%0 Journal Article
%T Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation
%A Ivan M. Uzunov
%A Zhivko D. Georgiev
%J Journal of Computational Methods in Physics
%D 2014
%R 10.1155/2014/308947
%X 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). 1. Introduction As is well known the complex cubic Ginzburg-Landau equation has been used to describe a variety of phenomena including second-order phase transitions, superconductivity, superfluidity, Bose-Einstein condensation, and strings in field theory [1, 2]. Perturbative methods are used for the description of spatiotemporal pattern formations in systems driven away from equilibrium near the threshold where the nonlinearities are weak and the spatial and temporal modulations of the unstable modes are slow [1]. One of them is the method of ¡°amplitude (model) equations¡± for the envelope function of unstable mode [1]. Model equations depend on the type of the linear instability. Universal amplitude equations are the real and complex cubic Ginzburg-Landau equation (CCGLE) as well as their generalizations, like the coupled complex cubic Ginzburg-Landau equation and the complex cubic-quintic Ginzburg-Landau equation (CCQGLE). CCGLE describes the evolution of the envelope function of unstable mode for any process exhibiting a Hopf bifurcation. As a model equation CCGLE is applied to the study of oscillatory uniform instability in lasers, oscillatory periodic instability in Rayleigh- B¨¦nard convection in binary mixtures as well as electrohydrodynamic instabilities in nematic liquid crystals (see [1, 2] and
%U http://www.hindawi.com/journals/jcmp/2014/308947/