Lattice solitons in cubic-quintic media

Solitons are localized nonlinear waves and occur in many branches of physics. Their properties have provided fundamental understanding of complex nonlinear systems. There has been considerable interest in studying solitons in lattices that can be generated in nonlinear optical materials. In nonlinear optics, the propagation of electromagnetic waves in photorefractive materials with intrinsic nonlinear resonance can be modelled by the nonlinear Schrödinger (NLS) equation containing both cubic and quintic terms. Recently, particular attention is drawn to nonlinear waves in parity-time (PT) symmetric potentials. In this thesis, the existence and stability properties of optical solitons on periodic and certain PT-symmetric lattices are investigated. The governing equation for the physical model is the cubic-quintic nonlinear Schrödinger (CQNLS) equation with an external potential: i u_z + Delta u + alpha |u|^2 u + beta |u|^4 u + V u = 0 . Here, u corresponds to the complex-valued, slowly varying amplitude of the electric field in the x (or xy) plane propagating in the z-direction, Delta u corresponds to diffraction, V is an external optical potential (i.e. lattice), alpha and beta are coefficients of the cubic and quintic nonlinearities, respectively. First, solutions to the CQNLS equation with two dimensional periodic potential V_N for N=4 V_N(x,y)= ... , one dimensional PT-symmetric potential V_PT(x)= ... and two dimensional PT-symmetric potential V_PT(x,y)= ... are obtained analytically whenever possible. Then, numerical solutions are calculated by means of the 'Spectral Renormalization Method' or 'Pseudo-Spectral Renormalization Method'. It is observed that the numerical solutions are in good agreement with the analytical solutions. Next, the stability properties of the obtained solitons are investigated. Linear stability analysis is conducted by linear spectrum analysis and linear evolution, whereas nonlinear stability analysis is conducted by nonlinear evolution. A split-step Fourier method is used for both evolutions.