Table of Contents
Journal of Computational Methods in Physics
Volume 2014, Article ID 308947, 13 pages
http://dx.doi.org/10.1155/2014/308947
Research Article

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

1Department of Applied Physics, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria
2Department of Theoretical Electrical Engineering, Technical University Sofia, 8 Kl. Ohridski Boulevard, 1000 Sofia, Bulgaria

Received 31 May 2014; Revised 4 September 2014; Accepted 5 September 2014; Published 15 October 2014

Academic Editor: Ivan D. Rukhlenko

Copyright © 2014 Ivan M. Uzunov and Zhivko D. Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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).