Thread: Differential equation dimension analysis

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)

Forhad, welcome to TPF. Can you re-write your post using standard LaTeX notation, since I couldn't really make heads or tails of your maths code. Enclose the LaTeX code between "tex" and "/tex", each of which between angle brackets [].

Originally Posted by Forhad

A differential equation of solitary wave oscillons is defined by,

What is a solitary wave oscillon? Do you mean a soliton wave oscillation?