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

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 Schrödinger 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 Schrödinger 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 Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation.

Dissipative structures in a parametrically driven dissipative lattice: chimera, localized disorder, continuous-wave, and staggered state

“…It is relevant to mention that, in addition to the direct realizations of Eq. (1) as the chain of coupled oscillators with the complex amplitudes, the same model can be derived as an asymptotic approximation for a parametrically driven discrete nonlinear Klein-Gordon equation or Frenkel-Kontorava model, i.e., a chain of nonlinear oscillators with real dynamical variables [78]. Figure 1 by parametrically driven damped discrete nonlinear Schrödinger equation (1).…”

“…In particular, the stability of discrete solitons in the parametrically driven DNLS equations, both conservative and dissipative ones, has been studied in Refs. [74][75][76][77][78].…”

Bloch oscillations, Landau–Zener transition, and topological phase evolution in an array of coupled pendula

“…Here, we report a direct measurement of a topological and geometric phase by tracking the dynamics in the bulk of a classical system: a one-dimensional array of coupled pendula. The analysis of the measurement is based on an approximate mapping between the time evolution of the classical coupled oscillators and the quantum tight-binding or discrete Schrödinger equation of electrons on lattice potentials, as was recently demonstrated for the nonlinear case ( 26 , 31 , 32 ). The mapping is in the spirit of the mapping in ref.…”