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

Article type: Research Article

Authors: Muda, Yuslenita^{a; b} | Akbar, Fiki T.^c | Kusdiantara, Rudy^{a; c; d} | Gunara, Bobby E.^c | Susanto, Hadi^{a; *}

Affiliations: [a] Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, United Kingdom. E-mails: ymuda@essex.ac.uk, hsusanto@essex.ac.uk | [b] Department of Mathematics, Faculty of Science and Technology, State Islamic University of Sultan Syarif Kasim Riau, Pekanbaru, 28294, Indonesia | [c] Theoretical Physics Laboratory, Theoretical High Energy Physics and Instrumentation Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, Indonesia. E-mails: ftakbar@fi.itb.ac.id, bobby@fi.itb.ac.id | [d] Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, Indonesia. E-mail: rudy@math.itb.ac.id

Correspondence: [*] Corresponding author. E-mail: hsusanto@essex.ac.uk.

Abstract: We consider a discrete nonlinear Klein–Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger 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 Schrödinger 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 Schrödinger equation are presented.

Keywords: Discrete nonlinear Schrödinger equation, discrete Klein–Gordon equation, small-amplitude approximation, discrete breather, discrete soliton

DOI: 10.3233/ASY-191579

Journal: Asymptotic Analysis, vol. 120, no. 1-2, pp. 73-86, 2020