A geo-numerical approach for the classification of fixed points in the reduced model of the cubic-quintic complex Ginzburg-Landau equation

Abstract In this paper we present a geo-numerical approach for classifying fixed points of the set of ordinary differential equations (ODEs) that represent the reduced model of the cubic-quintic complex Ginzburg–Landau equation (CQ CGL) in a two parameter plane. We will solve the set of ODEs representing the system using the Dormand–Prince method (DOPRI) then by analyzing the final stage of the solution a decision will be made which will result in the classification of solutions into three categories, fixed points, limit cycles or a completely instable solutions, this approach will reveal the variety of solutions and there bifurcation from stable to oscillatory to instable solutions. This study allows us to increase the reliability of the propagation and communication in fiber optic applications.

[1]  Jong-Hyung Lee,et al.  Analysis and Characterization of Fiber Nonlinearities with Deterministic and Stochastic Signal Sources , 2000 .

[2]  N. Akhmediev,et al.  Bifurcations from stationary to pulsating solitons in the cubic¿quintic complex Ginzburg¿Landau equation , 2005, nlin/0602030.

[3]  N. Akhmediev,et al.  Solitons of the Complex Ginzburg-Landau Equation , 2001 .

[4]  J. Moores.  On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term , 1993 .

[5]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[7]  U. Peschel,et al.  Solitons of singly resonant optical parametric oscillators , 1999 .

[8]  P. Grelu,et al.  Dissipative solitons for mode-locked lasers , 2012, Nature Photonics.

[9]  G. Town,et al.  Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber , 2002 .

[10]  P. Grelu,et al.  Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes. , 2007, Chaos.

[11]  Mustapha Tlidi,et al.  On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles , 2008 .

[12]  N. Akhmediev,et al.  Dissipative Solitons in the Complex Ginzburg-Landau and Swift Hohenberg Equations , 2005 .

[13]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[14]  Adrian Ankiewicz,et al.  Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Adrian Ankiewicz,et al.  Dissipative solitons : from optics to biology and medicine , 2008 .

[16]  C. Christov,et al.  Dissipative solitons , 1995 .

[17]  A. Maimistov,et al.  Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation , 1993 .

[18]  Three Sources and Three Component Parts of the Concept of Dissipative Solitons , 2008 .