Geodesic
On Replacements Reducing One-Dimensional Systems of Shallow-Water Equations to the Wave Equation with Sound Speed $c^2=x$
Matematičeskie zametki, Tome 93 (2013) no. 5, pp. 716-727.

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We obtain point transformations for three one-dimensional systems: shallow-water equations on a flat and a sloping bottom and the system of linear equations obtained by formal linearization of shallow-water equations on a sloping bottom. The passage of these systems to the Carrier–Greenspan parametrization is also obtained. For linear shallow-water equations on a sloping bottom, we obtain the solution in the form of a traveling wave with variable velocity. We establish the relationship between the resulting solution and the solution of the two-dimensional wave equation.
Keywords: shallow-water equations on a flat and a sloping bottom, two-dimensional wave equation, self-similar solution, traveling-wave solution, Carrier–Greenspan parametrization
Mots-clés : point transformation, hodograph transformation, Jacobian. 
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S. Yu. Dobrokhotov; S. B. Medvedev; D. S. Minenkov. On Replacements Reducing One-Dimensional Systems of Shallow-Water Equations to the Wave Equation with Sound Speed $c^2=x$. Matematičeskie zametki, Tome 93 (2013) no. 5, pp. 716-727. http://geodesic.mathdoc.fr/item/MZM_2013_93_5_a5/

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