Nonlinear Schrödinger Models And Modulational Instability In Real Electrical Lattices
Résumé
In nonlinear dispersive media, the propagation of modulated waves, such as envelope (bright) solitons or hole (dark) solitons, has been the subject of considerable interest for many years, as for example in nonlinear optics [A.C. Newell and J.V. Moloney, Nonlinear Optics (Addison-Presley, 1991)]. On the other hand, discrete electrical transmission lines are very convenient tools to study the wave propagation in 1D nonlinear dispersive media [A.C. Scott (Wiley-Interscience, 1970)]. In the present paper, we study the generation of nonlinear modulated waves in real electrical lattices. In the continuum limit, our theoretical analysis based on the Nonlinear Schrodinger equation (NLS) predicts three frequency regions with different behavior concerning the Modulational Instability of a plane wave. These predictions are confirmed by our experiments which show that between two modulationally stable frequency bands where hole solitons can be generated, there is a third band where spontaneous or induced modulational instability occurs and where envelope solitons exist. When lattice effects are considered the dynamics of modulated waves can be modeled by a discrete nonlinear Schrodinger equation which interpolates between the Ablowitz-Ladik and Discrete-self-trapping equations.