In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.

在这项工作中，引入了两种方法来求解线性阻尼非线性Schrödinger方程（NLSE），以对耗散无赖波（DRW）和耗散通气（DB）进行建模。线性阻尼NLSE被认为是不可积分的微分方程。因此，由于线性阻尼项的存在，它直到现在还不支持明确的解析解。因此，将得出两个详细的精确解。第一个解称为半解析解，第二个解为近似数值解。在这两种解决方案中，将使用标准NLSE的分析解决方案（即在没有阻尼项的情况下）作为求解线性阻尼NLSE的初始解决方案。关于近似数值解，借助有限差分法（FDM）的移动边界法（MBM）将致力于实现这一目的。将推导并获得半解析解的最大残留（局部和全局）误差公式。半解析解的最大残留局部误差和全局误差的数值将使用一些物理数据进行估算。此外，将使用基于Chebyshev逼近技术的非线性多项式来评估与半解析解的局部和全局误差有关的误差函数。此外，将进行近似解析解和数值解之间的比较，以检查两个解的准确性。作为对某些物理结果的现实应用；获得的解决方案将用于研究碰撞的未磁化对离子等离子体中的耗散无赖波（DRW）和耗散通气（DBs）的特性。最后，这项研究有助于我们解释和理解各种等离子体模型，流体力学，光纤，玻色-爱因斯坦凝聚物中的调制结构的动力学行为。