Justification of the discrete nonlinear Schrodinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisons

摘要

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 Schrodinger 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 Schrodinger 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 Schrodinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation. (C) 2019 Elsevier B.V. All rights reserved.