Statistical approach on modulational instability in nonlinear discrete systems

The modulational instability (Benjamin-Feir instability) in several nonlinear discrete systems (discrete NLS, Ablowitz-Ladik, discrete deformable NLS equation) is investigated using a statistical approach. A kinetic equation for a 2-point correlation function is obtained, and using a Wigner-Moyal transformation it is written in a mixed space-wave number representation. A linear stability analysis of the resulting equation is performed, and the obtained integral stability equation is discussed using several forms of the initial unperturbed spectrum (δ-spectrum, Lorentzian). The results are compared with the continuum limit (NLS equation) and previous results. Ten years ago Kivshar and Salerno have discussed the problem of modulational instability in a 'deformable nonlinear Schroedinger equation' (dNLS). A model suitable to investigate the effect of the interplay between the nonlinear on-site and inter-site interactions on the modulational instability is presented. For λ = 0 it becomes the discrete nonlinear Schroedinger equation (or discrete self-trapping equation) which is nonintegrable, while for γ = 0 it becomes the integrable Ablowitz-Ladik equation. The problem of modulational instability is discussed in a previous paper from a deterministic point of view. A linear stability of a small modulated plane wave is investigated. Besides the usual instability in the long wave length region, an instability in the short wave region is found, and this is the effect of the interplay mentioned above (γ < 2λ. If the deterministic approach of the modulational instability (DAMI) phenomenon is well known, the complementary statistical approach (SAMI) is less used. In this approach a kinetic equation for a two-point correlation function is written down and a linear stability analysis of it is performed. The aim is to investigate the influence of the statistical properties of the medium on the instability development. This can have important consequences (in hydrodynamics and plasma physics) and the influence is significant. The main result can be formulated in the following way: if the space correlation in the initial state is of too short range the instability is suppressed. Recently we applied this approach to discrete systems, like the discrete self-trapping and Ablowitz-Ladik equations. It is the aim of the present paper to discuss SAMI for dNLS equation and to see the effect of the interplay between on-site and inter-site interactions on the development of MI. In the present paper, we briefly review the DAMI for dNLS. The problem, is also carefully discussed in the literature, namely, the deformable nonlinear Schroedinger model discussed by Kivshar and Salerno is well suited to see the effect of the interplay between on-site and intersite nonlinearity on the modulational instability phenomenon. This problem was studied in the present paper both in a deterministic and statistical approach. In both approaches a new instability region is found, when k, the wave vector of the carrying wave, is near the boundary of the Brillouin zone (k ≅ π. This new instability region is a direct consequence of the interplay mentioned above. A complete treatment of MI from a statistical approach is done. An integral stability equation was obtained and it was solved for several initial spectral functions (δ-spectrum centered on k = 0 and k = ± π, a Lorentzian spectrum). In the case of Lorentzian spectrum the increment of the instability (the imaginary part of the frequency of the slow modulation) was calculated for the discrete NLS equation. The result emphasizes the strong influence of the statistical properties of the medium on the development of MI, namely the instability is possible only if a certain long range correlation exists between the particles in the initial state. If it is too short the instability is suppressed. The same conclusion is obtained also in other situations and seems to be generally valid. (author)