Trapped supercritical waves for the forced KdV equation with two bumps

The forced Korteweg–de Vries equation (fKdV) is considered to study free surface flows in a two-dimensional channel over an obstacle. The forcing is due to a bottom topography, where we consider one bump and two bumps. In the study of the wave motion of the free surface, there is an important parameter value, the Froude number (F  ), the ratio of the upstream velocity to the critical speed of shallow water wave. Our focus is on the supercritical case (F>1F>1), where solitary wave solutions can be observed. Interesting surface wave phenomena can be observed when a bottom topography is rather complicated. First, we revisit various stationary wave solutions of the forced KdV equation in the presence of two bumps. There are multiple trapped supercritical wave solutions between two bumps. Their numerical stability is investigated under different bump scenarios. Our numerical results show that the interplay between trapped solitary waves and two bumps plays a key role to determine the time evolution of those wave solutions. Interestingly, there are multiple trapped supercritical wave solutions, which are trapped between two bumps up to a certain time.