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Differential equation dimension analysis

  1. May 22, 2013 #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.
    Thanks in advance.
    To see details, please check the paper here in equations (21), (41)and (42)
     
    Last edited: May 22, 2013
  2. jcsd
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