Banca de QUALIFICAÇÃO: LUCIANO JOSE REGO BEZERRA JUNIOR

Uma banca de QUALIFICAÇÃO de DOUTORADO foi cadastrada pelo programa.
STUDENT : LUCIANO JOSE REGO BEZERRA JUNIOR
DATE: 02/05/2024
TIME: 10:00
LOCAL: Auditório do Instituto de Física
TITLE:

MODULATIONAL INSTABILITY IN ONE-DIMENSIONAL NONLINEAR SYSTEMS

KEY WORDS:

Modulation Instability; Nonlinear Schrödinger equation; Saturated Nonlinearity; Interacting particles; Solitons

PAGES: 73
BIG AREA: Ciências Exatas e da Terra
AREA: Física
SUBÁREA: Física da Matéria Condensada
SUMMARY:

The study of nonlinear phenomena in low-dimensional systems is of great importance for a better understanding of information transport in general. Modulational instability, solitons, and chaotic regimes are investigated in various areas of physics, such as nonlinear optics and electronic systems. Two works are presented in this thesis, both addressing the dynamics of particles in nonlinear systems. We determine the Schrödinger equation with both and use the eighth-order Runge-Kutta numerical method for numerical calculations. Participation function, centroid, and the square modulus of the wave function were physical quantities used to assist us in these studies. For both works, the initial condition used was uniformly extended particles plus a small perturbation added only at the initial instant at each site of the 1D lattice. In the first work, we investigate the dynamics of optical pulses or particles in a 1D lattice with saturated nonlinearity. Thus, we show how the parameter regulating the saturation of the nonlinear parameter modifies the particle dynamics in this system. We demonstrate how stable uniform solutions evolve into regular and irregular breathing regimes, where we identify these two as intermediate regimes and localized solutions (self-trapped). Furthermore, we show how saturation influences these regimes and their transitions. Phase diagrams are presented, highlighting the influence of saturation on the dynamics. For the regular breathing regime (breathing mode), we numerically show that the critical χ above which uniform solutions become regular breathing solutions increases with saturation. An analytical assessment corroborates this result. Moreover, the critical nonlinear force that separates the regimes of regular and chaotic breathing solutions exhibits a decreasing behavior as the saturation parameter increases. We observe clear signatures of rogue waves within the chaotic regime, as shown in long-tail statistics. The boundaries of this regime are increased by saturable nonlinearity. Thus, the regime of localized solutions, which we demonstrate to be well described by structures similar to bright solitons, becomes less accessible with increasing saturation. In the second work, we investigate the dynamics of two interacting particles in a one-dimensional nonlinear system. We show that interaction promotes the emergence of a breathing regime when we take into account the onsite interaction in the dynamics. We also present that the value of the critical nonlinear parameter, above which self-trapping occurs, is altered, being a smaller value compared to the case of non-interacting particles. With the emergence of the breathing regime, we calculate the critical threshold points for the three regimes. Thus, the critical nonlinear parameter for the regime of uniformly distributed solution (initial condition) to the breathing regime rapidly drops to small values of interaction but then decreases slowly for larger values of interaction. On the other hand, the critical nonlinear parameter that promotes the transition between the breathing regime and the self-trapped regime has a non-monotonic behavior with interaction, so that we have an abrupt drop of the critical point for small values of interaction, a rapid increase, and then slowly decreases with interaction. Finally, we show that for sufficiently large interaction values, particle separation is promoted during the dynamics of the wave function. 

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