Symmetry reductions and new functional separable solutions of nonlinear Klein—Gordon and telegraph type equations

Abstract

The paper is concerned with different classes of nonlinear Klein—Gordon and telegraph type equations with variable coefficients\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(x){u_{tt}} + d(x){u_t} = {[a(x){u_x}]_x} + b(x){u_x} + p(x)f(u),$$\\end{document}where f (u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using the composition of functions u = U (z) with z = φ(x, t). We show that f (u) and any four of the five functional coefficients a(x), b(x), c(x), d(x), and p(x) in such equations can be set arbitrarily, while the remaining coefficient can be expressed in terms of the others. The study investigates the properties and finds some solutions of the overdetermined system of PDEs for φ(x, t). Examples of specific equations with new exact functional separable solutions are given. In addition, the study presents some generalized traveling wave solutions to more complex, nonlinear Klein—Gordon and telegraph type equations with delay.