Instabilidade modulacional em cadeia com não linearidade saturada

Abstract

Nonlinear phenomena, such as modulational instability (MI) and self-trapping, are present in some physical systems such as in the areas of optics, Bose-Einstein condensate and plasma physics. We can observe MI in nonlinear media with dispersive effects, occurring when we add a small disturbance in the amplitude of the wave function and it is amplified. It is already established for a chain without saturation that the critical non-linear coupling parameter above which occurs MI decays with the inverse of the chain size. With this, using the Holstein model, our goal is to study MI in a one-dimensional nonlinear system, with saturated nonlinearity. To investigate the nature of this phenomenon, we solved the discrete nonlinear Schrödinger equation (DNLSE) numerically with the aid of the eighth-order numerical Runge-Kutta method. In our calculations, we used the participation function, a measure of the degree of localization, and calculated the square module of the wave function, to observe its profile. We solved the nonlinear continuous Schrödinger equation analytically, where we made a theoretical estimative for the critical values of the saturated nonlinearity. We found that when considering saturated nonlinearity, the behavior changes and this critical parameter decays with a term proportional to the inverse of the chain size plus another term that decays with the inverse of the chain size squared. We also show that the increase in saturation promotes the displacement of the critical parameters of intensity of the non-linearity of the transitions already known in the unsaturated case.