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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Exact Analytical Solutions : Physical and/or Mathematical Validity

Journal of Applied Nonlinear Dynamics 10(1) (2021) 95--109 | DOI:10.5890/JAND.2021.03.006

P.H. Kamdoum-Tamo$^{1,2}$ , E. Tala-Tebue$^{1,3}$, A. Kenfack-Jiotsa$^{1,2}$, T.C. Kofane$^{1}$

$^{1}$ Laboratory of Mechanics, Department of Physics, Faculty of Sciences, and African Center of Excellence in I.C.T (C.E.T.I.C) University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

$^{2}$ Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher Teachers Training College, University of Yaounde I, P.O. Box 47 Yaounde, Cameroon

$^{3}$ Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Cameroon

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Abstract

In this work, we use the alternative ($G'/G$)-expansion method, the sech method, the tanh method and the Painleve truncated approach to find solutions of the modified complex Ginzburg-Landau equation. We show that any mathematically acceptable solution is not necessarily physically suitable. Among the two types of obtained solutions, there is a category with null infinite branches, for which no direct numerical simulation can be carried out. This type of solutions is however mathematically well-grounded. The second type concerns new solutions with infinite non-zero branches. For this second type, direct numerical simulations are performed to show that they are physically valid.

References

  1. [1] Li, D.S. and Zhang, H.Q. (2004), A new extended tanh-function method and its application to the dispersive long wave equations in (2+1) dimensions, ph{Appl. Math. Comput.}, 147, 789-797.
  2. [2]  Bekir, A. (2008), New solitons and periodic wave solutions for some nonlinear physical models by using the sine-cosine method, ph{Phys. Scr.}, 77, 501-505.
  3. [3]  Mehdi, D. and Fatemeh, S. (2007), Solution of a partial differential equation subject to temperature overspecification by Hes homotopy perturbation method, ph{Phys. Scr.}, 75, 778-787.
  4. [4]  Hirota, R. (1971), Exact solution of the Korteweg-de-Vries equation for multiple collisions of solitons, ph{Phys. Rev. Lett.}, 27, 1192-1194.
  5. [5]  Zayed, E.M.E. and Abdelaziz, M.A.M. (2011), Exact solutions for the nonlinear Schr\"{o}dinger equation with variable coefficients using the generalized extended tanh-function, the sine-cosine and the exp-function method, ph{Appl. Math. Comput.}, 218, 2259-2268.
  6. [6]  Tsigaridas, G., Fragos, A., Polyzos, I., Fakis, M., Ioannou, A., Giannetas, V., and Persephonis, P. (2005), Evolution of near-soliton initial conditions in non-linear wave equations through their B\"{a}klund transforms, ph{Chaos Solitons Fract.}, 23, 1841-1854.
  7. [7]  Osman, M.S., Abdel-Gawad, H.I., and El Mahdy, M.A. (2018), Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, ph{Results in Physics}, 8, 1054-1060.
  8. [8]  Osman, M.S. (2018), On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients, ph{Optik}, 156, 169-174.
  9. [9]  Osman, M.S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., and Zhou, Q. (2018), The unified method for conformable time fractional Schr\"{o}dinger equation with perturbation terms, ph{Chin. J. of Phys.}, 56, 2500-2506.
  10. [10]  Rezazadeh, H., Osman, M.S., Eslami, M., Ekici, M., Sonmezoglu, A., Asma, M., Othman, W.A.M., Wong, B.R., Mirzazadeh, M., Zhou, Q., Biswas, A., and Belic, M. (2018), Mitigating Internet bottleneck with fractional temporal evolution of optical solitonshaving quadratic-cubic nonlinearity, ph{Optik}, 164, 84-92.
  11. [11]  Abdel-Gawad, H.I. and Osman, M.S. (2014), Exact solutions of the Korteweg-de-Vries equation with space and time dependent coefficients by the extended unified method, ph{Ind. J. Pure Appl. Math.}, 45, 1-12.
  12. [12]  Osman, M.S., Ghanbari, B., and Machado, J.A.T. (2019), New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity, ph{Eur. Phys. J. Plus}, 134, 20-30.
  13. [13]  Suzo, A.A. (2005), Intertwining technique for the matrix Schr\"{o}dinger equation, ph{Phys. Lett. A}, 335, 88-102.
  14. [14]  Abdul-Majid, W. and Osman, M.S. (2018), Analyzing the combined multi-waves polynomial solutions in a two-layer-liquid medium, ph{Comp. and Math. with Appl.}, 76, 276-283.
  15. [15]  Osman, M.S. and Machado, J.A.T. (2018), New monautonomous combined multi-wave solutions for (2+1)-dimensional variable coefficients KdV equation, ph{Nonlinear Dyn.}, 93, 733-740.
  16. [16]  Osman, M.S. and Machado, J.A.T. (2017), The dynamical behavior of mixed-type soliton solutions descibed by (2+1)-dimensional Bogoyavlensky-Konopelchenko equation with variable coefficients, ph{J. of Elect. Waves and Appl.}, 32, 1457-1464.
  17. [17]  Osman, M.S. (2016), Multi-soliton rational solutions for some nonlinear evolution equations, ph{Open Phys.}, 14, 26-36.
  18. [18]  Osman, M.S. (2017), Analitycal study of rational and double-soliton rational solutions governed by the KdV-Sawada-Kotera-Ramani equation with variable coefficients, ph{Nonlinear Dyn.}, 89, 2283-2289.
  19. [19]  Banerjee, R.S. (1998), Painlev\{e} analysis of the ${K(m, n)}$ equations which yield compactons, ph{Phys. Scr.}, 57, 598-600.
  20. [20]  Vakhnenko, V.O., Parkes, E.J., and Morrison, A.J. (2003), A B\"{a}klund transformation and the inverse scattering transform method for the generalised Vakhnenko equation, ph{chaos solitons Fractals}, 17, 683-692.
  21. [21]  Wang, M., Li, X., and Zhang, J. (2008), The (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, ph{Phys. Lett. A}, 372, 417-423.
  22. [22]  Newell and Whitehead (1971), ph{Review of the finite bandwidth concept. In: Instability of continuous systems}. Berlin: Springer-Verlag.
  23. [23]  Stewartson, K. and Stuart (1971), A non-linear instability theory for a wave system in plane Poiseuille flow, ph{J. Fluid Mech.}, 48, 529-545.
  24. [24]  Halperin, B.I., Lubensky, T.C., and shang-Keng (1974), First-oder phase transitions in superconductors and smetic-ph{A} liquid crystals, ph{Phys. Rev. Lett.}, 32, 292-295.
  25. [25]  Liu, F., Mondello, M., and Goldenfeld, N. (1991), Kinetics of the superconducting transition, ph{Phys. Rev. Lett.}, 66, 3071-3074.
  26. [26]  Igor Arangon, S., and Kramer, L. (2002), The world of the complex Ginzburg-Landau equation, ph{Rev. Mod. Phys.}, 74, 99-143.
  27. [27]  Hentschel, H.G.E. and Procaccia, I. (1983), Passive scalar fluctuations in intermittent turbulence with applications to wave propagation, ph{Phys. Rev. A}, 28, 417-428.
  28. [28]  Nozaki, K. and Bekki (1983), Pattern selectionand spatiotemporal transition to chaos in the Ginzburg-Landau equation, ph{Phys. Rev. Lett.}, 51, 2171-2174.
  29. [29]  Stenflo, L. (1988), A solution of the generalised non-linear Schr\"{o}dinger equation, ph{J. Phys. A: Math. Gen.}, 21, L499-L500.
  30. [30]  Kenfack-Jiotsa, A., Fewo, S.I., and Kofan, T.C. (2006), Effects of renormalization parameters on singularities and special soliton solutions of the modified complex Ginzburg-Landau equation, ph{Phys. Scr.}, 74, 499-502.
  31. [31]  Mohamadou, A., Ndzana II, F., and Kofan, T.C. (2006), Pulse solution of the modified cubic complex Ginzburg-Landau equation, ph{Phys. Scr.}, 73, 596-600.
  32. [32]  Yomba, E. and Kofan, T.C. (1996), On exact solutions of the generalized modified complex Ginzburg-Landau equation using the Weiss-Tabor-Carnevale method , ph{Phys. Scr.}, 54, 576-580.
  33. [33]  Yomba, E. and Kofan, T.C. (1999), On exact solutions of the modified complex Ginzburg-Landau equation, ph{Physica D}, 125, 105-122.