×

Symmetries and exact solutions of the rotating shallow-water equations. (English) Zbl 1181.35185

Summary: Lie symmetry analysis is applied to study the nonlinear 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
76E07 Rotation in hydrodynamic stability
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
35C05 Solutions to PDEs in closed form
35B10 Periodic solutions to PDEs

Software:

SYMMGRP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1017/S0022112081001882 · Zbl 0462.76023 · doi:10.1017/S0022112081001882
[2] DOI: 10.1017/S0022112063001270 · Zbl 0122.21202 · doi:10.1017/S0022112063001270
[3] Rogers, Nonlinear Boundary Value Problems in Science and Engineering (1989) · Zbl 0686.35001
[4] Pedlosky, Geophysical Fluid Dynamics (1979) · doi:10.1007/978-1-4684-0071-7
[5] Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean (2003) · Zbl 1278.76004 · doi:10.1090/cln/009
[6] Pavlenko, Siberian Electron. Math. Rep. 2 pp 291– (2005)
[7] Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 2: Applications in Engineering and Physical Sciences pp xix– (1995) · Zbl 0864.35002
[8] DOI: 10.1063/1.523237 · Zbl 0372.22008 · doi:10.1063/1.523237
[9] DOI: 10.1016/S0895-7177(97)00063-0 · Zbl 0898.34002 · doi:10.1016/S0895-7177(97)00063-0
[10] Ovsyannikov, Dokl. Akad. Nauk. 333 pp 702– (1993)
[11] Gill, Atmosphere–Ocean Dynamics (1982)
[12] Ovsyannikov, Group Analysis of Differential Equations (1982)
[13] DOI: 10.1063/1.528613 · Zbl 0698.35137 · doi:10.1063/1.528613
[14] Olver, Applications of Lie Groups to Differential Equations (1993) · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2
[15] DOI: 10.1006/jsco.1999.0299 · Zbl 0958.68543 · doi:10.1006/jsco.1999.0299
[16] DOI: 10.1093/qjmam/hbi033 · Zbl 1088.76059 · doi:10.1093/qjmam/hbi033
[17] DOI: 10.1017/S0022112065000952 · Zbl 0131.23701 · doi:10.1017/S0022112065000952
[18] DOI: 10.1016/0960-0779(95)00068-2 · Zbl 1080.86503 · doi:10.1016/0960-0779(95)00068-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.