Spatial Stationary Long Waves in Shear Flows

The system of integrodifferential equations describing the spatial stationary free-boundary shear flows of an ideal fluid in the shallow-water approximation is considered. The generalized characteristics of the model are found and the hyperbolicity conditions are formulated. A new class of exact solutions of the governing equations is obtained which is characterized by a special dependence of the desired functions on the vertical coordinate. The system of equations describing this class of solutions in the hyperbolic case is reduced to Riemann invariants. New exact solutions of the equations of motion are found.