This is a research activity only at a tool level, but a research interest at all levels

The main point of view is that coming from the continuous symmetry groups of differential equations [G.W. Bluman & S. Kumei "Symmetries and Differential Equations" (1989) Applied Mathematical Sciences 81. Springer-Verlag] [G.W. Bluman & J.D. Cole "Similarity Methods for Differential Equations" (1974) Applied Mathematical Sciences 13. Springer-Verlag] [P.J. Olver "Applications of Lie Groups to Differential Equations" (1986) Graduate Texts in Mathematics 107. Springer-Verlag]

A very simple (trivial) example of the application of the methods based on Continuous Groups of Transformations to the resolution of P.D.E., one can consider the non-dimensional standard heat equation in 1-D:

∂_tθ =∂²_xxθ

From this equation, one may extract all the continuous groups of symmetry, given some restrictions (ie: locality, …) which are generalizable. One of such groups is a continuous translation in x. That is, the equation above is invariant under the transformation (∀ ε ∈ ℜ):

x → x+ ε

t → t

θ → θ

As a consequence; one may infer that some solutions of the equation could verify the same symmetry. Consequently, there could be solutions of the form: θ =θ'(t). Putting it on the initial heat equation one converts the P.D.E. into a O.D.E.:

^{d θ'}

— = 0

_{d t}

Thus the only solution of the heat equation verifying this symmetry is a pure constant.

All this technique could be done in general in such a way it is possible to obtain non-trivial solutions to (usually) more complicated P.D.E. equations (see references above). In general, the technique allows to simplify (sometimes even to completely solve) the problem either by reducing the order of the equation, or by diminishing the number of independent variables, or by allowing to express the solution as a specific series expansion, among other possibilities.

In general, not all the symmetries of a P.D.E. (system) are inherited by the solutions (state), but usually (unfortunately, not always) one may find solutions that verify a specified symmetry of the equation. It is a major challenge to know (mathematically) whether for each solution of the equation, there exists a symmetry of the system verified by the considered solution or not. To my knowledge, this has not been proved, but I suspect that the answer is affirmative (of course, generalizing the kind of symmetries to non-local, discrete, …).