Wave front tracing and asymptotic stability of planar travelling waves for a two-dimensional shallow river model

The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcyʹs law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations.