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Symmetries and exact solutions of the rotating shallow-water equations

Published online by Cambridge University Press:  14 August 2009

A. A. CHESNOKOV
A. A. CHESNOKOV*
Affiliation:
Lavrentyev Institute of Hydrodynamics and Novosibirsk State University, Novosibirsk 630090, Russia email: chesnokov@hydro.nsc.ru
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Abstract

Lie symmetry analysis is applied to study the non-linear rotating shallow-water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow-water equations can be transformed to the classical shallow-water model. The derived symmetries are used to generate new exact solutions of the rotating shallow-water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 5 , October 2009 , pp. 461 - 477
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Ball, F. K. (1963) Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 17, 240–256.CrossRefGoogle Scholar
[2]Ball, F. K. (1965) The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J. Fluid Mech. 22, 529–545.CrossRefGoogle Scholar
[3]Bila, N., Mansfield, E. & Clarkson, P. (2006) Symmetry group analysis of the shallow water and semi-geostrophic equations. Quart. J. Mech. Appl. Math. 59, 95–123.CrossRefGoogle Scholar
[4]Carminati, J. & Vu, K. (2000) Symbolic computation and differential equations: Lie symmetries. J. Symb. Comput. 29, 95–116.CrossRefGoogle Scholar
[5]Clarkson, P. A. & Kruskal, M. D. (1989) New similarity solutions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213.CrossRefGoogle Scholar
[6]Gill, A. E. (1982) Atmosphere–Ocean Dynamics, Academic Press, New York.Google Scholar
[7]Hereman, W. (1997) Review of symbolic software for Lie symmetry analysis. Math. Comp. Model. 25, 115–132.CrossRefGoogle Scholar
[8]Ibragimov, N. H (editor) (1995) CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2: Applications in Engineering and Physical Sciences, CRC Press, Boca Raton, FL, pp. xix.Google Scholar
[9]Majda, A. (2003) Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Institute of Mathematical Sciences, New York.CrossRefGoogle Scholar
[10]Olver, P. J. (1993) Applications of Lie Groups to Differential Equations, Springer, New York.CrossRefGoogle Scholar
[11]Ovsyannikov, L. V. (1982) Group Analysis of Differential Equations, Academic Press, New York.Google Scholar
[12]Ovsyannikov, L. V. (1993) Optimal systems of subalgebras Dokl. Akad. Nauk. 333, 702–704.Google Scholar
[13]Patera, J., Sharp, R. T., Winternitz, P. & Zassenhaus, H. (1977) Continuous subgroups of the fundamental groups of physics. Part III. The De Sitter groups. J. Math. Phys. 18, 2259–2288.CrossRefGoogle Scholar
[14]Pavlenko, A. S. (2005) Symmetries and solutions of equations of two-dimensional motions of politropic gas. Siberian Electron. Math. Rep. 2, 291–307. URL: http://semr.math.nsc.ru/v2/p291-307.pdfGoogle Scholar
[15]Pedlosky, J. (1979) Geophysical Fluid Dynamics, Springer, New York.CrossRefGoogle Scholar
[16]Rogers, C. & Ames, W. F. (1989) Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, New York.Google Scholar
[17]Sachdev, P. L., Palaniappan, D. & Sarathy, R. (1996) Regular and chaotic flows in paraboloid basin and eddies. Chaos Solit. Fract. 7, 383–408.CrossRefGoogle Scholar
[18]Thacker, W. C. (1981) Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499–508.CrossRefGoogle Scholar