An introcludion to the formation of coherent structures in spatially discrete and continuous systems is given. Of special interest is the phenomenon of nonlinear localization, where the self-focusing of an excitation is balanced by a dispersive process. This leads to the existence of localized waves (solitons) in continuous systems, and under very general conditions to intrinsic localized modes (discrete breathers) in systems of coupled anharmonic oscillators. Focus is set on nonlinear Schrödinger equations. A discrete equation, describing the propagation of the electric field in an array of coupled optical waveguides embedded in a material with a nonlinear index of refraction of the Kerr-type, is derived. The equation also describes the evolution of weakly coupled Bose-Einstein condensates in a periodic potential. The model contains nonlinear coupling terms and an effort is made to understand the novel features introduced by these terms as well as the nonlinear dispersion arising from taking the continuum limit of the discrete equation.
Important contributions in the papers are the discovery of inversion of stability between stationary excitations localized, respectively, on and in between sites in the lattice model for waveguide arrays, leading to an enhanced mobility of highly localized modes. As these can be controlled by simple perturbations. they may have an important applicat ion for optical multiport switching. The nonlinear coupling terms also lead to existence of discrete breathers with compact support and to a new type of stationary, complex, phase-twisted modes not previously reported. Of interest is also the possibility of controlling the magnitude and direction of the norm (Poynting power) current flowing across the waveguide array by simple non-symmetry-breaking perturbations. For the continuum equation, the nonlinear dispersion leads to the formation of exotic solitons, i.e., localized waves with discontinuous derivatives. The emergence of short-wavelength instabilities due to the simultaneous presence of nonlocal and nonlinea.r dispersion is also explained.
Linköping: Linköpings universitet , 2005. , 56 p.