Theoretical and Mathematical Physics

, Volume 151, Issue 3, pp 752–761 | Cite as

New solutions of the Schwarzian Korteweg-de Vries equation in 2+1 dimensions based on weak symmetries

  • M. L. Gandarias
  • M. S. Bruzón
Article

Abstract

We consider the (2+1)-dimensional integrable Schwarzian Korteweg-de Vries equation. Using weak symmetries, we obtain a system of partial differential equations in 1+1 dimensions. Further reductions lead to second-order ordinary differential equations that provide new solutions expressible in terms of known functions. These solutions depend on two arbitrary functions and one arbitrary solution of the Riemann wave equation and cannot be obtained by classical or nonclassical symmetries. Some of the obtained solutions of the Schwarzian Korteweg-de Vries equation exhibit a wide variety of qualitative behaviors; traveling waves and soliton solutions are among the most interesting.

Keywords

weak symmetry partial differential equation solitary wave 

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References

  1. 1.
    E. Hille, Analytic Function Theory, Vol. 2, Ginn, Boston (1962); H. Schwerdtfeger, Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry, Dover, New York (1979).Google Scholar
  2. 2.
    I. M. Krichever and S. P. Novikov, Russ. Math. Surveys, 35, No. 6, 53 (1980).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Weiss, J. Math. Phys., 24, 1405 (1983).MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    N. Kudryashov and P. Pickering, J. Phys. A, 31, 9505 (1998).MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    K. Toda and S. Yu, J. Math. Phys., 41, 4747 (2000).MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    J. Weiss, J. M. Tabor, and G. Carnevale, J. Math. Phys., 24, 522 (1983).MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    P. J. Olver and E. M. Vorob’ev, CRC Handbook of Lie Group Analysis of Differential Equations, (N. H. Ibragimov, ed.), Vol. 3, Symmetries, Exact Solutions, and Conservation Laws, CRC, Boca Raton, Fl. (1994).Google Scholar
  8. 8.
    P. A. Clarkson, Chaos Solitons Fractals, 5, 2261 (1995).CrossRefMathSciNetGoogle Scholar
  9. 9.
    P. J. Olver and P. Rosenau, Phys. Lett. A, 144, 107 (1986).CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    P. J. Olver and P. Rosenau, SIAM J. Appl. Math., 47, 263 (1987).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. M. Vorob’ev, J. Nonlinear Math. Phys., 3, 330 (1996).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. V. Dzhamay and E. M. Vorob’ev, J. Phys. A, 27, 5541 (1994).MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    G. Saccomandi, Note Mat., 23, No. 2, 217 (2004).MathSciNetGoogle Scholar
  14. 14.
    G. Cicogna, Note Mat., 23, No. 2, 15 (2005).MathSciNetGoogle Scholar
  15. 15.
    J. Ramirez, M. S. Bruzon, C. Muriel, and M. L. Gandarias, J. Phys. A, 36, 1467 (2003).MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    J. Ramírez, J. L. Romero, M. S. Bruzón, and M. L. Gandarias, Chaos Solitons Fractals, 32, 682 (2007).MathSciNetGoogle Scholar
  17. 17.
    G. W. Bluman and J. D. Cole, J. Math. Mech., 18, 1025 (1969).MATHMathSciNetGoogle Scholar
  18. 18.
    B. Champagne, W. Hereman, and P. Winternitz, Comput. Phys. Comm., 66, 319 (1991).MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • M. L. Gandarias
    • 1
  • M. S. Bruzón
    • 1
  1. 1.Departamento de MatemáticasUniversidad de CádizPuerto Real, CádizSpain

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