We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schr\"odinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schr\"odinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schr\"odinger equation are presented.

中文翻译：

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将阻尼驱动 Klein-Gordon 方程化简为离散非线性薛定谔方程：证明和数值比较

我们考虑具有阻尼和外部驱动的离散非线性 Klein-Gordon 方程。使用小振幅 ansatz，通常使用阻尼驱动的离散非线性 Schr\"odinger 方程来近似方程。在这里，我们第一次通过使用能量估计找到误差界限来证明这种近似的合理性。此外，我们证明Schr\"odinger 方程的局部和全局存在。执行描述分析结果的数值模拟。介绍了 Klein-Gordon 方程的离散呼吸器与离散非线性 Schr\"odinger 方程的离散孤子之间的比较。