On the effects of the next nearest neighbor couplings on the exotic modulated solitons and the transverse stability in a two-dimensional nonlinear reaction–diffusion electrical transmission line

The influence of second neighbor interactions in a two-dimensional (2D) nonlinear reaction diffusion (NLRD) electrical network, with the intersite circuit elements acting as nonlinear resistances, is investigated. Using the Kirchhoff laws, the model equations for the circuit are derived, and they reduce to the 2D extended nonlinear Schrödinger (2D-ENLS) equation via the reductive perturbation method in the semi-discrete limit. This 2D-ENLS equation has the norm as a conserved quantity, and by using this equation, the equilibrium points are found, and their stability is investigated, predicting then the nature of its solutions. From the dispersion relation found, it is obvious that the consideration of the second neighbor couplings increases the angular frequency band of the system and modifies both the group velocity of the waves as well as the number of regions of stability of equilibrium points. The exact analytical expressions of the solutions of this 2D-ENLS equation are found, namely the modulated 2D pulse and dark compactons as well as the modulated 2D pulse soliton, which are a function of the coefficients of nonlinear dispersive terms, and it is found that the second neighbor’s interactions also modify the number of regions of their existence. Moreover, the modulational instability (MI) is investigated, and it is found that the second neighbor interactions affect both the MI rate and their region of existence.