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Particular solutions of shallow-water equations over a non-flat surface. (English) Zbl 1115.76312
Summary: It is shown that the generalization of elementary solutions of the classical shallow water equations to the case of a non-flat surface is possible only for the class of underlying surfaces for which simple wave solutions exist, namely for slopes of constant inclination. The simple selfsimilar solutions for the shallow-water equations over slopes are obtained and the principal nonexistence of simple wave solutions was shown for other surfaces. It is shown that the characteristics of the equations over an oblique plane are branches of parabolas that have second-order contact with the characteristics of an appropriate system of shallow-water equations over the flat surface. As a consequence, shallow water physics on slopes is essentially different. The coordinate transformation that transforms the one-dimensional Saint-Venant system of equations to that for the classical shallow-water equations is found and sufficient conditions for the existence of this transformation are obtained.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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