Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients

The paper deals with non-linear Klein–Gordon type equations c(x)utt=[a(x)f(u)ux]x+b(x)g(u).The direct method for constructing functional separable solutions in implicit form to non-linear PDEs is used. This effective method is based on the representation of solutions in the form ∫h(u)du=ξ(x)ω(t)+η(x),where the functions h(u), ξ(x), η(x), and ω(t) are determined further by analyzing the resulting functional-differential equations. Examples of specific Klein–Gordon type equations and their exact solutions are given. The main attention is paid to non-linear equations of a fairly general form, which contain several arbitrary functions dependent on the unknown u and /or the spatial variable x (it is important to note that exact solutions of non-linear PDEs, that contain arbitrary functions and therefore have significant generality, are of great practical interest for testing various numerical and approximate analytical methods for solving corresponding initial–boundary value problems). Many new generalized traveling-wave solutions and functional separable solutions (in closed form) are described. Solutions of several Klein–Gordon equations with delay are also given. •Non-linear Klein–Gordon type equations are considered.•Arbitrary functions are included in equations.•Exact solutions are sought in the implicit form.•New functional separable solutions are presented.•New generalized traveling-wave solutions are obtained.