Spectral stability and dynamics of solitary waves in a coupled nonlinear left-handed transmission line

We study the dynamics of modulated waves in a discrete coupled left-handed nonlinear transmission line, assuming a two-dimensional propagations variations. By means of semi-discrete approximation, we derive a two-dimensional nonlinear Schrödinger equation (2D NLSE) governing the propagation of modulated waves in the network. We derive linear boundary value problem governing the evolution of the perturbed system. We then compute numerically its eigenvalues using Fourier and finite difference differentiation matrices. Our results show that the system supports bright soliton in relatively frequency band in both transverse and longitudinal directions. We perform numerical simulations of both 2D NLSE and the nonlinear lattice model to study the stability (instability) and the propagation properties of an initial bright soliton. Our numerical results are in good agreement with the analytical predictions.