Solitons and Nonlinear wave Research Papers (original) (raw)

This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau CGL equations. In the simplest case, the system features linear gain, cubic nonlinearity possibly combined with cubic loss, and group-velocity dispersion GVD in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive lossy one. The model gives rise to exact analytical solutions for stationary solitary pulses SPs. The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons holes. In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic second-harmonic-generating nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers. Complex Ginzburg-Landau (CGL) equations are ubiquitous models of pattern formation in nonlinear dissipative media, with important applications in physics and chemistry (in particular nonlinear optics and reaction-diffusion systems). Among various patterns generated by the CGL equations, especially important are localized ones, i.e., solitary pulses (SPs). The simplest cubic CGL equation with the linear gain and nonlinear loss gives rise to the well-known exact analytical solution for the SP, but it is always unstable, as the linear gain makes its background (zero solution) unstable. Therefore, an important issue is to find physically relevant modifications of the CGL equations that would be physically relevant and admit stable (and, if possible, analytically tractable) SP solutions. A known phenomenological solution to the problem is offered by the CGL equation with the cubic-quintic (CQ), rather than cubic, nonlinearity. Another option is provided by a system of two linearly coupled CGL equations , one including the linear gain, group-velocity dispersion (GVD) and cubic nonlinearity, while the other equation is linear, featuring only loss. This system is a direct physical model of a dual-core fiber laser, composed of an active core parallel-coupled to an additional (stabilizing) passive one. An important property of the system is that it admits analytical solutions for SPs, in the case in which the zero solution is stable (hence the SP has a chance to be stable too). This article aims to give a short review of models of this type and their fundamental solutions, including the identification of stability regions for the SPs in the system's parameter space, modes of destabilization of the pulses (in particular, the model with the anomalous GVD features destabilization of the SP against oscillatory perturbations, which do not destroy the pulse but rather turn it into a stable localized breather), interactions between stable SPs, solutions of the same system in the form of dark solitons, and some other properties. Also briefly considered are recently studied models of a different but related type, which are based on linearly coupled symmetric CGL equations with the CQ nonlinearity; models of this type feature such effects as spontaneous symmetry breaking between components of the SP, and formation of stable breathers in a broad range of parameters.