The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficients c(x)utt + d(x)ut = [a(x)ux]x + b(x)ux + p(x) f (u), 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.

中文翻译：

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非线性 Klein-Gordon 和电报类型方程的对称约简和新的函数可分解

该论文涉及不同类别的非线性 Klein-Gordon 和电报类型方程，具有可变系数 c(x)utt + d(x)ut = [a(x)ux]x + b(x)ux + p(x) f (u)，其中 f (u) 是任意函数。我们使用函数 u = U (z) 和 z = φ (x, t) 的组合，通过对称约简的直接方法来寻求这些方程的精确解。我们表明 f (u) 和五个函数系数 a(x)、b(x)、c(x)、d(x) 和 p(x) 中的任意四个可以任意设置，而剩余系数可以用其他系数表示。该研究调查了 φ (x, t) 的 PDE 超定系统的性质并找到了一些解。给出了具有新的精确泛函可分解的特定方程的示例。此外，