New distributional travelling waves for the nonlinear Klein-Gordon equation

Abstract

The present paper concerns the study of distributional travelling waves in models ruled by the nonlinear Klein-Gordon equation $u_{tt}-c^{2}u_{xx} =\phi(u)$, where $c>0$ is a real number and $\phi$ is an entire function which takes real values on the real axis. For this purpose, we use a product of distributions that extends the meaning of $\phi(u)$ to certain distributions $u$ and that allows us to define a solution concept consistent with the classical solution concept. The phi-four equation and the sine-Gordon equation are examined as particular cases.