Justification of the discrete nonlinear Schr\"odinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisons

TL;DR

This paper rigorously justifies the reduction of a damped, driven discrete nonlinear Klein-Gordon equation to a discrete nonlinear Schr"odinger equation using energy estimates, and supports findings with numerical simulations.

Contribution

It provides the first rigorous error bounds for the approximation and proves existence of solutions to the reduced equation, extending the theoretical understanding of these models.

Findings

Error bounds for the approximation are established.

Numerical simulations confirm the accuracy of the reduced model.

Existence of solutions to the discrete nonlinear Schr"odinger equation is proven.

Abstract

We consider a damped, parametrically driven discrete nonlinear Klein-Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schr\"odinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of {solutions to the discrete nonlinear} Schr\"odinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schr\"odinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation.