  We show that in one space dimension Lipschitz solutions of extremal surface
equations are equivalent to entropy solutions in $L^\infty(\R)$ of a
non-strictly hyperbolic system of conservation laws. We obtain an explicit
representation formula and the uniqueness of the entropy solutions to the
Cauchy problem of the system. By using this formula, we also obtain the
convergence and convergence rates as $t \rightarrow +\infty$ of the entropy
solutions to explicit traveling waves in the $L^1(\R)$ norm. Moreover, when
initial data are constants outside of a finite space interval, the entropy
solutions become the explicit traveling waves after a finite time. Finally, we
prove $L^1$ stabilities of the entropy solutions.
