On interfacial waves over random topography

In this paper, we investigate effects of a weak and slowly varying random topography on interfacial long wave propagation near the critical depth level, where the cubic nonlinearity is comparable to the quadratic nonlinearity. The evolution equation is derived from the Euler equations for two fluid layers. This equation is completely integrable and can be transformed into the modified KdV equation. Hirota's method is used to find two-soliton and soliton-shock-like solutions. For a steady-wave propagating over a random topography with zero-mean Gaussian distribution, all its moments satisfy the same convection-diffusion equation. The randomness of the topography causes an averaged solitary wave to deform into a spreading Gaussian wavepacket with its height decreasing and width increasing at the same rate determined by the correlation function of the topography. The front of an averaged shock wave also increases at the same rate. Asymptotic behaviours of an averaged two-solitary wave and solitary-shock wave are also discussed and the results are generalized to an averaged N -solitary wave and N-solitary-shock wave.