Consider \begin{equation}\label{1} \partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1) \end{equation} the $\phi^4$ model.
I know that $$H(x)=\tanh\left(\frac{x}{\sqrt{2}}\right),\: \forall \; x \in \mathbb{R}$$ is a time-independent solution of the $(1)$ and is called kink. Taking into account that $$sn(u,1)=\tanh(u)\: \forall \; u \in \mathbb{R}$$ where $sn$ denote the elliptical snoidal function, and that $sn$ is periodic. I can say that $H$ is periodic? In addition, the $\phi^4$ model admit explicit traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ , where $c \in \mathbb{R}$? The answer to the last question I think is 'yes', because the solution a translation of $H$ also remains a solution of $(1)$.