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)