Solitons and antisolitons on bounded surfaces

We generalize the one-dimensional KdV equation for an inviscid incompressible irrotational fluid layer with free surface, finite depth, and finite boundary conditions. We study the nonlinear dynamics of a fluid of arbitrary depth in a bounded domain. By introducing a special relation between the asymptotic limit of the potential of velocities at the bottom and the surface equation we obtain an infinite order PDE. The dispersion relation of the linearized equation is the well known capillarity-gravity dispersion relation for arbitrary depth. This generalized equation can be written as a differential-difference expression, and a class of traveling waves solutions in terms of power series expansion with coefficients satisfying a nonlinear recursion relation is obtained. In the limit of infinite long shallow water we recover the KdV equation together with its one-soliton solution. This generalized equation provides higher order (nonlinear) dispersion terms that do not cancel in the limit B0=1/3. Consequently this equation can be used for investigation of soliton-antisoliton transition when the depth of the layer is closed to the critical one.