Skip to main content
SpringerLink
Log in

Traveling Wave Solutions to Kawahara and Related Equations

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg–de Vries (KdV) equation are found by using an elliptic function method which is more general than the \(\mathrm {tanh}\)-method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to the KdV equation with only a cubic dispersion term, while for the KE which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. By solving the elliptic equation with coefficients that depend on the boundary conditions, velocity of the traveling waves, nonlinear strength, and dispersion coefficients, in the case of KdV equation we find the well-known solitary waves (solitons) for zero boundary conditions, as well as wave-trains of cnoidal waves for nonzero boundary conditions. Both solutions are either compressive (bright) or rarefactive (dark), and either propagate to the left or right with arbitrary velocity. In the case of KE with nonzero boundary conditions and zero cubic dispersion, we obtain cnoidal wave-trains which represent solutions to the TL equation. For KE with zero boundary conditions and all the dispersion terms present, we obtain again solitary waves, while for KE with all coefficients present and nonzero boundary condition, the solutions are written in terms of Weierstrass elliptic functions. For all cases of the KE we only find bright waves that are propagating to the right with velocity that is a function of both dispersion coefficients.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149. Cambridge University Press, Cambridge, UK (1991)

    Book  MATH  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, Washington, D.C. (1964)

    MATH  Google Scholar 

  3. Albert, J.P.: Positivity properties and stability of solitary-wave solutions of model equations for long waves. Commun. Partial Differ. Equ. 17, 1–22 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assas, L.M.B.: New exact solutions for the Kawahara equation using Exp-function method. J. Comput. Appl. Math. 233(2), 97–102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cornejo-Pérez, O., Negro, J., Nieto, L.M., Rosu, H.C.: Traveling-wave solutions for Korteweg–de Vries–Burgers equations through factorizations. Found. Phys. 36(10), 1587–1599 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction, vol. 2. Cambridge University Press, Cambridge, UK (1989)

    Book  MATH  Google Scholar 

  7. Eremenko, A.: Meromorphic traveling wave solutions of the Kuramoto–Sivashinsky equation (2005) (preprint). arXiv:nlin/0504053

  8. Fan, E.: Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. J. Phys. A Math. Gen. 35(32), 6853 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277(4), 212–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, E., Zhang, J.: Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys. Lett. A 305(6), 383–392 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu, Z., Liu, S., Liu, S., Zhao, Q.: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 290(1), 72–76 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hasimoto, H.: Water waves. Kagaku 40, 401–408 (1970) (in Japanese)

  13. Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192 (1971)

    Article  MATH  Google Scholar 

  14. Kano, K., Nakayama, T.: An exact solution of the wave equation \(u_t+ uu_x-u_{5x}= 0\). J. Phys. Soc. Jpn. 50, 361 (1981)

    Article  Google Scholar 

  15. Kakutani, T., Ono, H.: Weak non-linear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Jpn. 26(5), 1305–1318 (1969)

    Article  Google Scholar 

  16. Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33(1), 260–264 (1972)

    Article  Google Scholar 

  17. Korteweg, D.J., De Vries, G.: XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Lond. Edinb. Dublin Philos. Mag. J. Sci. 39(240), 422–443 (1895)

    Article  MATH  Google Scholar 

  18. Kudryashov, N.A.: Painlevé analysis and exact solutions of the Kortewegde Vries equation with a source. Appl. Math. Lett. 41, 41–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kudryashov, N.A.: On new traveling wave solutions of the KdV and the KdV–Burgers equations. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1891–1900 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kudryashov, N.A.: A note on new exact solutions for the Kawahara equation using Exp-function method. J. Comput. Appl. Math. 234(12), 3511–3512 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kudryashov, N.A.: Meromorphic solutions of nonlinear ordinary differential equations. Commun. Nonlinear Sci. Numer. Simul. 15(10), 2778–2790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 24(5), 1217–1231 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kudryashov, N.A.: Exact solutions of the generalized Kuramoto–Sivashinsky equation. Phys. Lett. A 147(5–6), 287–291 (1990)

    Article  MathSciNet  Google Scholar 

  24. Kudryashov, N.A.: A note on the \(G^{\prime }/G\)-expansion method. Appl. Math. Comput. 217(4), 1755–1758 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, S., Fua, Z., Liua, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289(1), 69–74 (2001)

    Article  MathSciNet  Google Scholar 

  27. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma, W.: Travelling wave solutions to a seventh order generalized KdV equation. Phys. Lett. A 180(3), 221–224 (1993)

    Article  MathSciNet  Google Scholar 

  29. Mancas, S.C., Spradlin, G., Khanal, H.: Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation. J. Math. Phys. 54(8), 081502 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mancas, S.C., Adams, R.: Elliptic solutions and solitary waves of a higher order KdVBBM long wave equation. J. Math. Anal. Appl. 452(2), 1168–1181 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Miura, M.R.: Bäcklund Transformation. Springer, Berlin (1978)

    Google Scholar 

  32. Nagashima, H.: Experiment on solitary waves in the nonlinear transmission line described by the equation \(\frac{\partial u}{\partial \tau }+u\frac{\partial u}{\partial \xi }-\frac{\partial ^5 u}{\partial \xi ^5}=0\). J. Phys. Soc. Jpn. 47(4), 1387–1388 (1979)

    Article  Google Scholar 

  33. Natali, F.: A note on the stability for Kawahara–KdV type equations. Appl. Math. Lett. 23, 591–596 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Newell, A.C.: The Inverse Scattering Transform. Solitons. Springer, Berlin (1980)

    Google Scholar 

  35. Nickel, J.: Elliptic solutions to a generalized BBM equation. Phys. Lett. A 364(3), 221–226 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nizovtseva, I.G., Galenko, P.K., Alexandrov, D.V.: The hyperbolic Allen–Cahn equation: exact solutions. J. Phys. A Math. Theor. 49(43), 435201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Otwinowski, M., Paul, R., Laidlaw, W.G.: Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature. Phys. Lett. A 128(9), 483–487 (1988)

    Article  MathSciNet  Google Scholar 

  38. Parkes, E.J., Duffy, B.R.: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 98(3), 288–300 (1996)

    Article  MATH  Google Scholar 

  39. Porubov, A.V.: Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer. Phys. Lett. A 221(6), 391–394 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  40. Porubov, A.V.: Exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid. J. Phys. A Math. Gen. 26(17), L797 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Porubov, A.V., Parker, D.F.: Some general periodic solutions to coupled nonlinear Schrödinger equations. Wave Motion 29(2), 97–109 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Prelle, M.J., Singer, M.F.: Elementary first integrals of differential equations. Trans. Am. Math. Soc. 279(1), 215–229 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. Reyes, M.A., Gutièrrez-Ruiz, D., Mancas, S.C., Rosu, H.C.: Nongauge bright soliton of the nonlinear Schrödinger (NLS) equation and a family of generalized NLS equations. Mod. Phys. Lett. A 31(03), 1650020 (2016)

    Article  MATH  Google Scholar 

  44. Rosenau, P.: A quasi-continuous description of a nonlinear transmission line. Phys. Scr. 34(6B), 827 (1986)

    Article  Google Scholar 

  45. Rosenau, P.: Dynamics of dense discrete systems high order effects. Prog. Theor. Phys. 79(5), 1028–1042 (1988)

    Article  Google Scholar 

  46. Sajjadi, S.G., Mancas, S.C., Drullion, F.: Formation of three-dimensional surface waves on deep-water using elliptic solutions of nonlinear Schrödinger equation. Adv. Appl. Fluid Mech. 18(1), 81–112 (2015)

    MathSciNet  MATH  Google Scholar 

  47. Stokes, G.G.: On the theory of oscillatory waves. Trans. Camb. Philos. Soc. 8, 441–473 (1847)

    Google Scholar 

  48. Wang, M.: Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A 199(3–4), 169–172 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wazwaz, A.-M.: New travelling wave solutions of different physical structures to generalized BBM equation. Phys. Lett. A 355(4), 358–362 (2006)

    Article  Google Scholar 

  50. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24(3), 522–526 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  51. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Reprint of the fourth (1927) edition. Chap. XX. Cambridge University Press, Cambridge, UK (1958)

  52. Yamamoto, Y., Takizawa, É.I.: On a solution on non-linear time-evolution equation of fifth order. J. Phys. Soc. Jpn. 50(5), 1421–1422 (1981)

    Article  Google Scholar 

  53. Yan, Z., Zhang, H.: New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water. Phys. Lett. A 285(5), 355–362 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yuan, J.-M., Shen, J., Jiahong, W.: A dual-Petrov–Galerkin method for the Kawahara-type equations. J. Sci. Comput. 34(1), 48–63 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhang, S.: Exp-function method for Klein–Gordon equation with quadratic nonlinearity. J. Phys. Conf. Ser. 96(1), 012002 (2008) (IOP Publishing)

Download references

Acknowledgements

The author would like to acknowledge Professor Pisin Chen from Leung Center for Cosmology and Particle Astrophysics (LeCosPA) for support during his stay in Taipei, and Professors Juan-Ming Yuan and Haret C. Rosu for their helpful comments and discussions on Kawahara equation.

Author information

Authors and Affiliations

  1. Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL, 32114-3900, USA

    Stefan C. Mancas

Authors
  1. Stefan C. Mancas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan C. Mancas.

Appendix

Appendix

Lemma 1

Traveling wave solutions to KE (1), satisfy only the Weierstrass elliptic equation with cubic nonlinearity.

Proof

Letting

$$\begin{aligned} {Y_\xi }^2=\sum _{i=0}^M a_i Y^i, \quad a_M \ne 0. \end{aligned}$$
(69)

For KdV equation (14) \(u=Y\) and the leading term for the second order derivative term is \(u_{\xi \xi } =\frac{1}{2} M a_M Y^{M-1}\). By matching the terms \(u^2\) with \(u_{\xi \xi }\) in Eq. (15) we obtain \(M=3\). For KE (1) \(u=Y^2\) and the leading term for the fourth order derivative term is \(u_{\xi \xi \xi \xi }=\frac{1}{2} M(M+2)(3M-2) {a_M}^2 Y^{2(M-1)}\). By matching the terms \(u^2\) and \(u_{\xi \xi \xi \xi }\) in Eq. (7) we also obtain \(M=3\). Notice that this method will not work if an ODE contains both even and odd derivative terms. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mancas, S.C. Traveling Wave Solutions to Kawahara and Related Equations. Differ Equ Dyn Syst 27, 19–37 (2019). https://doi.org/10.1007/s12591-017-0367-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-017-0367-5

Keywords

Search

Navigation