Ginzburg–Landau models of nonlinear electric transmission networks

Abstract

Complex Ginzburg–Landau (CGL) equations serve as canonical models in a great variety of physical settings, such as nonlinear photonics, dynamical phase transitions, superconductivity, superfluidity, hydrodynamics, plasmas, Bose–Einstein condensates, liquid crystals, field-theory strings, etc. This article provides a review of one- and two-dimensional (1D and 2D) CGL-based models of single- and multi-coupled (bundled) electric nonlinear transmission networks (NLTNs), built of elements combining nonlinearity and dispersion. They are modeled by CGL equations in the framework of the continuum approximation. The presentation starts with a survey of experimental results for solitons in electrical transmission lines. Both lossless and dissipative networks are considered. Nonlinear models originating from NLTNs, which are treated in the review, include conservative and dissipative nonlinear Schrödinger (NLS) equations, cubic and cubic–quintic CGL equations (ones with derivative terms are included as well), and their extensions in the form of the Kundu–Eckhaus (KE) and generalized Chen–Lee–Liu (CLL) equations. These models produce a variety of analytical and numerical solutions for the propagation of nonlinear-wave modes in electric networks. We here focus on cnoidal waves, bright and dark solitons, kinks, rogue waves, and chirped W-shaped kinks (Lambert waves), some of which have been observed in experiments, while others call for experimental realization. A summary of applications of NLTNs is presented, including an especially important example that relies on NLTNs for emulation of various dynamical phenomena known in other physical and neural systems. Based on models distinct from equations of the CGL type, we also review bifurcations of traveling waves propagating in 2D electrical networks.