  We study an integro-differential equation that describes the slow erosion of
granular flow. The equation is a first order non-linear conservation law where
the flux function includes an integral term. We show that there exist unique
traveling wave solutions that connect profiles with equilibrium slope at
$\pm\infty$. Such traveling waves take very different forms from those in
standard conservation laws. Furthermore, we prove that the traveling wave
profiles are locally stable, i.e., solutions with monotone initial data
approaches the traveling waves asymptotically as $t\to+\infty$.
