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Solitary wave solutions of some nonlinear PDEs arising in electronics

  • Syed Tauseef Mohyud-Din
  • Amna Irshad
Article

Abstract

In the present article, an Extended Trial Equation method has been applied to derive the new exact solutions for generalized form of equations. We consider ZK equation and ZK-BBM equation to demonstrate the features and credibility of the suggested technique. As a result, many new exact soliton solutions are obtained, which includes rational solutions, soliton type solutions, and singular soliton solutions. These types of solutions might perform significant role in engineering domains.

Keywords

Extended Trial Equation method Rational solution Soliton type solutions Singular soliton solution Generalized ZK equation (gZK) Generalized ZK-BBM 

Notes

Acknowledgement

Authors are highly grateful to the unknown referees for their valuable comments.

References

  1. Abdou, M.A.: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, Solitons Fractals 31, 95–104 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. Bekir, A., Boz, A.: Exact solutions for nonlinear evolution equations using Exp-function method. Phys. Lett. A 372(10), 1619–1625 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. Borhanifar, A., Kabir, M.M.: New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations. Comput. Appl. Math. 229, 158–167 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. Borhanifar, A., Jafari, H., Karimi, S.A.: New solitons and periodic solutions for the Kadomtsev–Petviashvili equation. Nonlinear Sci. Appl. 4, 224–229 (2008)MathSciNetMATHGoogle Scholar
  5. Borhanifar, A., Jafari, H., Karimi, S.A.: New solitary wave solutions for the bad Boussinesq and good Boussinesq equations. Numer. Methods Partial Differ. Equ. 25, 1231–1237 (2009a)MathSciNetCrossRefMATHGoogle Scholar
  6. Borhanifar, A., Kabir, M.M., Maryam Vahdat, L.: New periodic and soliton wave solutions for the generalized Zakharov system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system. Chaos, Solitons Fractals 42, 1646–1654 (2009b)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. Borhanifar, A., Jafari, H., Karimi, S.A.: New solitary wave solutions for generalized regularized long-wave equation. Int. J. Comput. Math. 87, 509–514 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. Bulut, H.: Classification of exact solutions for generalized form of K(m, n) equation. Abstr. Appl. Anal. 2013, 11 (2013)MathSciNetGoogle Scholar
  9. Darvishi, M.T., Arbabi, S., Najafi, M., Wazwaz, A.M.: Traveling wave solutions of a (2 + 1)-dimensional Zakharov-like equation by the first integral method and the tanh method. Optik Int. J. Light Electron Opt. 127(16), 6312–6321 (2016)CrossRefGoogle Scholar
  10. Du, X.H.: An irrational trial equation method and its applications. Pramana J. Phys. 75(3), 415–422 (2010)ADSCrossRefGoogle Scholar
  11. Eslami, M., Mirzazadeh, M., Vajargah, B.F., Biswas, A.: Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method. Optik Int. J. Light Electron Opt. 125(13), 3107–3116 (2014)CrossRefGoogle Scholar
  12. Filiz Taşcan, Ahmet Bekir: Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the Sine–Cosine method. Appl. Math. Comput. 215(8), 3134–3139 (2009)MathSciNetMATHGoogle Scholar
  13. Guner, O., Atik, H.: Soliton solution of fractional-order nonlinear differential equations based on the Exp-function method. Optik Int. J. Light Electron Opt. 127(20), 10076–10083 (2016)CrossRefGoogle Scholar
  14. Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M.: Extended Trial Equation method to generalized nonlinear partial differential equations. Appl. Math. Comput. 219(10), 5253–5260 (2013)MathSciNetMATHGoogle Scholar
  15. Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869–877 (2010)MathSciNetMATHGoogle Scholar
  16. Khater, M.M.A.: Solitary wave solutions for the generalized Zakharov Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation. Global J. Sci. Front. Res. Phys. Space Sci. 16(4) (2016) (Ver. 1.0)Google Scholar
  17. Khater, M.M.A., Lu, D., Zahran, E.H.M.: Solitary wave solutions of the Benjamin–Bona–Mahoney–Burgers equation with dual power-law nonlinearity. Appl. Math. Inf. Sci. 11(5), 1–5 (2017)Google Scholar
  18. Kumar, R., Kaushal, R.S., Prasad, A.: Some new solitary and travelling wave solutions of certain nonlinear diffusion-reaction equations using auxiliary equation method. Phys. Lett. A 372, 3395–3399 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. Liu, C.S.: Trial equation method and its applications to nonlinear evolution equations. Acta Phys. Sin. Chin. 54(6), 2505–2509 (2005)MathSciNetMATHGoogle Scholar
  20. Liu, X.P., Liu, C.P.: Chaos, Solitons Fractals 39, 1915–1919 (2009)ADSCrossRefGoogle Scholar
  21. Lü, Z., Chen, Y.: Constructing rogue wave prototypes of nonlinear evolution equations via a extended tanh method. Chaos, Solitons Fractals 81, 218–223 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Mirzazadeh, M., Biswas, A.: Optical solitons with spatio-temporal dispersion by first integral approach and functional variable method. Optik Int. J. Light Electron Opt. 125(19), 5467–5475 (2014)CrossRefGoogle Scholar
  23. Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Zhou, Q., Triki, H., Moshokoa, S.P., Biswas, A., Belic, M.: Optical solitons in birefringent fibers by Extended Trial Equation method. Optik Int. J. Light Electron Opt. 127(23), 11311–11325 (2016)CrossRefGoogle Scholar
  24. Mohyud-Din, S.T., Yildirim, A., Demirli, G.: Analytical solution of wave system in with coupling controllers. Int. J. Numer. Method Heat Fluid Flow 21(2), 198–205 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. Pandir, Y.: New exact solutions of the generalized Zakharov–Kuznetsov modified equal-width equation. Pramana J. Phys. 82(6), 949–964 (2014)CrossRefGoogle Scholar
  26. Ren, Y.J., Zhang, H.Q.: A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation. Chaos, Solitons Fractals 27, 959–979 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. Shukri, S., Al-Khaled, K.: The extended tanh method for solving systems of nonlinear wave equations. Appl. Math. Comput. 217(5), 1997–2006 (2010)MathSciNetMATHGoogle Scholar
  28. Wang, M.L., Li, X.Z.: Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A 343, 48–54 (2005a)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. Wang, M.L., Li, X.Z.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons Fractals 24, 1257–1268 (2005b)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. Wang, M.L., Zhang, J.L., Li, X.Z.: The G′/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. Wazwaz, A.M.: A Sine–Cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004)MathSciNetCrossRefMATHGoogle Scholar
  32. Wazwaz, A.M.: Compact and noncompact physical structures for the ZK-BBM equation. Appl. Math. Comput. 169, 713–725 (2005)MathSciNetMATHGoogle Scholar
  33. Wazwaz, A.M.: The tanh method for travelling wave solutions to the Zhiber–Shabat equation and other related equations. Commun. Nonlinear Sci. Numer. Simul. 13(3), 584–592 (2008a)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. Wazwaz, A.M.: The extended tanh method for the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms. Commun. Nonlinear Sci. Numer. Simul. 13, 1039–1047 (2008b)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. Wu, H.X., He, J.H.: Exp-function method and its application to nonlinear equations. Chaos, Solitons Fractals 30, 700–708 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. Yan, Z.: Abunbant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional integrable Davey–Stewartson-type equation via a new method. Chaos, Solitons Fractals 18, 299–309 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. Yomba, E.: Construction of new soliton-like solutions for the (2 + 1) dimensional Kadomtsev–Petviashvili equation. Chaos, Solitons Fractals 22, 321–325 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  38. Yomba, E.: Construction of new solutions to the fully nonlinear generalized Camassa–Holm equations by an indirect F-function method. J. Math. Phys. 46, 123504–123512 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  39. Zahran, M.H.M., Khater, M.M.A.: Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Model. 40, 1769–1775 (2016)MathSciNetCrossRefGoogle Scholar
  40. Zayed, E.M.E.: A note on the modified simple equation method applied to Sharma Tasso–Olver equation. Appl. Math. Comput. 218, 3962–3964 (2011)MathSciNetMATHGoogle Scholar
  41. Zayed, E.M.E., Al-Nowehy, A.G.: The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schrödinger equations. J. Assoc. Arab Univ. Basic Appl. Sci. (2016) (article in press)Google Scholar
  42. Zayed, E.M.E., Hoda Ibrahim, S.A.: Exact solutions of nonlinear evolution equation in mathematical physics using the modified simple equation method. Chin. Phys. Lett. 29, 060201–060204 (2012)CrossRefGoogle Scholar
  43. Zayed, E.M.E., Hoda Ibrahim, S.A.: Modified simple equation method and its applications for some nonlinear evolution equations in mathematical physics. Int. J. Comput. Appl. 67, 39–44 (2013)Google Scholar
  44. Zhang, Z.Y.: New exact traveling wave solutions for the nonlinear Klein–Gordon equation. Turk. J. Phys. 32, 235–240 (2008)Google Scholar
  45. Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350, 103–109 (2006)ADSCrossRefMATHGoogle Scholar
  46. Zhang, S., Tong, J.L., Wang, W.: A generalized G′/G-expansion method for the mKdV equation with variable coefficients. Phys. Lett. A 372, 2254–2257 (2008)ADSCrossRefMATHGoogle Scholar
  47. Zhou, X.W., Wen, Y.X., He, J.H.: Exp-function method to solve the nonlinear dispersive k(m, n) equations. Int. J. Nonlinear Sci. Numer. Simul. 9, 301–306 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesHITEC UniversityTaxila CanttPakistan

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