A Unified Analysis of Exact Traveling Wave Solutions for the Fractional-Order and Integer-Order Biswas–Milovic Equation: Via Bifurcation Theory of Dynamical System

• Zhang, Bei ^[1] ; Zhu, Wenjing ^[2] ; Xia Yonghui ^[3] ; Bai Yuzhen
  1. [1] Huaqiao University

     Huaqiao University

     China

     
  2. [2] China Jiliang University

     China Jiliang University

     China

     
  3. [3] Zhejiang Normal University

     Zhejiang Normal University

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00352-x
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• Resumen

  • This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential equations. We use the conformable fractional derivatives. The method is based on the bifurcation theory of planar dynamical systems. To show the effectiveness of this method, we choose Biswas–Milovic (for short, BM) equation with conformable derivative as an application. Also comparison is presented for the exact traveling wave solutions between the integer-order BM equation and fractional-order BM equation. It is believed that this approach can be extended to other nonlinear fractional-order partial differential equations.

• Referencias bibliográficas
  • 1. Biswas, A., Milovic, D.: Bright and dark solitons of the generalized nonlinear Schröndinger equation. Commun. Nonlinear Sci. Numer. Simul....
  • 2. Eslami, M., Mirzazadeh, M., Neirameh, A.: New exact wave solutions for Hirota equation. Pramana 84, 3–8 (2015)
  • 3. Eslami, M., Neirameh, A.: New exact solutions for higher order nonlinear Schrödinger equation in optical fibers. Opt. Quant. Electron....
  • 4. Eslami, M., Neyrame, A., Ebrahimi, M.: Explicit solutions of nonlinear (2+1)-dimensional dispersive long wave equation. J. King Saud...
  • 5. Kara, A.H., Biswas, A., Zhou, Q., Moraru, L., Moshokoa, S.P., Belic, M.: Conservation laws for optical solitons with Chen–Lee–Liu equation....
  • 6. Li, B., Zhao, J., Triki, H., Ekici, M., Mirzazadeh, M., Zhou, Q., Liu, W.: Soliton interactions for optical switching systems with symbolic...
  • 7. Liu, J.G., Eslami, M., Rezazadeh, H., Mirzazadeh, M.: Rational solutions and lump dolutions to a nonisospectral and generalized variable-coefficient...
  • 8. Nazarzadeh, A., Eslami, M., Mirzazadeh, M.: Exact solutions of some nonlinear partial differential equations using functional variable...
  • 9. Zubair, A., Raza, N., Mirzazadeh, M., Liu, W., Zhou, Q.: Analytic study on optical solitons in paritytime-symmetric mixed linear and nonlinear...
  • 10. Liu, W.J., Liu, M.L., Lin, S., Liu, J.C., Lei, M., Wu, H., Dai, C.Q., Wei, Z.Y.: Synthesis of high quality silver nanowires and their...
  • 11. Liu, W.J., Liu, M.L., Liu, B., Quhe, R.G., Lei, M., Fang, S.B., Teng, H., Wei, Z.Y.: Nonlinear optical properties of MoS2 − W S2 heterostructure...
  • 12. Liu, W., Liu, M., Wang, X., Shen, T., Chang, G., Lei, M., Deng, H., Wei, Z., Wei, Z.: Thicknessdependent ultrafast photonics of SnS2 nanolayers...
  • 13. Liu, M., Liu, W., Wei, Z.: MoT e2 saturable absorber with high modulation depth for erbium-doped fiber laser. J. Lightwave Technol. 37,...
  • 14. Liu, M., Ouyang, Y., Hou, H., Liu, W., Wei, Z.: Q-switched fiber laser operating at 1.5 μm based on WTe2. Chin. Opt. Lett. 17, 020006...
  • 15. Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
  • 16. Wu, Q., Huang, J.: Fractional Differential Equations. Qinghua University, Beijing (2015)
  • 17. Ahmadiana, S., Darvishi, M.: A new fractional Biswas–Milovic model with its periodic soliton solutions. Optik 127, 7694–7703 (2016)
  • 18. Eslami, M.: Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations. Appl. Math. Comput. 285, 141–148...
  • 19. Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable timefractional derivative. Calcolo 53, 475–485...
  • 20. Khodadad, F.S., Nazzari, F., Eslami, M., Rezazadeh, H.: Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with...
  • 21. Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., Asghari, R.: Traveling wave solution of conformable fractional generalized reaction...
  • 22. Liu, C.: Counterexamples on Jumarie’s two basic fractional calculus formulae. Commun. Nonl. Sci. Numer. Simulat. 22, 92–94 (2015)
  • 23. Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput....
  • 24. Khalil, R., Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
  • 25. Foroutana, M., Kumarb, D., Manafiand, J., Hoquee, A.: New explicit soliton and other solutions for the conformable fractional Biswas–Milovic...
  • 26. Jafari, H., Sooraki, A., Khalique, C.M.: Dark solitons of the Biswas–Milovic equation by the first integral method. Optik 124, 3929–3932...
  • 27. Mirzazadeh, M., Arnous, A.H.: Exact solution of Biswas–Milovic equation using new efficient method. Electron. J. Math. Anal. Appl. 3,...
  • 28. Ma, W.X., Huang, T.W., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr....
  • 29. Eslami, M., Mirzazadeh, M.: Optical solitons with Biswas–Milovic equation for power law and dualpower law nonlinearities. Nonlinear Dyn....
  • 30. Najafi, M., Arbabi, S.: Dark soliton and periodic wave solutions of the Biswas–Milovic equation. Optik 127, 2679–2682 (2016)
  • 31. Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur....
  • 32. Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 61 (2015)
  • 33. Manafian, J., Lakestani, M.: Application of tan(phi/2)-expansion method for solving the Biswas– Milovic equation for Kerr law nonlinearity....
  • 34. Zhou, Q., Ekici, M., Sonmezoglu, A., Mirzazadeh, M., Eslami, M.: Analytical study of solitons to Biswas–Milovic model in nonlinear optics....
  • 35. Zhou, Q., Ekici, M., Sonmezoglu, A., Mirzazadeh, M.: Optical solitons with Biswas–Milovic equation by extended G /G-expansion method....
  • 36. Wang, M.L., Li, X.Z., Zhang, J.L.: The G /G-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical...
  • 37. Biswas, A., Mirzazadeh, M., Eslami, M.: Dispersive dark optical soliton with Schödinger–Hirota equation by G /G-expansion approach in...
  • 38. Mirzazadeh, M., Eslami, M., Biswas, A.: Soliton solutions of the generalized Klein–Gordon equation by using ( G G )-expansion method....
  • 39. Eslami, M.: Trial solution technique to chiral nonlinear Schrodinger’s equation in (1+2)-dimensions. Nonliear Dyn. 85, 813–816 (2016)
  • 40. Zhou, Q., Ekici, M., Sonmezoglu, A., Mirzazadeh, M., Eslami, M.: Optical solitons with Biswas– Milovic equation by extended trial equation...
  • 41. Zhou, Q.: Optical solitons for Biswas–Milovic model with Kerr law and parabolic law nonlinearities. Nonlinear Dyn. 84, 677–681 (2016)
  • 42. Kumar, S.: Invariant solutions of Biswas–Milovic equation. Nonlinear Dyn. 87(2), 1153–1157 (2017)
  • 43. Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for Biswas–Milovic Model in nonlinear optics by Sine-Gordon equation method....
  • 44. Rizvi, S.T.R., Bashir, S., Ali, K., Ahmad, S.: Jacobian elliptic periodic traveling wave solutions for Biswas–Milovic equation. Optik...
  • 45. Raza, N., Abdullah, M., Butt, A.R.: Analytical soliton solutions of Biswas–Milovic equation in Kerr and non-Kerr law media. Optik 157,...
  • 46. Arshed, S., Biswas, A., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with differential group delay and parabolic...
  • 47. Yu, J., Sun, Y.: Exact traveling wave solutions to the (2 + 1)-dimensional Biswas–Milovic equations. Optik 149, 378–383 (2017)
  • 48. Raza, N., Javid, A.: Optical dark and singular solitons to the Biswas–Milovic equation in nonlinear optics with spatio-temporal dispersion....
  • 49. Li, J.: Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions. Science, Beijing (2013)
  • 50. Li, J., Chen, G.: On a class of singular nonlinear traveling wave equations. Int. J. Bifurcat. Chaos 17, 4049–4065 (2007)
  • 51. Song, Y., Tang, X.: Stability, steady-state bifurcations, and turing patterns in a predator–prey model with herd behavior and prey-taxis....
  • 52. Zhang, B., Xia, Y., Zhu, W., Bai, Y.: Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon...
  • 53. Zhu, W., Xia, Y., Zhang, B., Bai, Y.: Exact traveling wave solutions and bifurcations of the time fractional differential equations with...
  • 54. Zhu, W., Li, J.: Exact traveling wave solutions and birfurcations of the Biswas–Milovic equation. Nonlinear Dyn. 84, 1973–1987 (2016)