On the solitary wave dynamics of complex Ginzburg–Landau equation with cubic nonlinearity

Article

Abstract

The complex Ginzburg–Landau equation with cubic nonlinearity is an ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. In this article the exact solutions for complex Ginzburg–Landau equation using first integral method and \((\frac{G'}{G})\)-expansion method are obtained. These methods can be applied to non-integrable equations as well as to integrable ones.

Keywords

First integral method \((\frac{G'}{G})\)-Expansion Nonlinear partial differential equations Exact solutions The complex Ginzburg–Landau equation 

References

  1. Akhmediev, N.N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)MATHGoogle Scholar
  2. Akram, G. Batool, F.: A Class of traveling wave solutions for space-time fractional biological population model in mathematical physics. Indian J. Phys. (2017) (accepted) Google Scholar
  3. Akram, G., Batool, F.: Solitary wave solutions of the Schafer–Wayne short-pulse equation using two reliable methods. Opt. Quantum Electron. 49, 1–9 (2017). doi: 10.1007/s11082-016-0856-8 CrossRefGoogle Scholar
  4. Bakonyi, Z., Michaelis, D., Peschel, U., Onishchukov, G., Lederer, F.: Dissipative solitons and their critical slowing down near a supercritical bifurcation. J. Opt. Soc. Am. B 19, 487–491 (2002)ADSCrossRefMATHGoogle Scholar
  5. Bourbaki, N.: Commutative Algebra. Addison-Wesley, Paris (1972)MATHGoogle Scholar
  6. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)ADSCrossRefGoogle Scholar
  7. Darvishi, M.T., Arbabi, S., Najafi, M., Wazwaz, A.M.: Traveling wave solutions of a (2+1)-dimensional Zakharov-like equation by the first integral method and the tanh method. Optik 127, 6312–6321 (2016)ADSCrossRefGoogle Scholar
  8. Ding, T.R., Li, C.Z.: Ordinary Differential Equations. Peking University Press, Peking (1996)Google Scholar
  9. Feng, Z.: On explicit exact solutions to the compound Burgers–KdV equation. Phys. Lett. A 293, 57–66 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. Gurefe, Y., Misirli, E.: Exp-function method for solving nonlinear evolution equations with higher order nonlinearity. Comput. Math. Appl. 61, 2025–2030 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. Ippen, E.P.: Principles of passive mode locking. Appl. Phys. B Lasers Opt. 58, 159–170 (1994)ADSCrossRefGoogle Scholar
  12. Kolodner, P.: Drifting pulses of traveling-wave convection. Phys. Rev. Lett. 66, 1165–1168 (1991)ADSCrossRefGoogle Scholar
  13. Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoullis equation approach. Nonlinear Dyn. 81, 1933–1949 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. Sakaguchi, H., Malomed, B.A.: Instabilities and splitting of pulses in coupled Ginzburg–Landau equations. Phys. D 154, 229–239 (2001)CrossRefMATHGoogle Scholar
  15. Triki, H., Wazwaz, A.M.: Bright and dark soliton solutions for a K(m, n) equation with t-dependent coefficients. Phys. Lett. A 373, 2162–2165 (2009)ADSCrossRefMATHGoogle Scholar
  16. Wang, M., Li, X., Zhang, J.: The \((\frac{G^{\prime }}{G})\)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)ADSMathSciNetCrossRefGoogle Scholar
  17. Wazwaz, A.M.: A sine–cosine method for handlingnonlinear wave equations. Math. Comput. Model. 40, 499–508 (2004)CrossRefMATHGoogle Scholar
  18. Wazwaz, A.M.: The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A 350, 367–370 (2006)ADSCrossRefMATHGoogle Scholar
  19. Zhang, H.: New application of the \((\frac{G^{\prime }}{G})\)-expansion method. Commun. Nonlinear Sci. 14, 3220–3225 (2009)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan

Personalised recommendations