Stationary Localized Solutions in the Subcritical Complex Ginzburg-Landau equation

The discovery of confined traveling waves in convection in binary fluids [Heinrichs et al., 1987; Kolodner et al., 1987; Kolodner et al., 1988; Niemela et al., 1990; Moses et al., 1987] has been a motivation for theoretical work on localized solutions of amplitude equations [Afanasjev et al., 1996; Akhmediev et al., 1996; Deissler & Brand, 1990, 1994, 1995; Fauve & Thual, 1990; Hakim & Pomeau, 1991; Malomed & Nepomnyashchy, 1990; Marcq et al., 1994; Soto-Crespo et al., 1997; Thual & Fauve, 1988; van Saarloos & Hohenberg, 1990; van Saarloos & Hohenberg, 1992]. Nevertheless most of this work has been devoted to numerical analysis. The aim of this article is to give an analytical approach to the study of stationary localized solutions in the subcritical complex Ginzburg– Landau equation. When the equilibrium state of an extended system loses stability through a subcritical Hopf bifurcation it is described near threshold by its normal form [Elphick et al., 1987] which is the quintic Ginzburg–Landau equation with complex coefficients for a complex amplitude A(x, t):