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 JAMP  Vol.6 No.10 , October 2018
Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation
Abstract: In this paper, we performed an investigation of the dissipative solitons of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE). Stationary to pulsating soliton bifurcation analysis of the 2D CSHE is displayed. The approach is based on the semi-analytical method of collective coordinate approach. This method is constructed on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. The reduced model helps to obtain approximately the boundaries between the stationary and pulsating solutions. We analyzed the dynamics and characteristics of the pulsating solitons. Then we obtained the bifurcation diagram for a definite range of the saturation of the Kerr nonlinearity values. This diagram reveals the effect of the saturation of the Kerr nonlinearity on the period pulsations. The results show that when the parameter of the saturation of the Kerr nonlinearity increases, one period pulsating soliton solution bifurcates to double period pulsations.
Cite this paper: Kamagaté, A. , Moubissi, A. (2018) Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation. Journal of Applied Mathematics and Physics, 6, 2127-2141. doi: 10.4236/jamp.2018.610179.
References

[1]   Swift, J. and Hohenberg, P.C. (1977) Hydrodynamic Fluctuations at the Convective Instability. Physics Review A, 15, 319-328
https://doi.org/10.1103/PhysRevA.15.319

[2]   Bestehorn, M. and Haken, H. (1990) Traveling Waves and Pulses in a Two-Dimensional Large-Aspect-Ratio System. Physics Review A, 42, 7195.
https://doi.org/10.1103/PhysRevA.42.7195

[3]   Malomed, B.A. (1984) Nonlinear Waves in Nonequilibrium Systems of the Oscillatory Type, Part I. Zeitschrift für Physik B Condensed Matter, 55, 241-248.
https://doi.org/10.1007/BF01329018

[4]   Peletier, L.A. and Williams, J.F. (2007) Some Canonical Bifurcations in the Swift-Hohenberg Equation. SIAM Journal on Applied Dynamical Systems, 6, 208-235.
https://doi.org/10.1137/050647232

[5]   Yoboue, P., Diby, A., Asseu, O. and Kamagate, A. (2016) Stability of Dissipative Optical Solitons in the 2D Complex Swift-Hohenberg Equation. International Journal of Physics, 4, 78-84.

[6]   Hoyuelos, M. (2005) Numerical Study of the Vector Complex Swift-Hohenberg Equation. Physica D: Nonlinear Phenomena, 223, 174-179.
https://doi.org/10.1016/j.physd.2006.09.002

[7]   Ippen, E.P. (1994) Principles of Passive Mode Locking. Applied Physics B, 58, 159-170.
https://doi.org/10.1007/BF01081309

[8]   Akhmediev, N. and Ankiewicz, A. (2005) Dissipative Solitons. Springer, Heidelberg.

[9]   Kamagaté, A., Grelu, Ph., Tchofo-Dinda, P., Soto-Crespo, J.M. and Akhmediev, N. (2009) Stationary and Pulsating Dissipative Light Bullets from a Collective Variable Approach. Physical Review E, 79, 026609.
https://doi.org/10.1103/PhysRevE.79.026609

[10]   Akhmediev, N. and Ankiewicz, A. (2008) Dissipative Solitons: From Optics to Biology and Medicine. Springer, Heidelberg.

[11]   Khairudin, N.I., Abdullah, F.A. and Hassan, Y.A. (2016) Stability of the Fixed Points of the Complex Swift-Hohenberg Equation. Journal of Physics: Conference Series, 693, 012003.
https://doi:10.1088/1742-6596/693/1/012003

[12]   Maruno, K., Ankiewicz, A. and Akhmediev, N. (2003) Exact Soliton Solutions of the One-Dimensional Complex Swift-Hohenberg Equation. Physica D, 176, 44-66.
https://doi.org/10.1016/S0167-2789(02)00708-X

[13]   Pedrosa, J., Hoyuelos, J.M. and Martel, C. (2008) Numerical Validation of the Complex Swift-Hohenberg Equation for Lasers. The European Physical Journal B, 66, 525-530.
https://doi.org/10.1140/epjb/e2008-00457-5

[14]   Arnous, A.H., Mirzazadeh, M., Moshokoa, S., Medhekar, S., Zhou, Q., Mahmood, M.F., Biswas, A. and Belic, M. (2015) Solitons in Optical Metamaterials with Trial Solution Approach and Backlund Transform of Riccati Equation. Journal and Computational and Theoretical Nanoscience, 12, 5940-5948.
https://doi.org/10.1166/jctn.2015.4739

[15]   Wang, H. and Yanti, L. (2011) An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation. Numerical Mathematics: Theory, Methods and Applications, 4, 237-254.

[16]   Soto-Crespo, J.M., Akhmediev, N.N. and Afanasjev, V.V. (1996) Stability of the Pulselike Solutions of the Quintic Complex Ginzburg-Landau Equation. Journal of the Optical Society of America B, 13, 1439-1449.
https://doi.org/10.1364/JOSAB.13.001439

[17]   Skarla, V. and Aleksic, N. (2006) Stability Criterion for Dissipative Soliton Solutions of the One- , Two- , and Three-Dimensional Complex Cubic-Quintic Ginzburg-Landau Equations. Physical Review Letters, 96, 013903.
https://doi.org/10.1103/PhysRevLett.96.013903

[18]   Asseu, O., Diby, A., Yoboué, P. and Kamagaté, A. (2016) Spatio-Temporal Pulsating Dissipative Solitons through Collective Variable Methods. Journal of Applied Mathematics and Physics, 4, 1032-1041.
https://doi.org/10.4236/jamp.2016.46108

[19]   Akhmediev, N. and Ankiewicz, A. (2001) Solitons of the Complex Ginzburg-Landau Equation. Springer, Berlin.
https://doi.org/10.1007/978-3-540-44582-1_12

[20]   Akhmediev, N., Soto-Crespo, J.M. and Town, G. (2001) Pulsating Solitons, Chaotic Solitons, Period Doubling, and Pulse Coexistence in Mode-Locked Lasers: Complex Ginzburg-Landau Equation Approach. Physical Review E, 63, 056602.
https://doi.org/10.1103/physreve.63.056602

[21]   Tchofo-Dinda, P., Moubissi, A.B. and Nakkeeran, K. (2001) Collective Variable Theory for Optical Solitons in Fibers. Physical Review E, 64, 016608.
https://doi.org/10.1103/PhysRevE.64.016608

[22]   Tsoy, E.N. and Akhmediev, N. (2005) Bifurcations from Stationary to Pulsating Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation. Physics Letters A, 343, 417-422.
https://doi.org/10.1016/j.physleta.2005.05.102

[23]   Wu, L., Guo, Z.J. and Song, L.-J. (2010) Properties of Pulsating Solitons in Dissipative Systems. Chinese Physics B, 19, 080512.
https://doi.org/10.1088/1674-1056/19/8/080512

 
 
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