# Differential equation dimension analysis

1. May 22, 2013

### Forhad3097

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.

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)

Last edited: May 22, 2013
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