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One- and two-soliton solutions to a new KdV evolution equation with nonlinear and nonlocal terms for the water wave problem

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Abstract

With the help of the Boussinesq perturbation expansion, a new basic equation describing the long, small-amplitude, unidirectional wave motion in shallow water with surface tension is derived to fourth order, namely a higher-order Korteweg–de Vries (KdV) equation. The procedure for deriving this equation assumes that the relation between the small parameter \(\alpha \), which measures the ratio of wave amplitude to undisturbed fluid depth, and the small parameter \(\beta \), which measures the square of the ratio of fluid depth to wave length, is taken in the form \(\beta = 0(\alpha ) = \varepsilon \), where \(\varepsilon \) is a small, dimensionless parameter which is the order of the amplitude of the motion. Hirota’s bilinear method is used to investigate one- and two-soliton solutions for this new higher-order KdV equation.

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References

  1. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)

    Article  Google Scholar 

  2. Porsezian, K., Nakkeeran, K.: Optical solitons in presence of Kerr dispersion and self-frequency shift. Phys. Rev. Lett. 76, 3955 (1996)

    Article  Google Scholar 

  3. Mihalache, D., Truta, N., Crasovan, L.C.: Painlevé analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term. Phys. Rev. E 56, 1064 (1997)

    Article  MathSciNet  Google Scholar 

  4. Gedalin, M., Scott, T.C., Band, Y.B.: Optical solitary waves in the higher order nonlinear Schrödinger equation. Phys. Rev. Lett. 78, 448 (1997)

    Article  Google Scholar 

  5. Radhakrishnan, R., Kundu, A., Lakshmanan, M.: Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media. Phys. Rev. E 60, 3314 (1999)

    Article  Google Scholar 

  6. Gatz, S., Hermann, J.: Soliton propagation and soliton collision in double-doped fibers with a non-Kerr-like nonlinear refractive-index change. Opt. Lett. 17, 484 (1992)

    Article  Google Scholar 

  7. Yanay, H., Khaykovich, L., Malomed, B.A.: Stabilization and destabilization of second-order solitons against perturbations in the nonlinear Schrödinger equation. Chaos 19, 033145 (2009)

    Article  MATH  Google Scholar 

  8. Chen, Y., Beckwitt, K., Wise, F., Aitken, B., Sanghera, J., Aggarwall, I.D.J.: Measurement of fifth- and seventh-order nonlinearities of glasses. Opt. Soc. Am. B. 23, 347 (2006)

    Article  Google Scholar 

  9. Mohamadou, A., Latchio, Tiofack, C.G., Kofané, T.C.: Wave train generation of solitons in systems with higher-order nonlinearities. Phys. Rev. E 82, 016601 (2010)

    Article  Google Scholar 

  10. Nguenang, J.P., Kofané, T.C.: Solitary waves in spin chains with biquadratic exchange and dipole–dipole interactions. Phys. Scr. 55, 367 (1997)

    Article  Google Scholar 

  11. Pushkarov, D.I., Pushkarov, KhI: Solitary magnons in one-dimensional ferromagnetic chain. Phys. Lett. A 61, 339 (1977)

    Article  MATH  Google Scholar 

  12. Shi, Z.P., Huang, G., Tao, R.: Solitonlike excitations in a spin chain with a biquadratic anisotropic exchange interaction. Phys. Rev. B 42, 747 (1990)

    Article  Google Scholar 

  13. Ursell, F.: The long-wave paradox in the theory of gravity waves. Proc. Camb. Philos. Soc. 49, 685 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ndohi, R., Kofané, T.C.: Solitary waves in a ferromagnetic chains near the marginal state of instability. Phys. Lett. A 154, 377 (1991)

    Article  Google Scholar 

  15. Dysthe, K.B.: Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. Ser. A 369, 105 (1979)

    Article  MATH  Google Scholar 

  16. Aspe, H., Depassier, M.C.: Evolution equation of surface waves in a convecting fluid. Phys. Rev. A 41, 6 (1990)

    Article  MathSciNet  Google Scholar 

  17. Guo, xiang, H., Velarde, M.G.: Dispersion-modified Burgers description for nonlinear surface wave excitations in Bénard–Marangoni convection. Commun. Theor. Phys. 34, 321 (2000)

    Article  Google Scholar 

  18. Dullin, H.R., Gottwald, G.A., Holm, D.D.: Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Burde, G.I.: Solitary wave solutions of the high-order KdV models for bi-directional water waves. Commun. Nonlinear Sci. Numer. Simul. 16, 1314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Burde, G.I., Sergyeyev, A.: Ordering of two small parameters in the shallow water wave problem. J. Phys. A Math. Theor. 46, 075501 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Karczewska, A., Rozmej, P., Infeld, E.: Shallow-water soliton dynamics beyond the Korteweg–de Vries equation. Phys. Rev. E 90, 012907 (2014)

    Article  Google Scholar 

  22. Pelap, F.B., Kofané, T.C.: The revised modulational instability criterion: I. The mono-inductance transmission line. Phys. Scr. 57, 410 (1998)

    Article  Google Scholar 

  23. Yemélé, D., Marqui, P., Bilbault, J.M.: Long-time dynamics of modulated waves in a nonlinear discrete LC transmission line. Phys. Rev. E 68, 016605 (2003)

    Article  MathSciNet  Google Scholar 

  24. Kliakhandler, I.L., Porulov, A.V., Velarde, M.G.: Localized finite-amplitude disturbances and selection of solitary waves. Phys. Rev. E 62, 4 (2000)

    Article  Google Scholar 

  25. Depassier, M.C., Letelier, J.A.: Fifth order evolution equation for long wave dissipative solitons. Phys. Lett. A 376, 452 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  27. Fokas, As: On a class of physically important integrable equations. Phys. D 87, 145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wazwaz, A.M.: The Hirotas direct method and the tanhcoth method for multiple-soliton solutions of the SawadaKoteraIto seventh-order equation. Appl. Math. Comput. 199, 133 (2008)

    Article  MathSciNet  Google Scholar 

  29. Martinez, J.J.: An introduction to the Hirota bilinear method. Department of Mathematics, Kobe University, Rokko

  30. Wazwaz, A.M.: Multiple-soliton solutions for the KP equation by Hirotas bilinear method and by the tanhcoth method. Appl. Math. Comput. 190, 633 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)

    Book  MATH  Google Scholar 

  32. Wadati, M.: Wave propagation in nonlinear lattice I. J. Phys. Soc. Jpn. 38, 673 (1975)

    Article  MathSciNet  Google Scholar 

  33. Yajima, T., Wadati, M.: The Rayleigh–Taylor instability and nonlinear waves. J. Phys. Soc. Jpn. 59, 3182 (1990)

    Article  MathSciNet  Google Scholar 

  34. Yajima, T., Wadati, M.: The Rayleigh–Taylor instability and nonlinear waves. J. Phys. Soc. Jpn. 59, 48 (1990)

    Article  MathSciNet  Google Scholar 

  35. El-Kalaawy, O.H.: Exact soliton solutions for some nonlinear partial differential equations. Chaos Solitons Fractals 14, 547 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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, 288 (1996)

    Article  MATH  Google Scholar 

  37. Parkes, E.J., Duffy, B.R.: Travelling solitary wave solutions to a compound KdV-Burgers equation. Phys. Lett. A 229, 217 (1997)

  38. Khater, A.H., Malfliet, W., Callebaut, D.K., Kamel, E.S.: The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction–diffusion equations. Chaos Solitons Fractals 14, 513 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Fan, E.: Travelling wave solutions for two generalized Hirota–Satsuma coupled KdV systems. Z. Naturforsch. A 56, 312 (2001)

    Article  Google Scholar 

  41. Elwakil, S.A., El-labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299, 179 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gao, Y.T., Tian, B.: Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics. Comput. Phys. Commun. 133, 158 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tian, B., Gao, Y.T.: Observable solitonic features of the generalized reaction duffing mode. Z. Naturforsch. A 57, 39 (2002)

    Article  Google Scholar 

  44. Tian, B., Zhao, K., Gao, Y.T.: Symbolic computation in engineering: application to a breaking soliton equation. Int. J. Eng. 35, 1081 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Al-Mdallal, Q.M., Syam, M.I.: SineCosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation. Chaos Soliton Fractals 33, 1610 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wazwaz, A.M.: A sine–cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wazwaz, A.M.: The sinecosine method for obtaining solutions with compact and non compact structures. Appl. Math. Comput. 159, 559 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yusufoglu, E., Bekir, A.: Solitons and periodic solutions of coupled nonlinear evolution equations by using the sinecosine method. Int. J. Comput. 83, 915 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Malfliet, W., Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54, 563 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wazwaz, A.M.: The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. Appl. Math. Comput. 154, 1196 (2005)

  51. Wazwaz, A.M.: The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation. Appl. Math. Comput. 167, 1179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wazwaz, A.M.: The tanhcoth and the sinecosine methods for kinks, solitons, and periodic solutions for the Pochhammer–Chree equations. Appl. Math. Comput. 195, 24 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Hereman, W., Zhuang, W.: Symbolic Computation of Solitons Via Hirotas Bilinear Method. Department of Mathematics and Computer Science, Colorado School of Mines, Golden, Colorado (1994)

    Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon

    M. Fokou & T. C. Kofane

  2. Centre d’Excellence Africain en Technologies de l’Information et de la Communication, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon

    M. Fokou, T. C. Kofane & A. Mohamadou

  3. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Maroua, P.O. Box 814, Maroua, Cameroon

    A. Mohamadou

  4. Department of Mathematics, California State University-Northridge, Northridge, CA, 91330-8313, USA

    E. Yomba

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  1. M. Fokou
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  4. E. Yomba
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Fokou, M., Kofane, T.C., Mohamadou, A. et al. One- and two-soliton solutions to a new KdV evolution equation with nonlinear and nonlocal terms for the water wave problem . Nonlinear Dyn 83, 2461–2473 (2016). https://doi.org/10.1007/s11071-015-2494-2

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