Propagation of nonlinear pulses near diffractive points of any order

TL;DR

This paper develops a method to construct approximate solutions for nonlinear hyperbolic equations near diffractive points of any order, extending geometric optics to low regularity and high frequency regimes.

Contribution

It introduces a novel approach to approximate pulse solutions near diffractive points of arbitrary order, with new low-regularity estimates and broader applicability than prior geometric optics results.

Findings

Constructed pulse-type solutions near diffractive points of any order.

Established low-regularity estimates for pulse and profile sizes.

Extended geometric optics to low regularity and high frequency limits.

Abstract

We construct pulse-type approximate solutions to nonlinear hyperbolic equations near diffractive points, allowing arbitrary (even infinite) order of grazing. We show that in low regularity spaces and the high frequency limit, such solutions can be approximated by a sum of incoming and reflected pulses constructed using incoming and reflected phases and profiles that satisfy transport equations. New low-regularity estimates comparing the size of pulses to the size of their profiles are required. Earlier geometric optics results for pulses assumed much higher regularity, and considered only propagation in free space or transversal reflection at boundaries.