Abstract
Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg–Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted.
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Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)
Bartuccelli, M., Constantin, P., Doering, C.R., Gibbon, J.D., Gisselfält, M.: Hard turbulence in a finite-dimensional dynamical system. Phys. Lett. A. 142, 349–356 (1989)
Stewartson, K., Stuart, J.T.: A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545 (1971)
Newell, A.C., Whitehead, J.A.: Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303 (1969)
Newell, A.C., Whitehead, J.A.: Review of the finite bandwidth concept. In: Leipholz, H. (ed.) Instability of Continuous Systems, pp. 279–303. Springer, Berlin (1971)
Segel, L.A.: Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224 (1969)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)
Choudhury, S.: Bifurcations and strongly amplitude-modulated pulses of the complex Ginzburg–Landau equation. In: Dissipative Solitons, Lecture Notes in Physics, vol. 661, pp. 429–443. Springer, Berlin (2005)
Doering, C., Gibbon, J., Holm, D., Nicolaenko, B.: Low-dimensional behavior in the complex Ginzburg–Landau equation. Nonlinearity 1, 279–309 (1988)
Ma, T., Park, J., Wang, S.: Dynamic bifurcation of the Ginzburg–Landau equation. SIAM J. Appl. Dyn. Syst. 3, 620–635 (2004)
Park, J.: Bifurcation and stability of the generalized complex Ginzburg–Landau equation. Pure Appl. Anal. 7(5), 1237–1253 (2008)
Doelman, A.: Traveling waves in the complex Ginzburg–Landau equation. J. Nonlinear Sci. 3, 225–266 (1993)
Tang, Y.: Numerical simulations of periodic travelling waves to a generalized Ginzburg–Landau equation. Appl. Math. Comput. 165, 155–161 (2005)
Wazwaz, A.: Exact Explicit and implicit solutions for the one-dimensional cubic and quintic complex Ginzburg–Landau equations. Appl. Math. Lett. 19, 1007–1012 (2006)
Zhong, Penghong, Yang, Ronghui, Yang, Ganshan: Exact periodic and blow up solutions for 2D Ginzburg–Landau equation. Phys. Lett. A 373, 19–22 (2008)
Liu, W.Y., Yu, Y.J., Chen, L.D.: Variational principles for Ginzburg–Landau equation by He’s semi-inverse method. Chaos Soliton. Fract. 33, 1801–1803 (2007)
Zhang, J.L., Wang, M.L., Gao, K.Q.: Exact solutions of generalized Zakharov and Ginzburg–Landau equations. Chaos Soliton Fract. 32, 1877–1886 (2007)
Mancas, S.C., Choudhury, S.R.: Traveling wave trains in the complex cubic-quintic Ginzburg–Landau equation. Chaos Soliton. Fract. 28, 834–843 (2006)
Mancas, S.C., Choudhury, S.R.: Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: plane wave (CW) solutions. Chaos Soliton Fract. 27, 1256–1271 (2006)
Mortazavi, M., Mirzazadeh, M.: Some new exact traveling wave solutions one dimensional modified complex Ginzburg–Landau equation. Comput. Methods Differ. Equ. 3(2), 70–86 (2015)
Bekir, A., El Achab, A.: Exact solutions of the 2D Ginzburg–Landau equation by the first integral method. Comput. Methods Differ. Equ. 2, 63–70 (2014)
Mirzazadeh, M., et al.: Optical solitons with complex Ginzburg–Landau equation. Nonlinear Dyn. 85(2), 1979–2016 (2016)
Kaplan, M., Bekir, A., Akbulut, A.: A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dyn. 85, 2843–2850 (2016)
Mirzazadeh, M., Eslami, M., Biswas, A.: 1-Soliton solution of KdV6 equation. Nonlinear Dyn. 80(1–2), 387–396 (2015)
Mirzazadeh, M., Eslami, M., Biswas, A.: Soliton solutions of the generalized Klein–Gordon equation by using \(\frac{G^{\prime }}{G}\)-expansion method. Comput. Appl. Math. 33(3), 831–839 (2014)
Bekir, A., Ayhan, B., Naci Özer, M.: Numerical simulations of periodic travelling waves to a generalized Ginzburg–Landau equation. Chin. Phys. B 22(1), 010202 (2013)
Bekir, A., Ayhan, B., Naci Özer, M.: New exact travelling wave solutions of nonlinear physical models. Chaos Solitons Fract. 41(4), 1733–1739 (2009)
Çevikel, A.C., Bekir, A., Akar, M., San, S.: A procedure to construct exact solutions of nonlinear evolution equations. Pramana: J. Phys. 79(3), 337–344 (2012)
Triki, H., Mirzazadeh, M., Bhrawy, A.H., Razborova, P., Biswas, A.: Solitons and other solutions to long-wave short-wave interaction equation. Rom. J. Phys. 60(1–2), 72–86 (2015)
El Achab, A.: Constructing of exact solutions to the nonlinear Schrödinger equation (NLSE) with power-law nonlinearity by the Weierstrass elliptic function method. Optik: Int. J. Light Electron Opt. 127(3), 1229–1232 (2016)
Guo-cheng, Wu, Tie-cheng, Xia: A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations. Phys. Lett. A 372, 604–609 (2008)
Guo-cheng, Wu, Tie-cheng, Xia: A new method for constructing soliton solutions to differential-difference equation with symbolic computation. Chaos Solitons Fract. 39, 2245–2248 (2009)
Jun-Min, W., Jie, J.: Algebraic method for constructing exact discrete soliton solutions of toda and mKdV lattices. Commun. Theor. Phys. (Beijing, China) 49, 1407–1409 (2008)
Batool, B., Akram, G.: On the solitary wave dynamics of complex Ginzburg–Landau equation with cubic nonlinearity. Opt. Quant. Electron. 49, 1 (2017)
Davey, A., Stewartson, K.: On three-dimensional packets of surfaces waves. Proc. R. Soc. Lond. Ser. A 338, 101–110 (1974)
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El Achab, A., Amine, A. A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg–Landau equation. Nonlinear Dyn 91, 995–999 (2018). https://doi.org/10.1007/s11071-017-3924-0
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DOI: https://doi.org/10.1007/s11071-017-3924-0