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

physics letters a

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