# 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,**

\langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle S^4\rangle=0 \tag{1}

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
$\int d^Dx[(\vec{\nabla}S)^2+S^2-S^4/2]$:

(D-2)\langle(\vec{\nabla}S)^2\rangle+D\langle S^2\rangle-\frac{D}{2}\langle S^4\rangle=0 \tag{2}

From Eqs. (1) and (2) one immediately finds

2\langle S^2\rangle+\frac{1}{2}(D-4)\langle S^4\rangle=0\,,

which equality can only be satisfied if $D<4$.
D= Refers dimension.

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 [].