Bright and dark soliton solutions for some nonlinear fractional differential equations

doi:10.1088/1674-1056/25/3/030203

Abstract In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona-Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann-Liouville sense.

Keywords: exact solutions ansatz method space-time fractional modified Benjamin-Bona-Mahoney equation time fractional mKdV equation

Received: 19 September 2015
Revised: 25 October 2015
Accepted manuscript online:

PACS:	02.30.Jr	(Partial differential equations)
	05.45.Yv	(Solitons)

Corresponding Authors: Ozkan Guner, Ahmet Bekir
E-mail: ozkanguner@karatekin.edu.tr;abekir@ogu.edu.tr

Cite this article:

Ozkan Guner, Ahmet Bekir Bright and dark soliton solutions for some nonlinear fractional differential equations 2016 Chin. Phys. B 25 030203

[1]	Podlubny I 1999 Fractional Differential Equations (California: Academic Press)
[2]	Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier)
[3]	Zheng B 2012 Commun. Theor. Phys. 58 623
[4]	Bekir A and Güner Ö 2013 Chin. Phys. B 22 110202
[5]	Lu B 2012 J. Math. Anal. Appl. 395 684
[6]	Zhang S and Zhang H Q 2011 Phys. Lett. A 375 1069
[7]	Tong B, He Y, Wei L and Zhang X 2012 Phys. Lett. A 376 2588
[8]	Bekir A, Güner Ö and Cevikel A C 2013 Abstr. Appl. Anal. 2013 426462
[9]	Güner Ö and Cevikel A C 2014 The Scientific World Journal 2014 489495
[10]	Liu W and Chen K 2013 Pramana-J. Phys. 81 3
[11]	Güner Ö and Eser D 2014 Adv. Math. Phys. 2014 456804
[12]	Bulut H, Baskonus H M and Pandir Y 2013 Abstr. Appl. Anal. 2013 636802
[13]	Younis M, Rehman H and Iftikhar M 2014 Appl. Math. Comput. 249 81
[14]	Güner Ö 2015 Chin. Phys. B 24 100201
[15]	Mirzazadeh M 2015 Pramana-J. Phys. 85 17
[16]	Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives: Theory and Applications (Switzerland: Gordon and Breach Science Publishers)
[17]	Caputo M 1967 Geophys. J. Royal Astronom. Soc. 13 529
[18]	Jumarie G 2006 Comput. Math. Appl. 51 1367
[19]	Jumarie G 2009 Appl. Maths. Lett. 22 378
[20]	He J H, Elegan S K and Li Z B 2012 Phys. Lett. A 376 257
[21]	Saad M, Elagan S K, Hamed Y S and Sayed M 2013 Int. J. Basic Appl. Sci. 13 23
[22]	Elghareb T, Elagan S K, Hamed Y S and Sayed M 2013 Int. J. Basic Appl. Sci. 13 19
[23]	Alzaidy J F 2013 British J. Math. Comput. Sci. 3 153
[24]	Ege S M and Misorli E 2014 Adv. Diff. Equ. 2014 135
[25]	Bekir A, Güner Ö and Unsal O 2015 J. Comput. Nonlinear Dyn. 10 021020
[26]	Yusufoglu E and Bekir A 2008 Chaos Solitons Fractals 38 1126
[27]	Wazwaz A M 2006 Phys. Lett. A 355 358
[28]	Yusufoglu E 2008 Phys. Lett. A 372 442
[29]	Abbasbandy S and Shirzadi A 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 1759
[30]	Cevikel A C, Bekir A, Akar M and San S 2012 Pramana-J. Phys. 79 337
[31]	Triki H and Wazwaz A M 2009 Phys. Lett. A 373 2162
[32]	Inc M, Ulutas E and Biswas A 2013 Chin. Phys. B 22 060204
[33]	Ebadi G and Biswas A 2011 Math. Comput. Model. 53 694
[34]	Biswas A, Triki H and Labidi M 2011 Physics of Wave Phenomena 19 24
[35]	Bekir A and Güner Ö 2013 Pramana-J. Phys. 81 203
[36]	Younis M and Ali S 2014 Appl. Math. Comput. 246 460
[37]	Bekir A, Aksoy E and Güner Ö 2012 Phys. Scr. 85 035009
[38]	Triki H and Wazwaz A M 2011 Appl. Math. Comput. 217 8846
[39]	Bekir A and Güner Ö 2013 Ocean Engineering 74 276
[40]	Kurulay M and Bayram M 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 1777
[41]	Zayed E M E and Amer Y A 2014 Int. J. Phys. Sci. 9 174
[42]	Zayed E M E and Arnous A H 2014 Eur. J. Acad. Essays 1 6
[43]	Güner Ö, Bekir A and Bilgil H 2015 Adv. Nonlinear Anal. 4 201
[44]	Abazari R 2011 Appl. Math. Sci. 5 2943
[45]	Alquran M and Al-Khaled K 2012 Math. Sci. 6 1
[46]	Qawasmeh A 2013 J. Math. Comput. Sci. 3 1475
[47]	Khan K and Akbar M A 2014 Ain Shams Eng. J. 5 247

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Bright and dark soliton solutions for some nonlinear fractional differential equations

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