as (1) here the unknown u(t, x) represents the water's free surface over a flat bed and κ is a constant related to the critical shallow-water wave speed (see also [37] for an alternative derivation as an hyperelastic-rod wave equation). We refer to this equation as to the Camassa-Holm equation, in honour to the first two authors which found a physical meaning stemming from the Euler equation. The second PDE we want to study is a system of hyperbolic equations with quadratic source       