Embedded solitons in dynamical lattices

We demonstrate that a discrete nonlinear Schrödinger (NLS) equation, which includes the next-nearest-neighbor linear couplings and cubic and quintic nonlinearities of the Ablowitz–Ladik type, gives rise to exact discrete-solitons solutions, both bright and dark, if a special constraint is imposed on coefficients of the equation. The bright lattice soliton is of the regular or embedded type, depending on the coefficients. Thus, this model produces the first explicit example of lattice solitons of the embedded type. The continuum limit of this model is an extended NLS equation which was recently shown to have an exact embedded-soliton (ES) solution. The discrete model may give rise to two different coexisting bright-soliton solutions, as well as to bright and dark ones, at the same values of coefficients in the equation. Evolution of the solitons under perturbations is studied numerically. It is shown that the perturbed regular lattice soliton oscillates without radiation loss, while the embedded soliton slowly loses energy, emitting phonon radiation at wave numbers which are very accurately predicted by the ES-phonon-band resonance condition.