Geodesic
Statistical Approach to Modulational Instability in Nonlinear Discrete Systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 56-63

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We use a statistical approach to investigate the modulational instability (Benjamin–Feir instability) in several nonlinear discrete systems: the discrete nonlinear Schrodinger (NLS) equation, the Ablowitz–Ladik equation, and the discrete deformable NLS equation. We derive a kinetic equation for the two-point correlation function and use a Wigner–Moyal transformation to write it in a mixed space-wave-number representation. We perform a linear stability analysis of the resulting equation and discuss the obtained integral stability condition using several forms of the initial unperturbed spectrum (Lorentzian and $\delta$-spectrum). We compare the results with the continuum limit (the NLS equation) and with previous results.
Keywords: modulational instability – nonlinear discrete systems. 
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     author = {D. Grecu and A. Visinescu},
     title = {Statistical {Approach} to {Modulational} {Instability} in {Nonlinear} {Discrete} {Systems}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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D. Grecu; A. Visinescu. Statistical Approach to Modulational Instability in Nonlinear Discrete Systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 56-63. http://geodesic.mathdoc.fr/item/TMF_2005_144_1_a5/