Bekir, A., El Achab, A. (2014). Exact solutions of the 2D Ginzburg-Landau equation by the first integral method. Computational Methods for Differential Equations, 2(2), 69-76.

Ahmet Bekir; Abdelfattah El Achab. "Exact solutions of the 2D Ginzburg-Landau equation by the first integral method". Computational Methods for Differential Equations, 2, 2, 2014, 69-76.

Bekir, A., El Achab, A. (2014). 'Exact solutions of the 2D Ginzburg-Landau equation by the first integral method', Computational Methods for Differential Equations, 2(2), pp. 69-76.

Bekir, A., El Achab, A. Exact solutions of the 2D Ginzburg-Landau equation by the first integral method. Computational Methods for Differential Equations, 2014; 2(2): 69-76.

Exact solutions of the 2D Ginzburg-Landau equation by the first integral method

^{1}Eskisehir Osmangazi University, Art-Science Faculty,
Department of Mathematics-Computer

^{2}University of Choua¨ıb Doukkali

Receive Date: 08 July 2014,
Revise Date: 12 September 2014,
Accept Date: 23 August 2014

Abstract

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to non integrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the 2D Ginzburg-Landau equation.

[1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1991. DOI 10.1017/CBO9780511623998. [2] A. Bekir, The exp-function method for ostrovsky equation, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2009), 735–740. [3] A. Bekir and A. El Achab, Traveling wave solutions to the k(m, n) equation with generalized evolution using the first integral method, New Trends in Mathematical Sciences, 2 (2014), 12–17. [4] P. J. Blennerhassett, On the generation of waves by wind, Philos. Trans. Roy. Soc. London Ser. A, 298 (1980/81), 451–494. DOI 10.1098/rsta.1980.0265. [5] N. Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. [6] C. Chun, Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method, Phys. Lett. A, 372 (2008), 2760–2766. DOI 10.1016/j.physleta.2008.01.005. [7] X. Deng, Travelling wave solutions for the generalized burgers–huxley equation, Applied Mathematics and Computation, 204 (2008), 733–737. [8] F. Fang and Y. Xiao, Stability of chirped bright and dark soliton-like solutions of the cubic complex ginzburg–landau equation with variable coefficients, Optics communications, 268 (2006), 305–310.

[9] Z. Feng, On explicit exact solutions to the compound Burgers-KdV equation, Phys. Lett. A, 293 (2002), 57–66. DOI 10.1016/S0375-9601(01)00825-8. [10] J.-H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equations using Expfunction method, Chaos Solitons Fractals, 34 (2007), 1421–1429. DOI 10.1016/j.chaos.2006.05. 072. [11] R. Hirota, Exact solution of the korteweg¯de vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192–1194. DOI 10.1103/PhysRevLett.27.1192. [12] C. Huai-Tang and Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation, Chaos Solitons Fractals, 20 (2004), 765– 769. DOI 10.1016/j.chaos.2003.08.006. [13] K.-i. Maruno, A. Ankiewicz, and N. Akhmediev, Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation, Phys. D, 176 (2003), 44–66. DOI 10.1016/S0167-2789(02) 00708-X. [14] M. Miura, B¨acklund Transformation, Springer, Berlin, 1978. [15] H. T. Moon, P. Huerre, and L. G. Redekopp, Three-frequency motion and chaos in the GinzburgLandau equation, Phys. Rev. Lett., 49 (1982), 458–460. DOI 10.1103/PhysRevLett.49.458. [16] K. R. Raslan, The first integral method for solving some important nonlinear partial differential equations, Nonlinear Dynam., 53 (2008), 281–286. DOI 10.1007/s11071-007-9262-x. [17] N. Taghizadeh, M. Mirzazadeh, and F. Farahrooz, Exact solutions of the nonlinear schr¨odinger equation by the first integral method, Journal of Mathematical Analysis and Applications, 374 (2011), 549–553. [18] F. Ta¸scan and A. Bekir, Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method, Appl. Math. Comput., 215 (2009), 3134–3139. DOI 10.1016/j.amc.2009.09.027. [19] F. Tascan, A. Bekir, and M. Koparan, Travelling wave solutions of nonlinear evolution equations by using the first integral method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 1810–1815. [20] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vol. 68 of Applied Mathematical Sciences, Springer-Verlag, New York, second ed., 1997. DOI 10.1007/ 978-1-4612-0645-3. [21] M. Wang, X. Li, and J. Zhang, The ( G′ G )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417–423. DOI 10.1016/j.physleta.2007.07.051. [22] A.-M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Modelling, 40 (2004), 499–508. DOI 10.1016/j.mcm.2003.12.010. [23] Y. Zhou, M. Wang, and T. Miao, The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations, Physics Letters A, 323 (2004), 77–88.