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 TMF, 2014, Volume 178, Number 3, Pages 322–345 (Mi tmf8607)

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

Yu. A. Chirkunovab, S. Yu. Dobrokhotovcd, S. B. Medvedeveb, D. S. Minenkovcd

a Novosibirsk State Technical University, Novosibirsk, Russia
b Institute for Computation Technologies, Siberian Branch, RAS, Novosibirsk, Russia
c Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia
d Moscow Institute for Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
e Lomonosov Moscow State University, Moscow, Russia

Abstract: We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on $21$ arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.

Keywords: shallow water equation, Euler–Poisson–Darboux equation, exact solution, invariant solution, symmetry group

DOI: https://doi.org/10.4213/tmf8607

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English version:
Theoretical and Mathematical Physics, 2014, 178:3, 278–298

Bibliographic databases:

Citation: Yu. A. Chirkunov, S. Yu. Dobrokhotov, S. B. Medvedev, D. S. Minenkov, “Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms”, TMF, 178:3 (2014), 322–345; Theoret. and Math. Phys., 178:3 (2014), 278–298

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8607
• https://doi.org/10.4213/tmf8607
• http://mi.mathnet.ru/eng/tmf/v178/i3/p322

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. P. Siriwat, Ch. Kaewmanee, S. V. Meleshko, “Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates”, Int. J. Non-Linear Mech., 86 (2016), 185–195
2. Yu. A. Chirkunov, “Conformal invariance and new exact solutions of the elastostatics equations”, J. Math. Phys., 58:3 (2017), 031502
3. A. V. Aksenov, S. Yu. Dobrokhotov, K. P. Druzhkov, “Exact Step-Like Solutions of One-Dimensional Shallow-Water Equations over a Sloping Bottom”, Math. Notes, 104:6 (2018), 915–921
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