Resonant One Dimensional Nonlinear Geometric Optics

In this paper, we study the existence and resonant interaction of oscillatory wave trains in one space dimension, giving a rigorous proof of the validity of the corresponding expansions of weakly nonlinear optics. We consider both semilinear and quasilinear systems, the latter before shock formation. Some important features of the study are the following. (1) We prove the existence of families of exact solutions which have asymptotic expansions governed by weakly nonlinear optics. Equations with variable coefficients, nonconstant background fields and nonlinear phases are permitted. Our weak transversality hypotheses allow us to justify expansions where even a formal theory did not exist before. (2) We make a detailed study of resonances. The geometry associated with such resonances is related to the theory of planar webs. (3) We study the smoothness of the profiles. Their regularity is ruled by a sum law analogous to that describing the propagation of singularities in one dimension. (4) The expansions are justified up to the breakdown of the profiles which coincides with a suitably defined breakdown for exact solutions.