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

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 × H s H^{s+1}\times H^s , s > 1 2 s>\frac 12 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 by Kowalczyk, Martel, and the second author, as well as a Lyapunov functional that controls the H ˙ 1 × L 2 \dot H^1 \times L^2 energy.