2D Ginzburg-Landau System of Complex Modulation for Coupled Nonlinear Transmission Lines

In this paper a coupled mono-inductance transmission line is considered and the envelope modulation is reduced to the generalized 2D-Ginzburg-Landau system (2D-GL system). The case of scalar equation is also considered and we deduce from the obtained 2D-GL system a 2D-cubic Ginzburg-Landau equation (2D-GL equation) containing the derivatives with respect to one spatial variable in the cubic terms. The obtained 2D-GL system and 2D-GL equation admit spatial wave solutions. The modulational instability of these spatial wave solutions is investigated. In the case of the 2D-GL system we have restricted ourselves in the zero wavenumber of the perturbations, and the obtained modulational instability conditions depends only on the system's coefficients and the wavenumber of the carriers. The modulational instability criterion is established for non zero wavenumber of the perturbation, and depends on both the wavenumber of the perturbation and the carrier. For the 2D-GL system we also study the coherent structures and present some numerical studies.