Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

DOI10.3233/ASY-191579zbMath1460.35330arXiv1911.01631MaRDI QIDQ5151334

Hadi Susanto, Y. Muda, Fiki T. Akbar, Bobby Eka Gunara, Rudy Kusdiantara

Publication date: 17 February 2021

Published in: Asymptotic Analysis (Search for Journal in Brave)

Abstract: We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schr"odinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schr"odinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schr"odinger equation are presented.

Full work available at URL: https://arxiv.org/abs/1911.01631

zbMATH Keywords

discrete nonlinear Schrödinger equationdiscrete breatherdiscrete solitondiscrete Klein-Gordon equationsmall-amplitude approximation

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) NLS equations (nonlinear Schrödinger equations) (35Q55) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Soliton equations (35Q51) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Soliton solutions (35C08)

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