Asymptotic behavior of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations

Abstract

We show decay estimates for the propagator of the discrete Schrödinger and Klein-Gordon equations in the form $\norm{U(t)f}{l^\infty}\leq C (1+|t|)^{-d/3}\norm{f}{l^1}$. This implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant $l^p$ norms. The analytical decay estimates are corroborated with numerical results.

arXiv comment13 pages, 4 figures

References in corpus (4)

• Josephson Junction Arrays with Bose-Einstein Condensates
• Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential
• Superfluid current disruption in a chain of weakly coupled Bose-Einstein Condensates
• Excitation Thresholds for Nonlinear Localized Modes on Lattices

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