Nonlinear Schrodinger-type systems: Complex lattices and non-paraxiality.

摘要

his thesis investigates nonlinear systems that are dispersive and conservative in nature and well-approximated by the nonlinear Schr"odinger (NLS) equation.;The NLS equation is the prototypical equation for describing such phenomena and it has been utilized in a large number of physical systems.;This work considers novel applications and exotic parameter regimes that fall inside the class of solutions well described by nonlinear Schr"odinger-type systems.;A brief historical, physical, and mathematical introduction to deriving the NLS equation and its variants is presented.;The topics considered in detail cover optical systems in various media and are naturally divided into two parts: non-paraxiality through the inclusion of higher-order dispersion/diffraction and beam propagation in the presence of complex lattices.;The higher-order dispersion/diffraction effects on soliton solutions are considered in detail. The propagation of a short soliton pulse as it travels down a fiber optic in the presence of a linear time-periodic potential is considered. Due to the short duration of the pulse fourth-order dispersive effects are relevant. The band gap structure is determined using Floquet-Bloch theory and the shape of its dispersion curves as a function of the fourth-order dispersion coupling constant ;Next the spectral transverse instabilities of one-dimensional solitary wave solutions to the two-dimensional NLS equation with biharmoinc diffraction and subject to higher-dimensional perturbations are studied. Physically, the inclusion of the biharmonic term corresponds to spatial beams with a narrow width in comparison to their wavelength. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed and a finite band of unstable transverse modes is found to exist. In the long wavelength limit an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The dynamics of a one-dimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations.;The second half of the dissertation is concerned with beam propagation in the presence of complex lattices, in particular lattices that possess parity-time (;In the final chapter a class of exact multi-component constant energy solutions to a Manakov system in the presence of an external