The q parameter and the intervals of initial beam power \(({ p}_{0})\) of q-Gaussian laser beams have been explored theoretically in the current study. In order to lead the propagation behaviour of the q-Gaussian laser beam, the BWP (beam-width parameter) differential equation was minimized into two variables: the q parameter and \({p}_{0}\) . The nonlinearity in a dielectric function used herein is mainly due to collisional plasma. Further investigation of the propagation behaviour of q-Gaussian laser beams in the collisional plasma makes use of the impact of intervals of \({p}_{0}\) and q parameters. The Akhmanov’s parabolic equation method via paraxial and WKB (Wentzel-Kramers-Brillouin) approximations utilised in the present study and the nonlinear second-order differential equation is set up for the BWP f and solved numerically using fourth-order Runga–Kutta method. Results found fascinating, are discussed and revealed graphically. The current study results may have potential applications in laser-based plasma technologies, such as particle acceleration and inertial confinement fusion.
