This thesis is mainly concerned with the properties of some discrete nonlinear Schrödinger equations. These naturally arise in many different physical contexts as the limiting form of general dynamical lattice equations that incorporate nonlinearity and coupling. Interest is focused on theoretical models of coupled optical waveguides constructed from materials with a nonlinear index of refraction. In arrays of waveguides the overlap of the evanescent electric field of the modes in neighbouring waveguides provides a coupling and the nonlinearity of the material provides a mechanism to halt the discrete diffraction that otherwise would spread localized energy across the array. In particular, waveguide structures where also a nonlinear coupling is taken into account are studied. It is noted that the equation for the evolution of the complex amplitudes of the electric field along an array of waveguides also can be used to describe the dynamics of Bose-Einstein condensates trapped in a periodic optical potential. Possible excitations in arrays in both one and two dimensions are considered, with emphasis on the effects of the nonlinear coupling.
Localized excitations are considered from the viewpoint of the theory of discrete breathers, or intrinsic localized modes, i.e., solutions of the dynamical equations that are periodic in time and have a spatial localization. The general theory of such solutions, that appear under very general circumstances in nonlinear lattice equations, is reviewed. In an array of waveguides this means that light can propagate along the array confined essentially to one or a few waveguides. In general a distinction is made between excitations that are centred on a waveguide, or site in the lattice, and excitations that are centred inbetween waveguides. Usually only the former give stable propagation. When the localized beam can be displaced to neighbouring waveguides the array can operate as an optical switch. With the inclusion of nonlinear coupling between the sites, as in the model derived in this thesis, the stability of the site-centred and bond-centred solutions can be exchanged. It is shown how this leads to the existence of highly localized mobile solutions that can propagate transversely in the one-dimensional array of waveguides. The inversion of stability of stationary solutions occurs also in the two-dimensional array, but in this setting it fails to give good mobility of localized excitations. The reason for this is also explained.
In a two-dimensional lattice a discrete breather can have the form of a vortex. This means that the phase of the complex amplitude will vary on a contour around the excitation, such that the phase is increased by 2πS, where S is the topological charge, on the completion of one turn. Some ring-like vortex excitations are considered and in particular a stable vortex with S=2 is found. It is also noted that the effect of charge flipping, i.e., when the topological charge periodically changes between -S and S, is connected to the existence of quasiperiodic solutions.
The nonlinear coupling of the waveguide model will also give rise to some more exotic and novel properties of localized solutions, e.g., discrete breathers with a nontrivial phase. When the linear coupling and the nonlinear coupling have opposite signs, there can be a decoupling in the lattice that allows for compact solutions. These localized excitations will have no decaying tail. Of interest is also the flexibility in controlling the transport of power across the array when it is excited with a nonlinear plane wave. It is shown how a change of the amplitude of a plane wave can affect the magnitude and direction of power flow in the array.
Also the continuum limit of the one-dimensional discrete waveguide model is considered with an equation incorporating both nonlocal and nonlinear dispersion. In general continuum equations the balance between nonlinearity and dispersion can lead to the formation of localized travelling waves, or solitons. With nonlinear dispersion it is seen that these solitons can be nonanalytic and have discontinuous spatial derivatives. The emergence of short-wavelength instabilities due to the simultaneous presence of nonlocal and nonlinear dispersion is also explained.
Institutionen för fysik, kemi och biologi , 2007.