What is the relationship between symmetry groups of differential equations like in works P. Olver, L. Ovsyannikov, P. Winternitz and kinematical invariance groups like in works U. Niederer, C. Boyer, W. Fushchich. Are there any articles with comparison these two concepts?
The kinematical invariance groups are a special case of the general symmetry groups considered in the books by Olver and Ovsyannikov.
S. C. Anco, thank you for your answer. I understand your answer. This became clear when I had read the article by U. Niederer. But I want something more, for example: is the kinematical invariance group a invariant subgroup in the general symmetry group? Do you know any articles about this relationship? Thank you.
All symmetry matters that you have mentioned are rather particular cases
of the general geometric theory of nonlinear PDEs developped by the Russian school. See ''Symmetries and Conservation Laws for Differential Equations of Mathematical Physics", AMS Translations of Math. Monographs, vol. 182. The problem is that this book is written in a higher level language than that is common for experts in math. physics. This is why resuts you can find in this book are as usual ignored. In particular, the Olver book presents some results of the Russian school in a lower level language, which is familiar for general experts in math. physics but is
unsufficient to face the highest complexity of problems related with nonlinear PDEs.
Alexandre Mikhailovich Vinogradov, thank you for your answer. I have been reading this book before the sixth capter(nonlocal symmetries) a couple of years ago. Its a very interesting and good book, but I dont remember any words about kinematical symmetries and their link, for example, with gauge symmetries. Maybe I should continue reading this book. Thanks a lot.
That's correct: you will not find any reference to kinematic symmetries in this book. This is because the peculiarity of kinematic and other particular kinds of symmetries is encoded in the form of their generating functions.
For instance, generating functions of point symmetries depend only on special combinations of first order derivatives.
So, in a practical search of various special symmetries you have to solve the equation $l_F(f)=0$ for the generated function $f$ of the corresponding special form.
Thanks, the generating function of point symmetry must be linear in first order derivatives and there are no higher order derivatives in it.
Not simply linear, but with some additional conditions if the number of unknown functions is > 1, ...
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