Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics

  • Mustafa Inc
  • Abdullahi Yusuf
  • Aliyu Isa Aliyu
  • Dumitru Baleanu


This research presents soliton solutions and stability analysis to some conformable nonlinear partial differential equations (CNPDEs). The CNPDEs equations in this paper are conformable Cahn–Allen and conformable Zoomeron equations. The powerful sine-Gordon method is used to carry out the soliton solutions for these equations. The aspects of stability analysis for the considered equations is investigated using the linear stability technique. The sine-Gordon method proves to be efficient and effective for the extraction of soliton solutions for different types of CNPDEs.


CA ZM SGEM Stability analysis 


  1. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279(1), 57–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Agrawal, G.P.: Nonlinear Fiber Optics, 5th edn. Elsevier, New York (2013)zbMATHGoogle Scholar
  3. Ahmed, B.S., Zerrad, E., Biswas, A.: Kinks and domain walls of the Zakharov–Kuznetsov equation in plasmas. Proc. Rom. Acad. Ser. A 14(4), 281–286 (2013)MathSciNetGoogle Scholar
  4. Bekir, A., Guner, O., Bhrawy, A.H., Biswas, A.: Solving nonlinear fractional differential equations using exp-function and \(\text{ G }^{\prime }/\text{ G }\)-expansion methods. Rom. J. Phys. 60(3–4), 360–378 (2015)Google Scholar
  5. Bhrawy, A.H., Abdelkawy, M.A., Kumar, S., Johnson, S., Biswas, A.: Soliton and other solutions to quantum Zakharov–Kuznetsov equation in quantum magneto-plasmas. Indian J. Phys. 87(5), 455–463 (2013)ADSCrossRefGoogle Scholar
  6. Bhrawy, A.H., Alzaidy, J.F., Abdelkawy, M.A., Biswas, A.: Jacobi spectral collocation approximation for multi-dimensional time fractional Schrödinger’s equation. Nonlinear Dyn. 84(3), 1553–1567 (2016)CrossRefGoogle Scholar
  7. Biswas, A., Song, M.: Soliton solution and bifurcation analysis of the Zakharov–Kuznetsov Benjamin–Bona–Mahoney equation with power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 18(7), 1676–1683 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Biswas, A., Zerrad, E.: Solitary wave solution of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear Anal. Ser. B Real World Appl. 11(4), 3272–3274 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chen, Y., Yan, Z.: A simple transformation for nonlinear waves. Chaos Solitons Fractals 26, 399–406 (2005)ADSMathSciNetCrossRefGoogle Scholar
  10. Ebadi, G., Biswas, A.: The \(\text{ G }^{\prime }/\text{ G }\) method and 1-soliton solution of Davey–Stewartson equation. Math. Comput. Model. 53(5–6), 694–698 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Ebadi, G., Mojaver, A., Milovic, D., Johnson, S., Biswas, A.: Solitons and other solutions to the quantum Zakharov–Kuznetsov equation. Astrophys. Space Sci. 341(2), 507–513 (2012)ADSCrossRefzbMATHGoogle Scholar
  12. Ekici, M., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Optical soliton pertubation with fractional temporal evolution by first integral method with conformabal fractional derivatives. Optik 127(22), 10659–10669 (2016a)ADSCrossRefGoogle Scholar
  13. Ekici, M., Mirzazadeh, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Solitons in optical metamaterials with fractional temporal evolution. Optik 127(22), 10879–10897 (2016b)ADSCrossRefGoogle Scholar
  14. Esen, A., Yagmurlu, N.M., Tasbozan, O.: Approximate analytical solution to time-fractional damped Burger and Cahn–Allen equations. Appl. Math. Inf. Sci. 7(5), 1951–1956 (2013)MathSciNetCrossRefGoogle Scholar
  15. Eslami, M., Mirzazadeh, M., Biswas, A.: Soliton solutions of the resonant nonlinear Schrödinger’s equation in optical fibers with time-dependent coefficients by simplest equation approach. J. Mod. Opt. 60(19), 1627–1636 (2013)ADSCrossRefGoogle Scholar
  16. 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 125(13), 3107–3116 (2014)ADSCrossRefGoogle Scholar
  17. Fabian, A.L., Kohl, R., Biswas, A.: Pertubation of topological solitons due to sine-Gordon equation and its type. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1227–1244 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. Güner, O., Bekir, A., Cevikel, A.C.: A variety of exact solutions for the time fractional Cahn–Allen equation. Eur. Phys. J. Plus 130, 146 (2015). CrossRefGoogle Scholar
  19. Hammad, M.A., Khalil, R.: Conformable fractional heat differential equation. Int. J. Pure Appl. Math. 94(2), 215–221 (2014)Google Scholar
  20. Hosseini, K., Bekir, A., Ansari, R.: New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik 132, 203–209 (2017). ADSCrossRefGoogle Scholar
  21. Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons and modulation instability analysis of an integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Superlatt. Microstruct. 112, 628–638 (2017a). ADSCrossRefGoogle Scholar
  22. Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons and modulation instability analysis with (3+1)-dimensional nonlinear Shrödinger equation. Superlatt. Microstruct. 112, 296–302 (2017b). ADSCrossRefGoogle Scholar
  23. Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitary waves, conservation laws and modulation instability analysis to the nonlinear Schrödinger’s equation in compressional dispersive Alven waves. Optik 155, 257–266 (2018)ADSCrossRefGoogle Scholar
  24. Johnpillai, A.G., Kara, A.H., Biswas, A.: Symmetry solutions and reductions of a class of generalized (2+1) dimensional Zakharov–Kuznetsov equation. Int. J. Nonlinear Sci. Numer. Simul. 12(1–8), 35–43 (2011)MathSciNetGoogle Scholar
  25. Khalil, R., Horani, A.L.M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kohl, R., Milovic, D., Zerrad, E., Biswas, A.: Optical solitons by He’s variational principle in a non-Kerr law media. J. Infrared Millim. Terahertz Waves 30(5), 526–537 (2009)CrossRefGoogle Scholar
  27. Krishnan, E.V., Biswas, A.: Solutions of the Zakharov–Kuznetsov equation with higher order nonlinearity by mapping and ansatz methods. Phys. Wave Phenom. 18(4), 256–261 (2010)ADSCrossRefGoogle Scholar
  28. Mirzazadeh, M., Ekici, M., Sonomezoglu, A., Eslami, M., Zhou, Q., Zerrad, E., Biswas, A., Belic, M.: Optical solitons in nano-fibers with fractional temporal evolution. J. Comput. Theor. Nanosci. 13(8), 5361–5374 (2016a)CrossRefGoogle Scholar
  29. Mirzazadeh, M., Ekici, M., Sonomezoglu, A., Ortakaya, S., Eslami, M., Biswas, A.: Solitons solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics. Eur. Phys. J. Plus. 131(6), 166–177 (2016b)CrossRefGoogle Scholar
  30. Morris, R., Kara, A.H., Biswas, A.: Soliton solution and conservation laws of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity. Nonlinear Anal. Model. Control 18(2), 153–159 (2013)MathSciNetzbMATHGoogle Scholar
  31. Rawashdeh, M.S.: A reliable method for the space–time fractional Burgers and time-fractional Cahn–Allen equations via the FRDTM. Adv. Differ. Equ. 2017, 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  32. Saha, M., Sarma, A.M.: Study of modulation instability and solitary waves in nonlinear optical systems. Ph.D. Thesis, Indian Institute of Guwahati (2013a)Google Scholar
  33. Saha, M., Sarma, A.K.: Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms. Commun. Nonlinear Sci. Numer. Simulat. 18, 2420–2425 (2013b)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. Seadawy, A.R., Arshad, M., Lu, D.: Stability analysis of new exact traveling-wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems. Eur. Phys. J. Plus 132, 162 (2017)CrossRefGoogle Scholar
  35. Suarez, P., Biswas, A.: Exact 1-soliton solution of the Zakharov–Kuznetsov equation in plasmas with power law nonlinearity. Appl. Math. Comput. 217(17), 7372–7375 (2011)MathSciNetzbMATHGoogle Scholar
  36. Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Physica A Stat. Mech. Appl. 473, 352–362 (2017). ADSMathSciNetCrossRefGoogle Scholar
  37. Tascan, F., Bekir, A.: Travelling wave solutions of the Cahn–Allen equation by using first integral method. Appl. Math. Comput. 207, 279–282 (2009)MathSciNetzbMATHGoogle Scholar
  38. Wazwaz, A.M.: The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188, 1467–1475 (2007)MathSciNetzbMATHGoogle Scholar
  39. Yan, C., Yan, Z.: New exact solutions of (2+1)-dimensional Gardnerequation via the new sine-Gordon equation expansion method. Phys. Lett. A 224, 77–84 (1996)ADSMathSciNetCrossRefGoogle Scholar
  40. Zhou, Y., Cai, S., Liu, Q.: Bounded traveling waves of the (2+1)-dimensional Zoomeron equation. Math. Probl. Eng. 2015, 163597 (2015). MathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Science FacultyFirat UniversityElazigTurkey
  2. 2.Department of Mathematics, Science FacultyFederal University DutseJigawaNigeria
  3. 3.Department of MathematicsCankaya UniversityAnkaraTurkey
  4. 4.Institute of Space SciencesMagurele, BucharestRomania

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