Solitary pulses in linearly coupled Ginzburg-Landau equations

Abstract

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.

From one- to two-dimensional solitons in the Ginzburg-Landau model of lasers with frequency-selective feedback

“…To this end, we consider a complex Ginzburg-Landau model with the cubic nonlinearity (CGL3), for which an analytical chirped-sech localized solution is well known in the one-dimensional (1D) setting [10,11]. While this solution is always unstable, it has been shown that an additional, linearly coupled, dissipative linear equation can lead to its stabilization in coupled-waveguide models, keeping the solution in the exact analytical form [8,12,13]. Selflocalized states in a wide variety of systems described by such coupled linear and nonlinear equations, in both 1D and 2D, was recently discussed in Ref.…”

“…The field of spatial pattern formation in nonlinear systems has grown significantly in the last decades (see reviews [1][2][3][4][5][6][7][8][9]). In particular, that growth was significantly contributed to by the interest in self-localized states ("solitons") and their stability in pattern-forming systems, both conservative and dissipative ones.…”

“…It was also noted that such a cubic approximation to the VCSEL-FSF would place it in the same class of models as those previously introduced for coupled optical waveguides with active (pumped) and passive (lossy) cores in Refs. [8,12,13,24,25], and for pulsed fiber lasers -in Ref. [26].…”

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

“…This possibility was first proposed in the context of nonlinear fiber optics in [1,2]; see also a review in [3]. More recently, a similar scheme was elaborated for the application of gain and the stabilization of solitons in plasmonics [4][5][6], as well as for the creation of stable two-dimensional dissipative solitons and solitary vortices in dual laser cavities [7,8].…”

“…More recently, a similar scheme was elaborated for the application of gain and the stabilization of solitons in plasmonics [4][5][6], as well as for the creation of stable two-dimensional dissipative solitons and solitary vortices in dual laser cavities [7,8]. Commonly-adopted models of dual-core nonlinear waveguides are based on linearly-coupled systems of nonlinear Schrödinger (NLS) equations, which include gain and loss terms [3]. One of the advantages provided by these systems for the theoretical analysis is the availability of exact analytical solutions for stable solitons [9].…”