New travelling wave solutions for fractional regularized long-wave equation and fractional coupled Nizhnik-Novikov-Veselov equation

Ozkan Guner


In this paper, solitary-wave ansatz and the (G′/G)-expansion methods have been used to obtain exact solutions of the fractional regularized long-wave (RLW) and coupled Nizhnik-Novikov-Veselov (NNV) equation. As a result, three types of exact analytical solutions such as rational function solutions, trigonometric function solutions, hyperbolic function solutions are formally derived these equations. Proposed methods are more powerful and can be applied to other fractional differential equations arising in mathematical physics.


Exact solution; ansatz method; (G′/G)-expansion method; fractional regularized long-wave equation; fractional coupled Nizhnik-Novikov-Veselov equation.

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Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.

Podlubny, I. (1999). Fractional Differential Equations, Academic Press, California.

Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam


Song, L., Wang, W. (2013). A new improved Adomian decomposition method and its application to fractional diff tial equations. Applied Mathematical Modelling. 37 (3) 1590–1598.

Wang, Q. (2007). Homotopy perturbation method for fractional KdV equation. Appl. Math. Comput.. 190 1795-802.

Wu, G.C. and Baleanu, D. (2013). Variational iteration method for fractional calculus - a universal approach by Laplace transform. Advances in Difference Equations. 2013 18.

Cui, M. (2009). Compact fi diff method for the fractional diff equation. Journal of Computational Physics. 228 (20) 7792–

Khan, N.A., Ara, A. and Mahmood, A. (2012). Numerical solutions of time-fractional Burgers equations: a comparison between generalized

diff tial transformation technique and homotopy perturbation method. Int. J. Num. Meth. Heat & Fl. Flow. 22 (2) 175-193.

Song, L. and Zhang, H. (2007). Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys. Lett. A. 367 88–94.

El-Ajou, A., Odibat, Z., Momani, S. and Alawneh, A. (2010). Construction of analytical solutions to fractional diff tial equations using homotopy analysis method. Int. J. Appl. Math. 40 (2) 43-51.

Tian, S.F. and Zhang, H.Q. (2012). On the integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation. Journal of Physics A: Mathematical and Theoretical. 45, 055203. .

Tian, S.F. and Zhang, H.Q. (2014). On the integrability of a generalized variable-coefficient forced Korteweg-de Vries equation in fl Stud. Appl. Math.132, 212–246.

Mohyud-Din, S., Yıldırım, A. and Yu¨lu¨klu¨, E. (2012). Homotopy analysis method for space- and time-fractional KdV equation, Int. J. Num. Meth. Heat & Fl. Flow. 22 (7), 928-941.

Sahoo, S. and Ray, S.S. (2015). Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV–Zakharov– Kuznetsov equations. Computers and Mathematics with Applications. 70, 158–166.

Zhang, S. and Zhang, H.Q. (2011). Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A. 375, 1069– 1073.

Tian, S.F. (2017). Initial–boundary value problems for the general coupled nonlinear Schro¨dinger equation on the interval via the Fokas method. J. Differential Equations, 262, 506-558.

Tian, S.F. (2016). The mixed coupled nonlinear Schro¨dinger equation on the half-line via the Fokas method. Proc. R. Soc. Lond. A. 472, 20160588.

Bekir, A., Guner, O. and Unsal, O. (2015). The fi integral method for exact solutions of nonlinear fractional diff tial equations, J. Comput. Nonlinear Dynam. 10(2), 021020-5.

Baleanu, D., U˘gurlu, Y. and Kilic, B. (2015). Improved (Gt/G)− expansion method for the time-fractional biological population model and Cahn–Hilliard equation. J. Comput. Nonlinear Dynam. 10 (5) 051016.

Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L. and Zhang, T.T. (2016).

On periodic wave solutions with asymptotic behaviors to a (3+1)- dimensional generalized B-type Kadomtsev–Petviashvili equation in fl dynamics. Comput. & Math. Appl. 72, 2486–2504.

Bekir, A., Guner, O., Bhrawy, A.H. and Biswas, A. (2015). Solving nonlinear fractional diff rential equations using exp-function and (Gt/G)-expansion methods. Rom. Journ. Phys. 60(3-4), 360-378.

Bulut, H., Baskonus, H M. and Pandir, Y. (2013). The modifi trial equation method for fractional wave equation and time fractional generalized Burgers equation. Abstract and Applied Analysis. 636802.

Guner, O. and Eser, D. (2014). Exact solutions of the space time fractional symmetric regularized long wave equation using diff t methods. Advances in Mathematical Physics. 456804 .

Zhang, S., Zong Q-A., Liu, D. and Gao, Q. (2010). A generalized exp-function method for fractional riccati differential equations. Communications in Fractional Calculus. 1, 48-51.

Guner, O., Bekir, A. and Bilgil, H. (2015). A note on exp-function method combined with complex transform method applied to fractional diff tial equations. Adv. Nonlinear Anal. 4(3), 201–208.

Kaplan, M., Bekir, A., Akbulut, A. and Aksoy, E. (2015). The modifi simple equation method for nonlinear fractional diff tial equations. Rom. Journ. Phys. 60(9-10), 1374–1383.

Kaplan, M., Akbulut, A. and Bekir, A. (2016). Solving space-time fractional diff tial equations by using modifi simple equation method, Commun. Theor. Phys. 65(5), 563–568.

Aksoy, E., Kaplan, M. and Bekir, A. (2016). Exponential rational function method for space-time fractional diff tial equations. Waves in Random and Complex Media. 26(2), 142-151.

Guner, O. (2015). Singular and non-topological soliton solutions for nonlinear fractional diff tial equations. Chin. Phys. B. 24(10), 100201.

Guner, O. and Bekir, A. (2016). Bright and dark soliton solutions for some nonlinear fractional diff tial equations. Chin. Phys. B. 25(3), 030203.

Tu, J.M., Tian, S.F., Xu, M.J. and Zhang, T.T. (2016). Quasi-periodic waves and solitary waves to a generalized KdV-Caudrey-Dodd-Gibbon equation from fl dynamics. Taiwanese J. Math. 20, 823-848.

Tu, J.M., Tian, S.F., Xu, M.J. and Zhang, T.T. (2016). On Lie

symmetries, optimal systems and explicit solutions to the Kudryashov– Sinelshchikov equation, Appl. Math. Comput. 275, 345–352.

Lin, S.D. and Lu, C.H. (2013). Laplace transform for solving some families of fractional diff tial equations and its applications. Advances in Difference Equations. 137.

Srivastava, H.M., Golmankhaneh, A.K., Baleanu, D. and Yang, X.J. (2014). Local fractional Sumudu transform with application to IVPs on Cantor sets. Abstract and Applied Analysis. 620529.

Wang, X.B., Tian, S.F., Xua, M.J. and Zhang T.T. (2016). On integrability and quasi-periodic wave solutions to a (3+1)-dimensional generalized KdV-like model equation. Appl. Math. Comput. 283, 216– 233.

Arnous, A.H. and Mirzazadeh, M. (2015). B¨acklund transformation of fractional Riccati equation and its applications to the space–time FDEs. Mathematical Methods in the Applied Sciences. 38(18), 4673–4678.

Feng, L.L., Tian, S.F., Wang, X.B. and Zhang, T.T. (2017). Rogue waves, homoclinic breather waves and soliton waves for the (2+1)- dimensional B-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97.

Tian, S.F., Zhang, Y.F., Feng, B.L. and Zhang, H.Q. (2015). On the Lie algebras, generalized symmetries and Darboux transformations of the fi evolution equations in shallow water. Chin. Ann. Math. 36B, 543–560.

Tian, S.F., Wang, Z. and Zhang, H.Q. (2010). Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources. J. Math. Anal. Appl. 366, 646-662.

Al-Shara, S. (2014). Fractional transformation method for constructing solitary wave solutions to some nonlinear fractional partial diff ential equations. Applied Mathematical Sciences. 8, 5751-5762.

Xu, M.J., Tian, S.F., Tu, J.M. and Zhang T.T. (2016). B¨acklund transformation, infi conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation. Nonlinear Anal.: Real World Appl. 31, 388–408.

Li, Z.B. and He, J. H. (2011). Application of the fractional complex transform to fractional diff tial equations. Nonlinear Sci. Lett. A. 2121-126.

Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Royal Astronom. Soc. 13, 529- 539.

Jumarie, G. (2006). Modifi Riemann–Liouville derivative and fractional Taylor series of nondiff tiable functions further results. Comput. Math. Appl. 51, 1367–1376.

Jumarie, G. (2009). Table of some basic fractional calculus formulae derived from a modifi Riemann-Liouvillie derivative for nondiff tiable functions. Appl. Maths. Lett.. 22, 378-385.

He, J H., Elegan, S.K. and Li, Z.B. (2012). Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A. 376, 257–259.

Biswas, A. (2008). 1-soliton solution of the K(m, n) equation with generalized evolution. Phys. Lett. A. 372, 4601-4602.

Triki, H., Wazwaz, A.M. (2009). Bright and dark soliton solutions for a K(m, n) equation with t-dependent coefficients. Phys. Lett. A. 373, 2162–2165.

Bekir, A., Guner, O. (2013). Bright and dark soliton solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation. Pramana J. Phys. 81, 203.

Triki, H., Milovic, D. and Biswas, A. (2013). Solitary waves and shock waves of the KdV6 equation. Ocean Engineering. 73, 119–125.

Younis, M. and Ali, S. (2015). Bright, dark, and singular solitons in magneto-electro-elastic circular rod. Waves in Random and Complex Media. 25(4), 549-555.

Bekir, A. and Guner, O. (2013). Topological (dark) soliton solutions for the Camassa–Holm type equations. Ocean Engineering. 74, 276–279.

Abdel-Salam, E.A.B., Hassan, G.F. (2016). Solutions to class of linear and nonlinear fractional differential equations. Commun. Theor. Phys.. 65, 127–135.

Peregrine, D.H. (1966). Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330.

Peregrine, D.H. (1967). Long waves on a beach. J. Fluid Mech. 27, 815– 827.

Benjamin, T.B., Bona, J.L. and Mahony, J. (1972). Model equations for waves in nonlinear dispersive systems. J. Philos. Trans. R. Soc. Lond. 227, 47–78.

Abdel-Salam, E.A.B., Yousif, E.A. (2013). Solution of nonlinear space-time fractional diff tial equations using the fractional riccati expansion method. Mathematical Problems in Engineering. 846283.

Esen, A. and Kutluay, S. (2006). Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174, 833–845.

Dag, I. (2000). Least square quadratic B-spline fi element method for the regularized long wave equation. Comp. Meth. Appl. Mech. Eng. 182, 205–215.

Dag, I. and Ozer, M.N. (2001). Approximation of RLW equation by least square cubic B-spline fi element method. Appl. Math. Model. 25, 221–231.

Saka, B., Dag, I. and Irk, D. (2008). Quintic B-spline collocation method for numerical solutions of the RLW equation. Anziam J. 49(3), 389–410.

Saka, B., Sahin, A., Dag, I. (2011). B-spline collocation algorithms for numerical solution of the RLW equation. Numer. Meth. Part. D. E. 27, 581–607.

Yusufoglu, E. and Bekir, A. (2007). Application of the variational iteration method to the regularized long wave equation. Computers and Mathematics with Applications. 54, 1154–1161.

Liu, Y. and Yan, L. (2013). Solutions of fractional Konopelchenko- Dubrovsky and Nizhnik-Novikov-Veselov equations using a generalized fractional subequation method. Abstract and Applied Analysis. 839613.

Hong, T., Wang, Y.Z. and Huo, Y.S. (1998). Bogoliubov quasiparticles carried by dark solitonic excitations in nonuniform Bose Einstein condensates. Chin. Phys. Lett. 15, 550 552.

Das, G.C. (1997). Explosion of soliton in a multicomponent plasma.

Phys. Plasmas. 4, 2095-2100.

Lou, S.Y. (1999). A direct perturbation method: Nonlinear Schrodinger equation with loss. Chin. Phys. Lett. 16, 659-661.

Shin, B.C., Darvishi, M.T. and Barati, A. (2009). Some exact and new solutions of the Nizhnik Novikov Vesselov equation using the Exp- function method. Computers and Mathematics with Applications. 58, 2147-2151.

Deng, C. (2010). New abundant exact solutions for the (2 + 1)- dimensional generalized Nizhnik–Novikov–Veselov system. Commun Nonlinear Sci Numer Simulat. 15, 3349–3357

Boubir, B., Triki, H. and Wazwaz, A.M. (2013). Bright solitons of the variants of the Novikov–Veselov equation with constant and variable coefficients. Applied Mathematical Modelling. 37, 420–431.

Wang, M.L., Li X. and Zhang, J. (2008). The (Gt/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A. 372(4), 417-423.



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