Integrable systems in the description of waves in straits and channels of variable width and depth

A theoretical model to explain data about the initiation and propagation of solitary waves in presented. Starting with the Euler equation for frictionless incompressible homogeneous fluids, and using an approximation considering almost one dimensional long waves and shallow water, a partial differential nonlinear equation of the Kadomstev-Petviashivili type is obtained, characterized by variable coefficients and boundary conditions. Once the vorticity is specified, and for given contour conditions, the equation can be exactly integrated.