Nonlinear Schrödinger-Type Systems

This thesis investigates nonlinear systems that are dispersive and conservative in nature and well-approximated by the nonlinear Schrödinger (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 Schrödinger-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 β is discussed. Several features not observed in the absence of high-order dispersion (β=0) are highlighted, such as a nonzero threshold value of potential strength below which there is no band gap and the formation of novel localized modes at large potential amplitudes. A higher order two band tight binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semi-infinite and first band gaps are constructed. Stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation. 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 (PT) symmetries. A new family of non-hermitian optical potentials given in terms of double exponential periodic functions is introduced. The center of PT-symmetry is not around zero and the potential satisfies a shifted PT-symmetry relation at two distinct locations. These novel refractive index modulations are examined from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics, abrupt phase transitions in the eigenvalue spectrum and exceptional points in the band structure are studied in detail. In addition, the nonlinear properties of coherent structures in Kerr media is probed. The spatial symmetries of such lattice solitons follow the shifted PT-symmetric relations. Furthermore, such lattice solitons have a power threshold and their linear and nonlinear stability is critically dependent on their spatial symmetry point. In the final chapter a class of exact multi-component constant energy solutions to a Manakov system in the presence of an external PT-symmetric complex potential is constructed. This type of uniform wave pattern displays a non-trivial phase whose spatial dependence is induced from the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogenous gain and loss. The constant-intensity continuous waves are then used to perform a modulational instability analysis in the presence of both non-hermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using Fourier-Floquet-Bloch theory. The stability of the self-focusing and defocusing cases is considered and compared to the zero-potential results. Our linear stability results are supplemented with direct (nonlinear) numerical simulations.