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

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.

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Cited by in corpus (8)

• On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice
• Stabilization of the coupled pendula chain under parametric PT-symmetric driving force
• Long-time stability of breathers in Hamiltonian $\mathcal{P}\mathcal{T}$-symmetric lattices
• Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisons
• Darboux's Theorem, Lie series and the standardization of the Salerno and Ablowitz-Ladik models
• Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms
• On small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting
• Reduction of damped, driven Klein-Gordon equations into a discrete nonlinear Schrödinger equation: justification and numerical comparisons