On the simulation of the energy transmission in the forbidden band-gap of a spatially discrete double sine-Gordon system

TL;DR

This paper introduces a numerical method for simulating energy transmission in a spatially discrete double sine-Gordon system, accurately capturing energy dynamics and examining nonlinear energy transmission phenomena like supratransmission.

Contribution

It develops a positivity-preserving finite-difference scheme that accurately approximates energy rates in damped double sine-Gordon chains, validated against known solutions.

Findings

The method accurately reproduces known solutions of sine-Gordon systems.

It demonstrates the nonlinear energy transmission and the influence of damping on supratransmission.

Simulations confirm the scheme's effectiveness in modeling energy dynamics.

Abstract

In this work, we present a numerical method to consistently approximate solutions of a spatially discrete, double sine-Gordon chain which considers the presence of external damping. In addition to the finite-difference scheme employed to approximate the solution of the difference-differential equations of the model under investigation, our method provides positivity-preserving schemes to approximate the local and the total energy of the system, in such way that the discrete rate of change of the total energy with respect to time provides a consistent approximation of the corresponding continuous rate of change. Simulations are performed, first of all, to assess the validity of the computational technique against known qualitative solutions of coupled sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used in the investigation of the phenomenon of nonlinear transmission of energy in double sine-Gordon systems; the qualitative effects of the damping coefficient on the occurrence of the nonlinear process of supratransmission are briefly determined in this work, too.