Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficientsстатья
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Аннотация:The paper deals with non-linear Klein-–Gordon type equations
$$
c(x)u_{tt}=[a(x)f(u)u_x]_x+b(x)g(u).
$$
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 form
$$
\int h(u)\,du=\xi(x)\omega(t)+\eta(x),
$$
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.