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Thread: Differential equation dimension analysis

  1. #1 Differential equation dimension analysis 
    Junior Member
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    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)
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  2. #2  
    Administrator Markus Hanke's Avatar
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    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 [].
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  3. #3  
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    Quote Originally Posted by Forhad View Post
    A differential equation of solitary wave oscillons is defined by,
    What is a solitary wave oscillon? Do you mean a soliton wave oscillation?
    The most important thing to keep in mind is that you don't know everything and nobody else does either.
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