Some Remarks on the Semilinear Wave Equation

Abstract

We study the following equation

$$\square \psi + W\prime (\psi ) = 0$$

(1)

where

$$\psi :{{R}^{4}} \to C; \square = \frac{1}{{{{c}^{2}}}}\frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}} - \Delta$$

and $W\prime (\psi ) = \frac{{\partial W}}{{\partial {{\psi }_{1}}}} + i\frac{{\partial W}}{{\partial {{\psi }_{2}}}}; \psi = {{\psi }_{1}} + i{{\psi }_{2}}$; W : C → R. We assume that

$$W({{e}^{{i\vartheta }}}\psi ) = W(\psi ) \vartheta \in R$$

so that $W\prime ({{e}^{{i\vartheta }}}\psi ) = {{e}^{{i\vartheta }}}W\prime (\psi )$ and W’(ς) is for ς real.