On the formation of shocks of electromagnetic plane waves in non-linear crystals

TL;DR

This paper revisits classical results on shock formation in nonlinear hyperbolic PDEs, providing detailed behavior descriptions and bounds, with applications to electromagnetic waves in nonlinear crystals.

Contribution

It offers a refined analysis of shock formation, including bounds and behavior of solutions, specifically applied to electromagnetic plane waves in nonlinear crystals.

Findings

Boundedness of John's singular quantity until shock time

Inverse density of characteristics tends to zero at shock

Established upper and lower bounds for shock formation time

Abstract

An influential result of F. John states that no genuinely non-linear strictly hyperbolic quasi-linear first order system of partial differential equations in two variables has a global $C^2$-solution for small enough initial data. Inspired by recent work of D. Christodoulou, we revisit John's original proof and extract a more precise description of the behaviour of solutions at the time of shock. We show that John's singular first order quantity, when expressed in characteristic coordinates, remains bounded until the final time, which is then characterised by an inverse density of characteristics tending to zero in one point. Moreover, we study the derivatives of second order, showing again their boundedness when expressed in appropriate coordinates. We also recover John's upper bound for the time of shock formation and complement it with a lower bound. Finally, we apply these results to electromagnetic plane waves in a crystal with no magnetic properties and cubic electric non-linearity in the energy density, assuming no dispersion.