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Quasi-periodic wave solutions with asymptotic analysis to the Saweda-Kotera-Kadomtsev-Petviashvili equation

Regular Article

Abstract

In this paper, the (2+1)-dimensional Saweda-Kotera-Kadomtsev-Petviashvili (SK-KP) equation is investigated, which can be used to describe certain situations from the fluid mechanics, ocean dynamics and plasma physics. With the aid of generalized Bell’s polynomials, the Hirota’s bilinear equation and N-soliton solution are explicitly constructed to the SK-KP equation, respectively. Based on the Riemann theta function, a direct and lucid way is presented to explicitly construct quasi-periodic wave solutions for the SK-KP equation. The two-periodic waves admit two independent spatial periods in two independent horizontal directions, which are a direct generalization of one-periodic waves. Finally, the relationships between soliton solutions and periodic wave solutions are strictly established, which implies the asymptotic behaviors of the periodic waves under a limited procedure.

References

  1. 1.
    G.W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer-Verlag, New York, 1989).Google Scholar
  2. 2.
    P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd edition (Springer, New York, 1993).Google Scholar
  3. 3.
    N.H. Ibragimov (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1 (CRC Press, Boca Raton, 1994).Google Scholar
  4. 4.
    V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons (Springer, 1991).Google Scholar
  5. 5.
    M.J. Ablowitz, P.A. Clarkson, Solitons: Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, 1991).Google Scholar
  6. 6.
    J.J.C. Nimmo, Darboux Transformations from Reductions of the KP Hierarchy (World Scientific, Singapore, 1995).Google Scholar
  7. 7.
    C. Rogers, W.K. Schisf, Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, in Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002).Google Scholar
  8. 8.
    R. Hirota, Direct Methods in Soliton Theory (Springer, 2004).Google Scholar
  9. 9.
    E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its, V.B. Matveev, Algebro-Geometric Aproach to Non-Linear Integrable Equations (Springer, 1994).Google Scholar
  10. 10.
    S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons: The Inverse Scattering Methods (Consultants Bureau, New York, 1984).Google Scholar
  11. 11.
    P.D. Lax, Commun. Pure Appl. Math. 28, 141 (1975).CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    H.P. Mckean, P. Moerbeke, Invent Math. 30, 217 (1975).CrossRefADSMathSciNetMATHGoogle Scholar
  13. 13.
    F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions (Cambridge University Press, 2003).Google Scholar
  14. 14.
    F. Gesztesy, H. Holden, J. Michor, G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions, Vol. II, $(1+1)$-Dimensional Discrete Models, in Cambridge Studies in Advanced Mathematics, Vol. 114 (Cambridge University Press, Cambridge, 2008).Google Scholar
  15. 15.
    Z.J. Qiao, Commun. Math. Phys. 239, 309 (2003).CrossRefADSMATHGoogle Scholar
  16. 16.
    E.G. Amosenok, A.O. Smirnov, Lett. Math. Phys. 96, 157 (2011).CrossRefADSMathSciNetMATHGoogle Scholar
  17. 17.
    C.W. Cao, Y.T. Wu, X.G. Geng, J. Math. Phys. 40, 3948 (1999).CrossRefADSMathSciNetMATHGoogle Scholar
  18. 18.
    Y.C. Hon, E.G. Fan, J. Math. Phys. 46, 032701 (2005).CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    A. Nakamura, J. Phys. Soc. Jpn 48, 1365 (1980).CrossRefADSGoogle Scholar
  20. 20.
    X.B. Hu, C.X. Li, J.J.C. Nimmo, G.F. Yu, J. Phys. A 38, 195 (2005).CrossRefADSMathSciNetMATHGoogle Scholar
  21. 21.
    E.G. Fan, Y.C. Hon, Phys. Rev. E 78, 036607 (2008).CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    E.G. Fan, J. Phys. A 42, 095206 (2009).CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    E.G. Fan, Y.C. Hon, Rep. Math. Phys. 66, 355 (2010).CrossRefADSMathSciNetMATHGoogle Scholar
  24. 24.
    W.X. Ma, R. Zhou, L. Gao, Mod. Phys. Lett. A 24, 1677 (2009).CrossRefADSMathSciNetMATHGoogle Scholar
  25. 25.
    W.X. Ma, Rep. Math. Phys. 72, 41 (2013).CrossRefADSMathSciNetMATHGoogle Scholar
  26. 26.
    K.W. Chow, J. Math. Phys. 36, 4125 (1995).CrossRefADSMathSciNetMATHGoogle Scholar
  27. 27.
    A.M. Wazwaz, Appl. Math. Comput. 149, 103 (2004).CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    A.M. Wazwaz, Partial Differential Equations: Methods and Applications (Balkema Publishers, The Netherlands, 2002).Google Scholar
  29. 29.
    S.F. Tian, P.L. Ma, Commun. Theor. Phys. 62, 245 (2014).CrossRefADSMathSciNetMATHGoogle Scholar
  30. 30.
    S.Y. Lou, Z. Naturforsch. 53a, 251 (1998).ADSGoogle Scholar
  31. 31.
    B. Tian, Y.T. Gao, Eur. Phys. J. B 30, 97 (1995).MathSciNetMATHGoogle Scholar
  32. 32.
    M. Eslami, M. Mirzazadeh, Eur. Phys. J. Plus 128, 140 (2013).CrossRefGoogle Scholar
  33. 33.
    M. Eslami, A. Neirameh, Eur. Phys. J. Plus 129, 54 (2014).CrossRefGoogle Scholar
  34. 34.
    Y. Chen, Z.Y. Yan, H.Q. Zhang, Theor. Math. Phys. 132, 970 (2002).CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Z.Y. Yan, Appl. Math. Comput. 168, 1065 (2005).CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    E. Tala-Tebue, D.C. Tsobgni-Fozap, A. Kenfack-Jiotsa, T.C. Kofane, Eur. Phys. J. Plus 129, 136 (2014).CrossRefGoogle Scholar
  37. 37.
    Y. Amadou, G. Betchewe, Douvagai, M. Justin, S.Y. Doka, K.T. Crepin, Eur. Phys. J. Plus 130, 13 (2015).CrossRefGoogle Scholar
  38. 38.
    S.F. Tian, H.Q. Zhang, J. Math. Anal. Appl. 371, 585 (2010).CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    S.F. Tian, H.Q. Zhang, J. Phys. A: Math. Theor. 45, 055203 (2012).CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    S.F. Tian, H.Q. Zhang, Commun. Nonlinear Sci. Numer. Simulat. 16, 173 (2011).CrossRefADSMathSciNetMATHGoogle Scholar
  41. 41.
    S.F. Tian, H.Q. Zhang, Chaos Solitons Fractals 47, 27 (2013).CrossRefADSMathSciNetMATHGoogle Scholar
  42. 42.
    S.F. Tian, H.Q. Zhang, Stud. Appl. Math. 132, 212 (2014).CrossRefMathSciNetMATHGoogle Scholar
  43. 43.
    E.T. Bell, Ann. Math. 35, 258 (1834).CrossRefGoogle Scholar
  44. 44.
    F. Lambert, I. Loris, J. Springael, Inverse Probl. 17, 1067 (2001).CrossRefADSMathSciNetMATHGoogle Scholar
  45. 45.
    C. Gilson, F. Lambert, J. Nimmo, R. Willox, Proc. R. Soc. London A 452, 223 (1996).CrossRefADSMathSciNetMATHGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.Center of Nonlinear EquationsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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