Espalhamento de gaussons em acopladores ópticos direcionais - Teixeira, Rafael Marques Paes (2017)
This dissertation presents the research that was focused on the study of solitons in nonintegrable systems governed by effective models of partial differential equations (PDEs) of (1+1) dimensions, that is, one spatial and one time dimension. In the main research line, we considered a model of two fields given by nonlinear Schrödinger equations with logarithmic nonlinearity and linear coupling, the main aspects of the collision dynamics of two Gaussian solitons (gaussons) were investigated, addressing the stability of the solutions, the interaction mechanisms, and the regularity of the scattering. The study of gaussons collisions in this model was performed via two approaches: semianalytic (variational/reduced model) and direct numerical simulation. For the system considered in this work, both approaches provided similar results which revealed the existence of a chaotic and fractal dynamics involving the initial conditions and the post-collisional properties of the solitons, which were verified by the analysis of the correlations between the input and output parameters. The fractal patterns are constituted by regularity windows that are positioned accordingly to a well defined rule, which was obtained via a linearization model that was found to be very similar in both approaches (i.e., the windows arrangement was well described by the variational model). In such windows the solitons collision dynamic turns out to be not chaotic (regular collision), it becomes predictable and very well defined, hence the width of these windows provides the conditions in which the control of the interaction is possible. Besides, by analyzing the dynamics of the variational parameters of the reduced model and also the dynamics of the most relevant physical quantities in the exact approach via PDEs (namely the total kinetic and potential energies), we attested that these windows arise due to a resonant mechanism of energy exchange between translational and vibrational soliton modes. By studying the linear stability of the solitary waves, we proved that the vibrational modes originate from shape oscillations induced by internal modes and also, possibly, by instability modes excited during the collisions. We verified that these oscillations are well defined in all regular collisions. The viability of the variational method in the system studied is due to the suitable description of the vibrational modes by the reduced model, and also the low influence of the emitted nonlinear radiation in the regular collision processes found in the direct simulations.