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

Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street, Nizhny Novgorod, 603950, Russia

E-mail Address: [email protected]

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Tiziano Penati

Department of Mathematics “F.Enriques”, Milano University, via Saldini 50, Milano, 20133, Italy

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, and

Simone Paleari

Department of Mathematics “F.Enriques”, Milano University, via Saldini 50, Milano, 20133, Italy

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https://doi.org/10.1142/S0129055X1650015XCited by:13 (Source: Crossref)

Abstract

Small-amplitude weakly coupled oscillators of the Klein–Gordon lattices are approximated by equations of the discrete nonlinear Schrödinger 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 the applications of the discrete nonlinear Schrödinger equation in the context of existence and stability of breathers of the Klein–Gordon lattice.

Keywords:

AMSC: 37K40, 37K55, 37K60, 70K45

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