# Thread: Differential equation dimension analysis

1. 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.
To see details, please check the paper here in equations (21), (41)and (42)  2. 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 [].  3. 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?  Tags for this Thread Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Forum Rules