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| (1) |
Consider a PDE problem with two independent variables and one dependent variable, , and consider the list of infinitesimals of a symmetry group.
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| (2) |
In the input above you can also enter the symmetry without labels for the infinitesimals, as in , or use the corresponding infinitesimal generator
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A is invariant under the transformations generated by in that , where in this formula represents the prolongation necessary to act on (see InfinitesimalGenerator).
The transformation and its inverse, from the original variables to new coordinates, say , that reduces by one the number of independent variables of a PDE system invariant under above is obtained via
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where in the above you can equally pass instead of . To express this transformation using jet notation use
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| (5) |
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| (6) |
That this transformation reduces the number of independent variables of any PDE system invariant, under above, is visible in the fact that it transforms the infinitesimals into ; to verify this you can use ChangeSymmetry
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| (7) |
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| (8) |
So to this list of infinitesimals corresponds, written in terms of , this infinitesimal generator
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Any PDESYS invariant under will also be invariant under the operator above, that is, will be independent of after you change variables in it using computed with SimilarityTransformation above.
If the new variables - here - are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced
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| (10) |