Damped quantum wave equation from non-standard Lagrangians and damping terms removal

Abstract

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.