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).
Highlights
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]
Having applied the Melnikov method to the equation of Duffing-Van der Pol oscillator we have proved the existence of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the unperturbed system (see (40))
We have studied the dynamics of the localized pulsating solutions of generalized cubic-quintic complex GinzburgLandau equation (CCQGLE) in the presence of intrapulse Raman scattering
Summary
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]. The second aim of this paper is to explore analytically the influence of the intrapulse Raman scattering on the localized pulsating solutions of the particular case of generalized CCQGLE, namely, (1) with ε = μ = ] = 0 This equation describes the bandwidth limited amplification and has recently been investigated in [30, 31]. Equation of the strongly nonlinear Duffing-Van der Pol oscillator has been introduced [30] in order to describe the influence of IRS on the soliton solutions. We apply the Melnikov method to the equation of strongly nonlinear Duffing-Van der Pol oscillator and prove the existence of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the unperturbed system. Appendix C gives the condition of existence of perturbed solitary wave solution [30] derived by the hyperbolic perturbation method and the hyperbolic Lindstedt-Poincare method as well as the relationship between the equilibrium amplitude and the velocity of the perturbed soliton solution, obtained by means of soliton perturbation theory [30]
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