Gravity Waves on the Free Surface of an Incompressible Perfect Fluid of Finite Depth

We consider the nonstationary waves on the surface of an incompressible perfect fluid of finite depth above the almost horizontal bottom in the case of two dimensional irrotational motion. We assume that the density of mass is equal to one, the gravitational field to (0, — 1) and at the time f^O the fluid occupies the domain 0 ( t ) ^ { ( y l 9 y 2 ) \ y l e R l 9 -h + b(yl)^y2^fi(t9 y,)} where h is a positive constant. We denote by Fb the bottom y2= — /i + Kj'i) and by Fs the free surface y2 = rf(t, J'i)The motion of the fluid occupying at t = 0 the given domain Q is described by the velocity v = (vl9 v2)9 the pressure p of the fluid and rj satisfying the equations