Abstract
A point transformation determining the equivalence of systems of equations of two-dimensional shallow water over horizontal and slo** bottoms is obtained. The symmetries of these systems of equations are found.
REFERENCES
J. J. Stoker, Water Waves: The Mathematical Theory with Applications (Interscience, New York, 1957).
L. V. Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978).
G. F. Carrier and H. P. Greenspan, ‘‘Water waves of finite amplitude on a slo** beach,’’ J. Fluid Mech. 4, 97–109 (1958). https://doi.org/10.1017/s0022112058000331
E. O. Tuck and L.-S. Hwang, ‘‘Long wave generation on a slo** beach,’’ J. Fluid Mech. 51, 449–461 (1972). https://doi.org/10.1017/S0022112072002289
E. N. Pelinovsky and R. Kh. Mazova, ‘‘Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles,’’ Nat. Hazards 6, 227–249 (1992). https://doi.org/10.1007/bf00129510
S. Yu. Dobrokhotov and B. Tirozzi, ‘‘Localized solutions of one-dimensional non-linear shallow-water equations with velocity \(c=\sqrt{x}\),’’ Russ. Math. Surv. 65, 177–179 (2010). https://doi.org/10.1070/RM2010v065n01ABEH004668
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Publisher’s Note.
Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Aksenov, A.V. Equivalence of Systems of Two-Dimensional Shallow-Water Equations over Horizontal and Slo** Bottom. Moscow Univ. Mech. Bull. 78, 156–158 (2023). https://doi.org/10.3103/S002713302306002X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S002713302306002X