Solutions to initial boundary value problems are constructed for the shallow water equations in the form of series locally convergent in the neighbourhood of a movable water-land boundary for an arbitrary bottom relief. The motion law and the velocity of this boundary are determined for various wave-shore interaction modes. The obtained results of analytic study of the solutions are used for the development of approximations of boundary conditions on the movable shoreline. Test problems are numerically solved using an explicit predictor-corrector scheme of the second order of approximation on adaptive grids retracing the position of the water-land boundary. The results of these calculations are presented.

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The problems of modeling three-dimensional flows adjacent to vacuum were regarded earlier (see, for example, [1–5]). In works [1–3] onedimensional and multi-dimensional flows of polytropic and normal gas adjacent to vacuum were investigated. In works [4, 5] symmetric swirl upward flows of polytropic gas around of the vertical located contact characteristic separating the gas and vacuum were considered. It is shown that even in case the gas abuts vacuum, it swirls counterclockwise. It is also found that the vortex itself moves westward, shifting slightly northward. The present paper considers the evolution of the asymmetric gas flow at the initial time continuously adjacent to vacuum. An equation system of gas dynamics under the action of gravity and Coriolis force is adopted as a mathematical model. For the equation system of gas dynamics the initial boundary value problem is set on the multiplicity characteristic of four. The solution to the problem is created in the form of a power series and the existence and uniqueness theorem of the solution around the free surface «gas-vacuum» is proved.