Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access February 15, 2014

Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation

  • Rodica Cimpoiasu EMAIL logo and Radu Constantinescu
From the journal Open Physics

Abstract

The paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.

[1] W. F. Ames, Nonlinear Partial Differential Equations in Engineering II, (Academic Press, New York, 1972) Search in Google Scholar

[2] G. W. Bluman, S. Kumei, Symmetries and Differential Equations, (Springer-Verlag, New York, 1989) http://dx.doi.org/10.1007/978-1-4757-4307-410.1007/978-1-4757-4307-4Search in Google Scholar

[3] P. E. Hydon, Symmetry Methods for Differential Equations, (Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000) http://dx.doi.org/10.1017/CBO978051162396710.1017/CBO9780511623967Search in Google Scholar

[4] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, (Dordrecht: Reidel, 1985) http://dx.doi.org/10.1007/978-94-009-5243-010.1007/978-94-009-5243-0Search in Google Scholar

[5] F. Kangalgil, F. Ayaz, Phys. Lett. A 372, 1831 (2008) http://dx.doi.org/10.1016/j.physleta.2007.10.04510.1016/j.physleta.2007.10.045Search in Google Scholar

[6] R. Cimpoiasu, R. Constantinescu, J. Nonlin. Math. Phys. 13, 285 (2006) http://dx.doi.org/10.2991/jnmp.2006.13.2.1010.2991/jnmp.2006.13.2.10Search in Google Scholar

[7] C. Korögluand, T. Özis, Comput. Math. Appl. 58, 2142 (2009) http://dx.doi.org/10.1016/j.camwa.2009.03.02810.1016/j.camwa.2009.03.028Search in Google Scholar

[8] P. J. Olver, Application of Lie Groups to Differential Equations, (Springer, New-York, 1986) http://dx.doi.org/10.1007/978-1-4684-0274-210.1007/978-1-4684-0274-2Search in Google Scholar

[9] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, Dokl. Akad. nauk SSSR 295, 288 (1987) Search in Google Scholar

[10] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, Commun. Math. Phys. 115, 1 (1988) http://dx.doi.org/10.1007/BF0123885010.1007/BF01238850Search in Google Scholar

[11] Y. Kodama, A. V. Mikhailov, Obstacles to Asymptotic Integrability in Algebraic aspects of Integrability, (Birkhauser, Boston Basel, Berlin, 1996) 10.1007/978-1-4612-2434-1_9Search in Google Scholar

[12] R. J. LeVeque, Numerical Methods for Conservation Laws, (Lectures in Mathematics: ETH Zurich, Birkhauser Verlag, 1992) http://dx.doi.org/10.1007/978-3-0348-8629-110.1007/978-3-0348-8629-1Search in Google Scholar

[13] B. Cockburn, S. Hou, C. Wang Shu, Math. Comput. 54, 545 (1990) 10.2307/2008501Search in Google Scholar

[14] S. C. Anco, G. Bluman, J. Math. Phys. 37, 2361 (1996) http://dx.doi.org/10.1063/1.53151510.1063/1.531515Search in Google Scholar

[15] A. H. Kara, F. M. Mahomed, Int. J. Theor. Phys. 39, 23 (2000) http://dx.doi.org/10.1023/A:100368683152310.1023/A:1003686831523Search in Google Scholar

[16] N. H. Ibragimov, J. Phys. A: Math. Theor. 44, 432002 (2011) http://dx.doi.org/10.1088/1751-8113/44/43/43200210.1088/1751-8113/44/43/432002Search in Google Scholar

[17] M. S. Bruzon, M. L. Gandarias, N. H. Ibragimov, J. Math. Anal. Appl. 357, 307 (2009) http://dx.doi.org/10.1016/j.jmaa.2009.04.02810.1016/j.jmaa.2009.04.028Search in Google Scholar

[18] N. H. Ibragimov, Archives of ALGA 7/8, 1 (2011) Search in Google Scholar

[19] M. Torrisi, R. Tracina, Nonlinear Anal-Theor 14, 1496 (2013) http://dx.doi.org/10.1016/j.nonrwa.2012.10.01310.1016/j.nonrwa.2012.10.013Search in Google Scholar

[20] Y. Lee, L. M. Smith, Physica D 179, 53 (2003) http://dx.doi.org/10.1016/S0167-2789(03)00010-110.1016/S0167-2789(03)00010-1Search in Google Scholar

[21] D. A. Siegel, Nature 409, 576 (2001) http://dx.doi.org/10.1038/3505465910.1038/35054659Search in Google Scholar PubMed

[22] D. Luo, F. Huang, Y. Diao, J. Geophys. Res. Atmos. 106, 31795 (2001) http://dx.doi.org/10.1029/2000JD00008610.1029/2000JD000086Search in Google Scholar

[23] J. Pedlosky, Geophysical Fluid Dynamics, (Springer, New-York, 1979) http://dx.doi.org/10.1007/978-1-4684-0071-710.1007/978-1-4684-0071-7Search in Google Scholar

[24] W. I. Fushchych, W. M. Shtelen, S. L. Slavutsky, J. Phys. A: Math. Gen. 24, 971 (1991) http://dx.doi.org/10.1088/0305-4470/24/5/01210.1088/0305-4470/24/5/012Search in Google Scholar

[25] G. E. Swaters, Geophys. Astrophys. Fluid Dyn. 36, 85 (1986) http://dx.doi.org/10.1080/0309192860820879810.1080/03091928608208798Search in Google Scholar

[26] F. Huang, S. Y. Lou, Phys. Lett. A 320, 428 (2004) http://dx.doi.org/10.1016/j.physleta.2003.11.05610.1016/j.physleta.2003.11.056Search in Google Scholar

[27] R. Constantinescu, J. Math. Phys. 38, 2786 (1997) http://dx.doi.org/10.1063/1.53201910.1063/1.532019Search in Google Scholar

Published Online: 2014-2-15
Published in Print: 2014-2-1

© 2014 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.2.2024 from https://www.degruyter.com/document/doi/10.2478/s11534-014-0430-6/html
Scroll to top button