Reduction of damped, driven Klein-Gordon equations into a discrete nonlinear Schr\"odinger equation: justification and numerical comparisons

TL;DR

This paper justifies approximating damped, driven discrete Klein-Gordon equations with discrete nonlinear Schr"odinger equations using energy estimates, and supports findings with numerical simulations comparing their solutions.

Contribution

First rigorous justification of the approximation with error bounds, including proofs of local and global existence, supported by numerical comparisons.

Findings

Error bounds for the approximation are established.

Numerical simulations confirm analytical results.

Discrete breathers and solitons are compared successfully.

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.