Modulated solitons and transverse stability in a two-dimensional nonlinear reaction diffusion electrical network

Abstract

We investigate the propagation of modulated solitons in a two-dimensional (2D) nonlinear reaction diffusion electrical network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and they reduce from the reductive perturbation technique to the 2D nonlinear dissipative Schrödinger equation governing the propagation of the small dissipative amplitude signals in the network. This equation does’nt have conserved quantities and it admits as solutions the 2D dissipative pulse and dark solitons, according to the sign of the product of dispersive and nonlinearity coefficients, with amplitude which narrows as the time increases. The exactness of the analytical analysis is confirmed by numerical simulations. Then by using the method of constants variation, the train of periodic pulse and dark solitons are also found, with their existence constraints also connected to the sign of the product of dispersive and nonlinearity coefficients, as found for standard pulse and dark solitons. Then the modulational instability (MI) criterion in system is found and is connected to the existence of modulated solitons.