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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


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) in Google Scholar

[3] P. E. Hydon, Symmetry Methods for Differential Equations, (Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000) in Google Scholar

[4] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, (Dordrecht: Reidel, 1985) in Google Scholar

[5] F. Kangalgil, F. Ayaz, Phys. Lett. A 372, 1831 (2008) in Google Scholar

[6] R. Cimpoiasu, R. Constantinescu, J. Nonlin. Math. Phys. 13, 285 (2006) in Google Scholar

[7] C. Korögluand, T. Özis, Comput. Math. Appl. 58, 2142 (2009) in Google Scholar

[8] P. J. Olver, Application of Lie Groups to Differential Equations, (Springer, New-York, 1986) 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) 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) 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) in Google Scholar

[15] A. H. Kara, F. M. Mahomed, Int. J. Theor. Phys. 39, 23 (2000) in Google Scholar

[16] N. H. Ibragimov, J. Phys. A: Math. Theor. 44, 432002 (2011) in Google Scholar

[17] M. S. Bruzon, M. L. Gandarias, N. H. Ibragimov, J. Math. Anal. Appl. 357, 307 (2009) 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) in Google Scholar

[20] Y. Lee, L. M. Smith, Physica D 179, 53 (2003) in Google Scholar

[21] D. A. Siegel, Nature 409, 576 (2001) in Google Scholar PubMed

[22] D. Luo, F. Huang, Y. Diao, J. Geophys. Res. Atmos. 106, 31795 (2001) in Google Scholar

[23] J. Pedlosky, Geophysical Fluid Dynamics, (Springer, New-York, 1979) in Google Scholar

[24] W. I. Fushchych, W. M. Shtelen, S. L. Slavutsky, J. Phys. A: Math. Gen. 24, 971 (1991) in Google Scholar

[25] G. E. Swaters, Geophys. Astrophys. Fluid Dyn. 36, 85 (1986) in Google Scholar

[26] F. Huang, S. Y. Lou, Phys. Lett. A 320, 428 (2004) in Google Scholar

[27] R. Constantinescu, J. Math. Phys. 38, 2786 (1997) in Google Scholar

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

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