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Modulational instability of zone boundary mode in nonlinear lattices: rigorous results.

We study the modulational instability of the zone boundary mode in nonlinear lattices with generic polynomial potentials. We present an exact expression of the instability growth rate in the high energy limit. The unstable wave number range and the most unstable wave number are obtained. Relevance of the present results on an energy localization state, which appears after growth of the modulational instability, is also discussed.