Generalised solitary waves of finite-difference nonlinear Schrödinger lattice equations

Generalised solitary waves of several variants of the nonlinear Schrödinger equation are examined. Generalised solitary waves have exponentially small oscillations that extend into the far field. The generalised solitary waves are found to occur in nonlinear Schrödinger equations that are singularly perturbed with higher order dispersion. Exponential asymptotic techniques are used to obtain the behaviour of the generalised solitary waves, it is found that special curves known as Stokes curves produce generalised solitary waves when they cross the real axis. Other studies have determined the generalised solitary waves and under what conditions they occur in many singularly perturbed nonlinear Schrödinger equations. Motivated by these studies, we investigate the Karpman equation which is a nonlinear Schrödinger equation perturbed with an additional fourth order dispersion. These results are used as a foundation for studying nonlinear Schrödinger equations with arbitrary order dispersive perturbations. Lattice Karpman equations are generated using finite difference discretisation which produce infinite order singularly perturbed differential equations. The lattice generalised solitary waves are calculated and the conditions necessary for them to occur is found. Finally the different Stokes curves that occur in the various lattice Karpman equations and the impact they have on the generalised solitary waves is investigated.