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We consider a scalar theory in a $1+D$ dimensional flat Minkowski space-time, with a general self-interaction potential, whose action can be written as \begin{equation} A=\int dt\, d^D\! x \left[\frac12(\partial_t\phi)^2-\frac12(\partial_i\phi)^2 -U(\phi)\right] \,, \end{equation} where $\phi$ is a real scalar field, $\partial_t=\partial/\partial t$, $\partial_i=\partial/\partial x^i$ and $i=1,2,\ldots,D$. The equation of motion following from \eqref{action} is a non-linear wave equation (NLWE) which is given as $$ -\phi_{,tt} + \Delta \phi = U'(\phi)=\phi +\sum\limits_{k=2}^{\infty}g_k\phi^k \tag{1}$$ where $$\quad{\Delta}=\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}\,. $$ I just want to know the logic, that how they transformed the equation (1)

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They just expanded the "potential" derivative $U'(\phi)$ in a Taylor series in powers of $\phi$. Kind of polynomial self-interaction.

The coefficient $g_1$ can always be made to be unity with help of redefinition of $t$ and $x$, so they wrote $g_1=1$.

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