scholarly journals Bifurcation and stability of the generalized complex Ginzburg--Landau equation

2008 ◽  
Vol 7 (5) ◽  
pp. 1237-1253 ◽  
Author(s):  
Jungho Park ◽  
Keyword(s):  
Landau Equation ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Generalized Complex ◽  
Bifurcation And Stability

Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

2017 ◽  
Vol 89 (4) ◽  
pp. 2933-2939 ◽  
Author(s):  
Wenjun Liu ◽  
Weitian Yu ◽  
Chunyu Yang ◽  
Mengli Liu ◽  
Yujia Zhang ◽  
...  
Keyword(s):  
Fiber Lasers ◽  
Landau Equation ◽  
Analytic Solutions ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Generalized Complex

scholarly journals Bifurcation to traveling waves in the cubic–quintic complex Ginzburg–Landau equation

2015 ◽  
Vol 13 (04) ◽  
pp. 395-411 ◽  
Author(s):  
Jungho Park ◽  
Philip Strzelecki
Keyword(s):  
Traveling Waves ◽  
Landau Equation ◽  
Basic Solution ◽  
Ginzburg Landau ◽  
Center Manifold Reduction ◽  
Modulation Equation ◽  
Ginzburg Landau Equation ◽  
The One ◽  
Manifold Reduction ◽  
Bifurcation And Stability

We consider the one-dimensional complex Ginzburg–Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg–Landau equation, Pure Appl. Anal. 7(5) (2008) 1237–1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves.

scholarly journals Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan M. Uzunov ◽  
Zhivko D. Georgiev
Keyword(s):  
Limit Cycle ◽  
Solitary Wave Solution ◽  
Landau Equation ◽  
Melnikov Method ◽  
Van Der Pol Oscillator ◽  
Unperturbed System ◽  
Van Der Pol ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Generalized Complex

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

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