Involution and Symmetry Reductions

Reference: Mathematical and Computer Modelling, 25 (1997) 63-73 (invited contribution)

Description: Like [7] this paper applies results from [5] to the symmetry theory of PDEs. This time the emphasis is on non-classical methods like the Bluman-Cole method or the method of differential constraints. For the former one a simple generalisation is presented (unfortunately, it was already discovered earlier by Pucci and Saccomandi). For the latter one I tried to use the formal theory to find criteria for the selection of useful constraints. However, the results have been disappointing from the point of view of applications. But in the meantime Peter Vassiliou has shown me that the results are actually quite interesting, if one reads them in a different way. Namely, the formal theory automatically picks out a special class of second-order PDEs in the plane already discussed by Cartan and solvable by the method of Darboux. We hope to extend these results in the future to higher dimensions.

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