Modulational instability in nonlinear bi-inductance transmission line

A nonlinear dissipative transmission line is considered and by performing the complex demodulation technique of the signal which allows, in particularly, to separate the right traveling and left traveling waves, we show that the amplitudes of these waves can be described by a complex coupled Ginzburg-Landau equations (CG-LE). The so-called phase winding solutions of the constructed CG-LE is analyzed. We also study the coherent structures in the obtained complex Ginzburg-Landau system. We show that the constructed CG-LE possesses nonlinear plane wave solutions and the modulational instability of these solutions is analyzed. The condition of the modulational instability is given in term of the coefficients of the constructed CG-LE and then in term of the wavenumber of the two right traveling and left traveling waves in the considered transmission line. The results obtained here show that the nonlinear plane wave solutions of the CG-LE under perturbation with zero wavenumber cannot be stable under modulation.