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D.E. Pelinovsky, T. Penati, and S. Paleari

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Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrodinger equations

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Reviews in Mathematical Physics 28 (2016), 1650015 (25 pages)

**Abstract:**

Small-amplitude weakly coupled oscillators of the Klein-Gordon
lattices are approximated by equations of the discrete nonlinear
Schrodinger type. We show how to justify this approximation by
two methods, which have been very popular in the recent
literature. The first method relies on a priori energy estimates and
multi-scale decompositions. The second method is based on a resonant
normal form theorem. We show that although the two methods are
different in the implementation, they produce equivalent results as
the end product. We also discuss applications of the discrete nonlinear Schrodinger
equation in the context of existence and stability of breathers of the Klein-Gordon lattice.

**Keywords:**

discrete Klein-Gordon equation, discrete nonlinear Schrodinger equation, multi-site breathers,
instability, justification of amplitude equations, energy methods, normal form theorem.