Soliton dynamics in the discrete nonlinear Schrodinger equation

Using a variational technique, based on an effective Lagrangian, we analyze static and dynamical properties of solitons in the one-dimensional discrete nonlinear Schrodinger equation with a homogeneous power nonlinearity of degree 2 sigma + 1. We obtain the following results. (i) For sigma < 2 there is no threshold for the excitation of a soliton; solitons of arbitrary positive energies, W = Sigma\u(n)\(2), exist. (ii) Range of multistability: there is a critical value of sigma, sigma(cr) approximate to 1.32, such that for sigma(cr) < sigma < 2, there exist three soliton-like states in a certain finite intermediate range of energies, two stable and one unstable (while there is no multistable regime in the continuum NLS equation). For energies below and above this range, there is a unique soliton state which is stable. (iii) For sigma greater than or equal to 2, there exists an energy threshold for formation of the soliton. For all sigma > 2 there exist two soliton states, one narrow and one broad. The narrow soliton is stable, while the broad one is not. (iv) We find an energy criterion for the excitation of solitons by initial configurations which are narrowly concentrated in few lattice sites.