Weakly nonlinear geometric optics for hyperbolic systems of conservation laws with shock waves

We consider in this paper a strictly hyperbolic system of conservation laws, in one space dimension. We suppose that a shock wave solution to this system is given and superimpose to it small amplitude fast oscillations. These oscillations are described by sets of phase functions, and resonances are taken into account. We justify the asymptotic expansions given by the weakly non-linear geometric optics: the oscillating part of the perturbed solution is approximated by an almost periodic function (a profile) which is solution to a non-linear integro-differential mixed problem. We give at the same time an asymptotics to the shock front.