Ginzburg-Landau models of nonlinear electric transmission networks

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 non-linearity 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 Schrodinger (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. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

This article reviews the application of one- and two-dimensional CGL models based on the complex Ginzburg-Landau equations in nonlinear transmission networks. These models generate a variety of analytical and numerical solutions for the propagation of solitons, kinks, and other wave modes. In addition, the article discusses bifurcations of traveling waves in 2D electrical networks, which are distinct from the CGL equations.