Hyperbolic equations are used in several physical phenomena to describe dynamical processes where information propagates with a finite speed. Recently, the wave equations with damping terms turned out to be fundamental hyperbolic equations in certain branches of physics mainly scattering processes and fractal medium. The aim of the present study is double shooting, first to prove that damped quantum wave equations may be obtained using the notion of non-standard Lagrangians and second to show that linear and nonlinear damping terms may be obtained if the concept of ‘two-occurrences of integrals’ is used, hence reducing the damped quantum wave equation to the conventional quantum wave equation known as the Klein-Gordon equation. This study supports the idea of non-standard Lagrangians and its usefulness in the theory of partial differential equations.

双曲方程在多种物理现象中用于描述信息以有限速度传播的动态过程。最近，带有阻尼项的波动方程被证明是物理学某些分支（主要是散射过程和分形介质）中的基本双曲方程。本研究的目的是双射，首先证明可以使用非标准拉格朗日概念获得阻尼量子波方程，其次证明如果使用“两次出现”的概念，可以获得线性和非线性阻尼项使用“积分”，因此将阻尼量子波动方程简化为称为克莱因-戈登方程的传统量子波动方程。