We analyze the existence and uniqueness of monotone traveling wavefront for a generalized nonlinear Klein–Gordon model

$$\frac{{\partial }^2\varphi }{\partial t^2}-\left(p+{\left(\frac{\partial \varphi }{\partial x}\right)}^2\right)\frac{{\partial }^2\varphi }{\partial x^2}+V^{\prime }(\varphi )=0,$$

using classical arguments of ordinary differential equations, with V(x) a potentials family containing the ϕ-four potential $V(x)=M_0{(1-x^2)}^2$ and the sine-Gordon-type potential $V(x)=(1/2)(1+\cos (\pi x))$. Also for these specific potentials, we give estimations of their monotone kink and anti-kink solutions.

中文翻译：

---

在广义 Klein-Gordon 方程中存在单调行波解的基本证明

我们分析了广义非线性 Klein-Gordon 模型的单调行波前的存在性和唯一性

$$\frac{{\partial }^2\phi }{\partial {吨}^2}-\left(磷+{\left(\frac{\partial \phi }{\partial X}\right)}^2\right)\frac{{\partial }^2\phi }{\partial X^2}+{伏}^{\prime }(\phi )=0,$$

使用常微分方程的经典论证，其中V ( x ) 是包含ϕ -4 势的势族$伏(X)={米}_0{(1-X^2)}^2$ 和正弦戈登型势 $伏(X)=(1/2)(1+\cos (\pi X))$. 同样对于这些特定的势，我们给出了它们的单调扭结和反扭结解决方案的估计。