A differential equation of solitary wave oscillons is defined by,
$$ \Delta S -S +S^3=0 $$
**How can we write this equation as,**
\begin{equation}
\langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle S^4\rangle=0 \tag{1}
\end{equation}
where $\langle f\rangle:=\int d^Dx f(x)$. Furthermore, another virial identity
can be found
from the scaling transformation ($\vec{x}\to \mu \vec{x}$)
by extremizing the scaled ($\vec{x}\to\mu \vec{x}$)
of the action corresponding to
$ \int d^Dx[(\vec{\nabla}S)^2+S^2-S^4/2]$:
\begin{equation}
(D-2)\langle(\vec{\nabla}S)^2\rangle+D\langle S^2\rangle-\frac{D}{2}\langle S^4\rangle=0 \tag{2}
\end{equation}
From Eqs. (1) and (2) one immediately finds
\begin{equation}
2\langle S^2\rangle+\frac{1}{2}(D-4)\langle S^4\rangle=0\,,
\end{equation}
which equality can only be satisfied if $D<4$.
D= Refers dimension.
If you have any Query then ask me please.
Thanks in advance.
To see details, please check the paper here in equations (21), (41)and (42)