Skip to main content
SpringerLink
Log in

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (1+1)-dimensional and higher dimensional systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ablowitz, M.J., Clarkson, P.A. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge, 1991

    Book  MATH  Google Scholar 

  2. Bluman, G.W., Cole J.D. The general similarity solution of the heat equation. J. Math. Mech., 18: 1025–1042 (1969)

    MathSciNet  MATH  Google Scholar 

  3. Bona, J. On solitary waves and their role in the evolution of long waves. Applications of nonlinear analysis. MA: Pitman, Boston, 1981

    Google Scholar 

  4. Cao, C.W. Nonlinearization of the Lax system for AKNS hierarchy. Sci. Chin. (Serices A), 33: 528–536 (1990)

    MATH  Google Scholar 

  5. Coely, A. et al., editors. Backlund and Darboux transformations. Providence, RI: American Mathematical Society, 2001

    Google Scholar 

  6. Doyle, P.W., Vassiliou, P.J. Separation of variables for the 1-dimensional non-linear diffusion equation. Int. J. Nonlin. Mech., 33: 315–326 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, E.G. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett., A277: 212–218 (2000)

    Article  Google Scholar 

  8. Fan, E.G. Soliton solutions for a generalized HirotaCSatsuma coupled KdV equation and a coupled MKdV equation. Phys. Lett. A, 282: 18–22 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu, Z.T., Liu, S.K. Solving Nonlinear Wave Equations by Elliptic Equation. Commun. Theor. Phys., 39: 531–536 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Gardner, C.S., Green, J.M., Kruskal, M.D., Miura, R.M. Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett., 19: 1095–1097 (1967)

    Article  Google Scholar 

  11. Hirota, R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett., 27: 1192–1194 (1971)

    Article  MATH  Google Scholar 

  12. Hirota, R. Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn., 33: 1456–1458 (1972)

    Article  Google Scholar 

  13. Khater, A.H., EI-Kalaawy, O H., Callebaut, D.K. Backlund transformations and exact solutions for alfven solitons in a relativistic electron-positron plasma. Phys. Scr., 58: 545–548 (1998)

    Article  Google Scholar 

  14. Komatsu, T.S., Sasa, S.I. Kink soliton characterizing traffic congestion. Phys. Rev. E, 52: 5574–5582 (1995)

    Article  Google Scholar 

  15. Lai, D.W.C., Chow, K.W. Coalescence of Ripplons, breathers, dromions and dark solitons. J. Phys. Soc. Jpn., 70: 666–677 (2001)

    Article  MATH  Google Scholar 

  16. Lonngren, K.E. Ion acoustic soliton experiments in a plasma. Opt. Quant. Electron, 30: 615–630 (1998)

    Article  Google Scholar 

  17. Lou, S.Y. Localized excitations of the (2+1)-dimensional sine-Gordon system. J. Phys. A: Math. Gen., 36: 3877–3892 (2003)

    Article  MATH  Google Scholar 

  18. Lou, S.Y., Chen, L.L. A formally variable separation approach to solve nonintegrable nonlinear systems. J. Math. Phys., 40: 6491–6500 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lou, S.Y., Ma, H.C. Finite symmetry transformation groups and exact solutions of Lax integrable systems. Chaos, Soliton & Fractals, 30: 804–821 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lou, S.Y., Ruan, H.Y. Revisitation of the localized excitations of the 2+1 dimensional KdV equarion. J. Phys. A: Math. Gen., 34: 305–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, H.C. A simple method to generate Lie point symmetry groups of (3+1)-dimensional Jimbo-Miwa equation. Chin. Phys. Lett., 22: 554–557 (2005)

    Article  Google Scholar 

  22. Ma, H.C., Lou, S.Y. Non-Lie symmetry groups of (2+1)-dimensional nonlinear systems. Commun. Theor. Phys., 46: 1005–1010 (2006)

    Article  MathSciNet  Google Scholar 

  23. Ma, H.C., Lou, S.Y. Solutions generated from the symmetry group of the (2+1)-dimensional Sine-gordon system. Z. Natureforsch, 60a: 229–236 (2005)

    Google Scholar 

  24. Malfeit, W. Solitary wave solutions of nonlinear wave equations. Am. J. Phys., 60: 650–654 (1992)

    Article  Google Scholar 

  25. Micu, S. On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim., 39(6): 1677–1696 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Olver, P.J. Application of Lie Group to Differential Equation. Springer-Verlag, New York, 1986

    Book  Google Scholar 

  27. Olver, P.J., Rosenau, P. The construction of special solutions to partial differential equations. Phys. Lett., 114A: 107–112 (1986)

    Article  MathSciNet  Google Scholar 

  28. Ono, H. Soliton fission in anharmonic lattices with reflectionless inhomogeneity. J. Phys. Soc. Jpn., 61: 4336–4343 (1992)

    Article  Google Scholar 

  29. Ovsiannikov, L.V. Group analysis of differential equations. Translated by W.F. Ames, Academic Press, New York, 1982

    MATH  Google Scholar 

  30. Pucci, E., Saccomandi, G. Evolution equations, invariant surface conditions and functional separation of variables. Physica D, 139: 28–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qu, C.Z., Zhang, S.L., Liu, R.C. Separation of variables and exact solutions to quasilinear diffusion equations with the nonlinear source. Physica D, 144: 97–123 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Raisinariu, C., Sukhutame U., Khare, K. Negaton and positon solutions of the KdV and mKdV hierarchy. J. Phys. A: Math. Gen., 29: 1803–1823 (1996)

    Article  Google Scholar 

  33. Stahlhofen, A. Supertransparent potentials for the Dirac equation. J. Phys. A: Math. Gen., 27: 8279–8290 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wadati, M. Introduction to solitons. Pramana: J. Phys., 57(5–6): 841–847 (2001)

    Article  Google Scholar 

  35. Wadati, M., Ohkuma, K. Multi-pole solutions of the modified Korteweg-de Vries equation. J. Phys. Soc. Jpn., 51: 2029–2035 (1981)

    Article  MathSciNet  Google Scholar 

  36. Wadati, M., Sanuki, H., Konno, K. Relationships among inverse method, Backlund transformation and an infinite number of conservation laws. Prog. Theor. Phys., 53: 419–436 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, M.L. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A, 213: 279–287 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yan, C.T. A simple transformation for nonlinear waves. Phys. Lett. A, 224: 77–84 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, S.L., Lou, S.Y., Qu, C.Z. Variable Separation and Exact Solutions to Generalized Nonlinear Diffusion Equations. Chin. Phys. Lett., 19: 1741–1744 (2002)

    Article  Google Scholar 

  40. Zhang, S.L., Lou, S.Y., Qu, C.Z. New variable separation approach: application to nonlinear diffusion equations. J. Phys. A: Math. Gen., 36: 12223–12242 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, J.F., Ren, D.F., Wang, M.L., et al. The Periodic Wave solutions for the generalized Nizhnik-Novikov-Veselov equation. Chin. Phys., 12(8): 825–830 (2003)

    Article  Google Scholar 

  42. Ziegler, V., Dinkel, J., Setzer, C., Lonngren, K.E. On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line. Chaos, Solitons & Fractals, 12: 1719–1728 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Donghua University, Shanghai, 201620, China

    Hong-cai Ma & Ai-ping Deng

  2. School of Mathematics and Information Sciences, Henan University, Kaifeng, 475000, China

    Zhi-Ping Zhang

Authors
  1. Hong-cai Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Ping Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ai-ping Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hong-cai Ma.

Additional information

Supported by the National Natural Science Foundation of China (No. 10647112) and the Foundation of Donghua University.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, Hc., Zhang, ZP. & Deng, Ap. A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation. Acta Math. Appl. Sin. Engl. Ser. 28, 409–415 (2012). https://doi.org/10.1007/s10255-012-0153-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-012-0153-7

Keywords

2000 MR Subject Classification

Search

Navigation