Shallow Water Waves Propagating along Undulation of Bottom Surface

The effect of undulated bottom on the shallow water waves is studied on the assumption that change in depth is sufficiently small and its characteristic length is much larger than the wave length. For small amplitude waves, an eigenvalue equation is obtained to give the conditions for appearance of trapped mode. The system of equations governing finite amplitude waves is derived. It is reduced to the modified Korteweg-de Vries equation with two additional terms. The stability of soliton for two dimensional perturbation is also studied.