Discrete embedded solitons

Abstract

We address the existence and properties of discrete embedded solitons (ESs), i.e., localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both the quadratic (second-harmonic generation) and cubic nonlinearities. A rich family of ESs was previously known in the continuum limit of the model. First, a simple motivating problem is considered, in which the cubic 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 properties of such maps, it is shown that (unlike ordinary gap solitons), discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific 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.

A revtex4 text file and 51 eps figure files. To appear in Nonlinearity

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• Variational approximations for travelling solitons in a discrete nonlinear Schrödinger equation
• Weyl solitons in three-dimensional optical lattices
• Anomalous Thermostat and Intraband Discrete Breathers
• Critical States Embedded in the Continuum
• Invisible surface defects in a tight-binding lattice
• Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices
• The fractional saturable impurity
• The fractional nonlinear impurity: A Green function approach