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. 
@article{MZM_2013_93_5_a5,
     author = {S. Yu. Dobrokhotov and S. B. Medvedev and D. S. Minenkov},
     title = {On {Replacements} {Reducing} {One-Dimensional} {Systems} of {Shallow-Water} {Equations} to the {Wave} {Equation} with {Sound} {Speed} $c^2=x$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {716--727},
     publisher = {mathdoc},
     volume = {93},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_5_a5/}
}
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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/