Influência da dispersão de quarta ordem na formação de estados localizados e na quebra espontânea de simetria

Abstract

This thesis investigates the role of fourth-order dispersion in the formation of localized states and spontaneous symmetry breaking in systems described by the Generalized Nonlinear Schrödinger Equation. Initially, a model including second- and fourth-order dispersion, in the absence of an external potential, and a spatially modulated defocusing nonlinearity was analyzed to determine the existence and dynamical stability of fundamental and excited stationary states. Subsequently, a double-well potential with attractive nonlinearity was considered to examine the influence of fourth-order dispersion on bifurcation diagrams and the spontaneous symmetry-breaking phenomenon. Stationary solutions were obtained using the imaginary-time propagation method, while excited states were obtained using the Gram–Schmidt orthogonalization procedure. Their dynamical stability was investigated by real-time propagation using the Split-Step Fourier method. The results show that fourth-order dispersion significantly modifies the properties of localized states, affecting their peak amplitude, spatial width, and stability regions. Furthermore, it alters the bifurcation structure by shifting the thresholds for spontaneous symmetry breaking and promoting the stabilization of localized solutions within specific parameter regimes. These findings contribute to a deeper understanding of higher-order dispersion effects in nonlinear optical systems and may provide valuable insights for the design of photonic devices based on the control of localized states and symmetry-breaking phenomena.