Abstract
In this work, we investigate the possibility of approximating saturable nonlinearity, which is commonly used in complex Ginzburg–Landau equation (CGLE) for modelling resonant interaction of an electromagnetic field with nonlinear media, with cubic-quintic (CQ) nonlinearity. To validate the suggested approximations, we use variational method to estimate 2D analytical solutions of the CGLE with both saturable and CQ nonlinearity. The paper compares three ways to determine parameters of the CQ approximation and discusses the obtained results in terms of accuracy.
Highlights
IntroductionAfter the introduction of the cubic “master equation” (Haus et al 1991, 1992), there have been derived many refinements in the form of CQ complex Ginzburg–Landau equation (CGLE), as a model of resonant interaction of electromagnetic field with nonlinear media
The class of complex Ginzburg–Landau equation (CGLE) represents a good model for describing a wide variety of phenomena in dissipative systems, such as superconductivity (Ginzburg et al 1950), nonequilibrium phenomena (Aranson and Kramer 2002), phase transitions (Nato Advanced Study Institute 1975), binary fluid convection (Kolodner 1992), laser generated spatiotemporal dissipative structures, soliton propagation in optical fiber systems (Kodama and Hasegawa 1992), biological systems (Morales et al 2015), etc.The CQ CGLE, in particular, is often utilized to describe competitive effects of linear and nonlinear gain and loss
Analytical solutions of dissipative structures in cubic CGLE have been obtained, but all of them are unstable except the class of solutions with arbitrary amplitude in an “exotic” medium with zero linear loss/gain (Akhmediev et al 1996)
Summary
After the introduction of the cubic “master equation” (Haus et al 1991, 1992), there have been derived many refinements in the form of CQ CGLE, as a model of resonant interaction of electromagnetic field with nonlinear media. These refinements have been done assuming small intensities, which allows the expansion of the Bloch equations in the series of intensities. If the loss saturates faster than the gain (b > 1) , coefficients a and a are positive
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