Polarized high-frequency wave propagation beyond the nonlinear Schrödinger approximation

Abstract: This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein--Gordon equations and the Maxwell--Lorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear Schr\"odinger equation. The given construction of polarized solutions is explicit, uses in addition a linear Schr\"odinger equation for each further order of approximation, and is accessible to direct numerical approximation.