Traveling-wave solutions of a generalized damped wave equation with time-dependent coefficients through the trial equation method

In this note, we investigate the existence of exact solutions of a nonlinear partial differential equation with time-dependent coefficients that generalizes the well-known nonlinear wave model with damping. The model under consideration generalizes other classical models from physics, like the nonlinear Klein–Gordon equation, the (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 + 1)$$\end{document}-dimensional ϕ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi ^4$$\end{document}-theory, the Fisher–Kolmogorov equation from population dynamics and the Hodgkin–Huxley model used in the description of the propagation of electric signals through the nervous system. An extension of the trial equation method (also known as the direct integral method) for partial differential equations with non-constant coefficients is used in this work in order to derive traveling-wave solutions in exact form.