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Could anyone please suggest related papers or article about the topic related to my one question below?

Reduce PDE to ODE by dilation symmetry

I also cite a paper in the link above.

We know that if there are number of $n$ states, we need to find $"n-1"$ symmetries to reduce the PDE to ODE. For each iteration (there are $n-1$ iterations), we need to do some tedious but not difficult calculation.

Is there any advanced method based on symmetries reduction (of course, it might based on some special condition of the structure of problems) such that maybe we can take fewer steps?

Thanks in advance.

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  • $\begingroup$ We really want to find invariant differential relations between the invariant functions. The symmetries might not commute, so the reduction cannot necessarily be carried out in the stages that your problem suggests. $\endgroup$
    – Ben McKay
    Commented May 31, 2020 at 14:40
  • $\begingroup$ @BenMcKay Yes! But I mean if the equations do have some nice form (please let me know which form is qualified) then we can reduce the PDE to ODE more efficiently. $\endgroup$
    – sleeve chen
    Commented May 31, 2020 at 16:40
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    $\begingroup$ As you know from your previous question, Peter Olver's book is probably the main source of information on symmetry methods for differential equations. You could also look at the books of Ibrahimov. $\endgroup$
    – Ben McKay
    Commented May 31, 2020 at 20:21

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