Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation

Stationary to pulsating soliton bifurcation analysis of the complex Ginzburg–Landau equation (CGLE) is presented. The analysis is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Stationary solitons, with constant amplitude and width, are associated with fixed points in the model. For the first time, pulsating solitons are shown to be stable limit cycles in the finite-dimensional dynamical system. The boundaries between the two types of solutions are obtained approximately from the reduced model. These boundaries are reasonably close to those predicted by direct numerical simulations of the CGLE.  2005 Elsevier B.V. All rights reserved. PACS: 04.30.Nk; 05.45.Yv; 42.65.Sf; 42.65.Tg The complex Ginzburg–Landau equation (CGLE) is one of the basic equations for modelling modulated amplitude waves [1], spatio-temporal dynamics and spontaneous development of coherent structures in a variety of nonlinear dissipative systems [2,3]. Examples include pulse generation by passively modelocked soliton lasers [4], signal transmission in all* Corresponding author. E-mail address: [email protected] (E.N. Tsoy). 1 Also at: Physical-Technical Institute of the Uzbek Academy of Sciences, Tashkent, Uzbekistan. 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.102 optical communication lines [5], travelling waves in binary fluid mixtures [6], and also pattern formation in many other physical systems [7]. Complicated patterns consist of simpler localized solutions like fronts, pulses, sources and sinks [8]. Pulsating soliton solutions of dissipative systems have attracted a great deal of attention in recent years. They have been found numerically [9–11] and observed experimentally [11] in a fiber laser. Pulsating solitons form one set of possible localized solutions of the CGLE, and they exist on an equal basis with stationary solitons. Such localized waves exist, in various forms in biology, chemistry and physics. 418 E.N. Tsoy, N. Akhmediev / Physics Letters A 343 (2005) 417–422 A pulsating soliton can be described as a limit cycle of an infinite-dimensional dissipative dynamical system [12]. It is different from the higher-order solitons that are usually connected with an integrable model [13]. Although numerical simulations show clearly the existence of pulsating solutions and their bifurcations from stationary solitons, so far there has been no progress in finding analytic expressions for pulsating solutions and bifurcation boundaries. The problem is not simple as there are several parameters of the CGLE that define the regions of existence for both stationary and pulsating solitons. Hence, the bifurcation boundaries are surfaces in this multi-dimensional space of the parameters. In this work we use a reduction from an infinitedimensional to a five-dimensional model, and we aim to find localized solutions of the CGLE and the transformations that they are subjected to when the system parameters are varied. Although exact solutions of the CGLE do exist [3], they can be presented explicitly only for certain relations between the parameters of the equation. Furthermore, only stationary solutions can be found. Hence, we are faced with the necessity of finding an efficient approximation to tackle the problem. We have found that the method of moments, originally developed by Maimistov [14] for the perturbed nonlinear Schrödinger equation (NLSE) can be used for solving our problem. The moments are the integral characteristics of the field under consideration. In principle, there are an infinite number of equations for moments. One can obtain exact results by using the complete set of these equations. However, in practice, one uses a trial function with a finite number of parameters, and this is the way to obtain a significant reduction in the number of variables used for the description of the dynamics. The cubic–quintic complex Ginzburg–Landau equation, in dimensionless form, is written as iψt + D 2 ψxx + |ψ |2ψ =−ν|ψ |4ψ + iδψ + i |ψ |2ψ + iβψxx + iμ|ψ |4ψ