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

摘要

The paper deals with non-linear Klein-Gordon type equationsc(x)u(11) = [a(x)f (u)u(x)](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 formintegral h(u)du = xi(x)omega(t)+ eta(x),where the functions h(u), xi(x), eta(x), and omega(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.