Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics

TL;DR

This paper demonstrates that small solutions to the 1+1 dimensional Born-Infeld equations decay in space within a subset of the light cone, despite the absence of global decay due to solutions traveling at light speed.

Contribution

It introduces a novel Virial identity and Lyapunov functional to prove spatial decay of small solutions in a nonlinear electromagnetism model.

Findings

Small solutions decay in space inside a proper subset of the light cone.

Constructed Virial identity related to a momentum law.

Developed a Lyapunov functional controlling the energy.

Abstract

We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well-known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small $H^{s+1}\times H^s$, $s>\frac12$ solutions do decay to the zero background state in space, inside a strictly proper subset of the light cone. We prove this result by constructing a Virial identity related to a momentum law, in the spirit of works \cite{KMM,KMM1}, as well as a Lyapunov functional that controls the $\dot H^1 \times L^2$ energy.