Diffractive Nonlinear Geometric Optics for Short Pulses

Abstract

This paper considers the behavior of pulse-like solutions of length $\varepsilon \ll 1$ to semilinear systems of hyperbolic partial differential equations on the time scale $t=O(1/\eps)$ of diffractive geometric optics. The amplitude is chosen so that nonlinear effects influence the leading term in the asymptotics. For pulses of larger amplitude so that the nonlinear effects are pertinent for times t=O(1), accurate asymptotic solutions lead to transport equations similar to those valid in the case of wave trains (see [D. Alterman and J. Rauch, J. Differential Equations, {178} (2002), pp. 437--465]). The opposite is true here. The profile equation for pulses for $t=O(1/\eps)$ is different from the corresponding equation for wave trains. Formal asymptotics leads to equations for a leading term in the expansion and for correctors. The equations for the correctors are in general not solvable, being plagued by small divisor problems in the continuous spectrum. This makes the construction of accurate approximations subtle. We use low-frequency cutoffs depending on $\eps$ to avoid the small divisors.