Effects of nonlocal dispersive interactions on self-trapping excitations

Abstract

A one-dimensional discrete nonlinear Schrödinger ͑NLS͒ model with the power dependence r Ϫs on the distance r of the dispersive interactions is proposed. The stationary states 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 s cr , there is an interval of bistability where two stable stationary states exist at each excitation number Nϭ ͚ n ͉ n ͉ 2 . For cubic nonlinearity the bistability of on-site solitons may occur for dipole-dipole dispersive interaction (sϭ3), while s cr for intersite solitons is close to 2.1. For increasing degree of nonlinearity , s cr increases. The long-distance behavior of the intrinsically localized states depends on s. For sϾ3 their tails are exponential, while for 2ϽsϽ3 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 sϽϩ1.

Discrete breathers in an one-dimensional array of magnetic dots

“…The decay law is close to the power law if we are not far from the anti-continuum limit. Indeed, we observe almost power law decay for = 0.018 with The power-law decay is in accord with other models that possess long-range interaction [42][43][44]. As increases and the dipole-dipole interaction becomes more prominent.…”

Fractional dynamics of coupled oscillators with long-range interaction

“…Discrete breathers have been widely studied in systems with short-range interactions (for a review, see [57,53]). Energy and decay properties of discrete breathers in systems with long-range interactions have also been studied in the framework of the Klein-Gordon [54,58], and the discrete nonlinear Schrodinger equations [59]. Therefore, it is interesting to consider breathers solution in systems with long-range interactions in infrared approximation.…”

Stabilization of nonlinear excitations by disorder

“…1 is very similar to the bistability that occurs when long-range effects are included in the NLS framework. An example of this has been studied recently by Gaididei et al 23 who also showed that the bistability occurred due to competition between two different length scales of the problem, one length scale being defined by the relation between the nonlinearity and the dispersion, while the range of the nonlocal interaction defines the other length scale. The same effect is present in Eq.…”