Effects of nonlocal dispersive interactions on self-trapping excitations

A one-dimensional discrete nonlinear Schrödinger (NLS) model with the power dependence ${\mathrm{r}}^{\mathrm{-}\mathrm{s}}$ on the distance r of the dispersive interactions is proposed. The stationary states ${\mathrm{\psi }}_{\mathrm{n}}$ of the system are studied both analytically and numerically. Two types of stationary states are investigated: on-site and intersite states. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the NLS model with a nearest-neighbor interaction. For s less than some critical value ${\mathrm{s}}_{\mathrm{cr}}$, there is an interval of bistability where two stable stationary states exist at each excitation number N=${\sum }_{\mathrm{n}}$|${\mathrm{\psi }}_{\mathrm{n}}$ ${\mathrm{|}}^2$. For cubic nonlinearity the bistability of on-site solitons may occur for dipole-dipole dispersive interaction (s=3), while ${\mathrm{s}}_{\mathrm{cr}}$ for intersite solitons is close to 2.1. For increasing degree of nonlinearity σ, ${\mathrm{s}}_{\mathrm{cr}}$ increases. The long-distance behavior of the intrinsically localized states depends on s. For s>3 their tails are exponential, while for 2<s<3 they are algebraic. In the continuum limit the model is described by a nonlocal NLS equation for which the stability criterion for the ground state is shown to be s<σ+1.