Critical Steady Surface Waves of Idea Fluid over a Bump with Surface Tension

The paper deals with steady forced surface waves propagating on a two-dimensional incompressible and inviscid fluid with a small bump placed on a rigid flat bottom. If the surface tension coefficient T on the free surface is not zero and the wave is moving with a constant speed C, the wave motion is determined by two non-dimensional constants, F =√gh and τ = T/(ρgh2), where g is the gravity constant and h is the height of the fluid at infinity. It has been known that F = 1 and τ = 1/3 are the critical values of F and τ , respectively. In the critical case F = 1 + λ1ϵ2 and τ = 1/3 + τ1ϵ with ϵ > 0 a small parameter, a time-dependent forced Kawahara (F-Kawahara)equation is derived to model the wave propagation on the free surface and the steady F-Kawahara equation is studied both theoretically and merically. It is shown that the steady F-Kawahara equation has many different kinds of one and multi-hump solutions when τ1 and λ1 vary. In particular, for a fixed τ1, there is a λ0 < 0 such that if λ1 < λ0, two one-hump steady solutions can be obtained, one with small amplitude and the other with large amplitude. By using the unsteady F-Kawahara equation, it appears that the small one-hump solution is stable while the large one is nstable. In addition, two-hump solutions are unstable.