Stability and interactions of pulses in simplified Ginzburg-Landau equations

Abstract

We consider in detail two special types of the parameter-free Ginzburg-Landau equation, viz., the ones that combine the bandwidth-limited linear gain and nonlinear dispersion, or the broadband gain, linear dispersion, and nonlinear losses. The models have applications in nonlinear fiber optics and traveling-wave convection. They have exact solitary-pulse solutions which are subject to a background instability. In the former model, we find that the solitary pulse is much more stable than a "densely packed" multi-pulse array. On the contrary to this, a multi-pulse array in the latter model is destroyed by the instability very slowly. Considering bound states of two pulses, we conclude that they may form a robust bound state in both models. Conditions which allow for formation of the bound states qualitatively differ in the two models.

Solitary pulses in linearly coupled Ginzburg-Landau equations

“…3 is shown in Fig. 19,33 On the other hand, weak instability, instead of producing multiple pulses, may generate a small-amplitude background field featuring regular oscillations. In region III of Fig.…”

Two-dimensional structures in the quintic Ginzburg–Landau equation

“…Many basic properties of the equation and its solutions are reviewed in [10][11][12][13], together with applications to a vast variety of phenomena including nonlinear waves [8,10], superconductivity [25,27], Bose-Einstein condensation [47], intra-pulse Raman scattering [49], liquid crystals and string theory. In particular, in nonlinear optics it describes the pulse generation and signal transmission through an optical fiber [30,31].…”

2D Novel Structures Along an Optical Fiber

“…Such as the nonlinear Schrödinger equation (NLSE), the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) is one of the most intensively studied equation describing weakly nonlinear phenomena in dissipative systems [3]. Thus, much work has been done about the features and the properties of this equation as well as its numerous applications such as nonlinear waves, superconductivity, superfluidity [6], Bose-Einstein condensation, Bénard convection [1] and nonlinear optics [7]. Solitary waves (or solitons) are self localized solutions of certain nonlinear PDEs describing the evolution of a dissipative system [8].…”