  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.
