We address the existence and properties of discrete embedded solitons (ESs), that is localised waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both Chi(2) (second-harmonic generation) and Chi(3) (Kerr) nonlinearities, for which a rich family of ESs are known to occur in the continuum limit. First, a simple motivating problem is considered, in which the Chi(3) nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From the properties of such maps, it is shown that (unlike ordinary gap solitary waves), discrete ESs have the same codimension as their continuum counterparts. A speci c numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a speci c reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.
Additional information: Preprint of a paper later published by IOP Publishing (2005), Nonlinearity, 18(6), pp.2591-2613, ISSN 0951-7715