Circuit-Based Numerical Solutions of Transmission Lines: Application to Korteweg-de Vries Equations

Transmission lines, devices employed for the transmission of electrical signals, can be used for the approximation of non-linear partial differential equations (PDEs) also in the nonlinear case. To this end, transmission lines are used to spatially discretize PDEs allowing the problem to be solved numerically even for intricate boundary conditions - e.g. at intersection of many wires in the system. Using transmission lines to solve the Korteweg-de Vries Equation (KdV) allows for efficient and accurate numerical solutions, as it provides a method to investigate the propagation of wave-like information in nonlinear and dispersive media with multiple dimensions. In the present paper a software developed in C# is presented, the latter employs discretization by transmission lines and the Runge-Kutta 4–5 integration algorithm to numerically solve the KdV equation in the one-dimensional case. The implemented program, named “WireExplorer," is able to simulate the propagation of solitonic pulses on different types of circuits composed of one or more wires, even in the case of intersections. Such conditions are not canonically solvable by resolution of the KdV equations, while approximate numerical resolution allowed the evaluation of these intricate cases. In particular, the obtained results showed the subdivision of the wave into smaller components, at intersections between wires, which propagate in different directions showing also the formation of dispersive tails propagating in the direction opposite to that of the main wave.