In this work, we study at the $L^2$ – level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrödinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrödinger equation and we prove its convergence and exponential stability of the discrete energy.

中文翻译：

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具有局部耗散的广义高阶非线性薛定谔方程的适定性和渐近行为

在这项工作中，我们在 ${升}^2$– 在有界区间上提出的初始边界值问题的水平全局适定性和长期稳定性，用于广义高阶非线性薛定谔方程，模拟光纤中的脉冲传播，具有局部阻尼项. 此外，我们实现了一个精确有效的代码来研究高阶非线性薛定谔方程的能量衰减，并证明了离散能量的收敛性和指数稳定性。