Integrable Nonlinear Equations for Water Waves in Straits of Varying Depth and Width

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one-dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three-dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi-one-dimensional long-wave and shallow-water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in the straits.

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