Home Lie Symmetries and Exact Solutions of Shallow Water Equations with Variable Bottom
Article
Licensed
Unlicensed Requires Authentication

Lie Symmetries and Exact Solutions of Shallow Water Equations with Variable Bottom

  • Manoj Pandey EMAIL logo
Published/Copyright: November 6, 2015

Abstract

In the present paper, Lie symmetries of nonlinear shallow water equations with variable shapes of the bottom that include horizontal, inclined plane and a parabolic bottom are obtained. Exact particular solutions of the governing system are then obtained using the invariance of the system under these symmetries using Lie’s method. The evolutionary behaviour of the C1 discontinuity wave, influenced by the amplitude of the discontinuity wave and the geometry of the bottom, is discussed in detail and some contrasting observations are made.

MSC® (2010): 35Q53 (35A30; 35C05; 76B15)

Acknowledgements

The author thanks one of the referees for the valuable suggestion and for making certain point more explicit.

References

[1] C. Zoppou and S. Roberts, Catastrophic collapse of water supply reservoirs in urban areas, J. Hydraul. Eng. 125 (1999), 686–695.10.1061/(ASCE)0733-9429(1999)125:7(686)Search in Google Scholar

[2] H. J. M. Ogink, J. G. Grijsen and A. J. H. Wijbenga, Aspects of flood level computations, in: International Symposium on Flood Frequency and Risk Analysis, May 14–17 Baton Rouge, USA, 1986.10.1007/978-94-009-3957-8_11Search in Google Scholar

[3] C. Synolakis, E. Okal and E. Bernard, The megatsunami of December 26, 2004, The Bridge, National Academy of Engineering, 35 (2), 26–35.Search in Google Scholar

[4] P. Watts, S. T. Grilli, J. T. Kirby, G. J. Fryer and D. R. Tappin, Landslide tsunami case studies using a Boussinesq model and a fully nonlinear Tsunami generation model. Nat. Hazards Earth Syst. Sci. 3 (2003), 391–402.10.5194/nhess-3-391-2003Search in Google Scholar

[5] G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer, Berlin, 1974.10.1007/978-1-4612-6394-4Search in Google Scholar

[6] G. W. Bluman and S. Kumei, Symmetries and differential equations, Springer, New York, 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[7] A. Donato and F. Oliveri, When non-autonomous equations are equivalent to autonomous ones, Appl. Anal. 58 (1995), 313–323.10.1080/00036819508840379Search in Google Scholar

[8] M. Pandey and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas, Wave Motion 44 (2007), 346–354.10.1016/j.wavemoti.2006.12.002Search in Google Scholar

[9] M. Pandey, R. Radha and V. D. Sharma, Symmetry analysis and exact solutions of magneto-gas-dynamic equations, Q. J. Mech. Appl. Math. 61 (2008), 291–310.10.1093/qjmam/hbn011Search in Google Scholar

[10] M. Pandey, Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas, Z. Angew. Math. Phys. 61 (2010), 87–94.10.1007/s00033-009-0030-2Search in Google Scholar

[11] W. Thacker, Some exact solutions to the nonlinear shallow water wave equations, J. Fluid Mech. 107 (1981), 499–508.10.1017/S0022112081001882Search in Google Scholar

[12] K. K. Ghosh and L. Debnath, Some exact solutions of non-linear shallow water equations, Int. J. Nonlinear Mech. 32 (1997), 633–636.10.1016/S0020-7462(96)00072-8Search in Google Scholar

[13] J. Jena, Group theoretic method for analyzing interaction of a weak discontinuity wave with a bore in shallow water waves, Appl. Anal. 84 (2005), 37–48.10.1080/00036810412331297226Search in Google Scholar

[14] T. Raja Sekhar and V. D. Sharma, Evolution of weak discontinuities in shallow water equations, Appl. Math. Lett. 23 (2010), 327–330.10.1016/j.aml.2009.10.003Search in Google Scholar

[15] S. Szatmari and A. Bihlo, Symmetry analysis of a system of modified shallow-water equations, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 530–537.10.1016/j.cnsns.2013.06.030Search in Google Scholar

[16] Y. Akyildiz, The shallow water equations: explicit solutions and superposition principle, J. Phys. A Math. Gen. 16 (1983), 2133–2136.10.1088/0305-4470/16/10/009Search in Google Scholar

[17] A. Jeffrey, Quasilinear hyperbolic systems and waves, Pitman, London, 1976.Search in Google Scholar

[18] V. D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, CRC Press, London, 2010.10.1201/9781439836910Search in Google Scholar

[19] A. Mentrelli, T. Ruggeri, M. Sugiyama and N. Zhao, Interaction between a shock and an acceleration wave in a prefect gas for increasing shock strength, Wave Motion. 45 (2008), 498–517.10.1016/j.wavemoti.2007.09.005Search in Google Scholar

[20] R. Radha and V. D. Sharma, Interaction of weak discontinuity with elementary waves of Riemann problem, J. Math. Phys. 53 (2012), 013506, 12 pp.10.1063/1.3671383Search in Google Scholar

Received: 2015-2-10
Accepted: 2015-10-20
Published Online: 2015-11-6
Published in Print: 2015-12-1

©2015 by De Gruyter

Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2015-0093/html
Scroll to top button