  We introduce some analytic relations on the set of partial differential
equations of two variables. It relies on a new comparison method to give rough
asymptotic estimates for solutions which obey different partial differential
equations. It uses a kind of scale transform called tropical geometry which
connects automata with real rational dynamics. Two different solutions can be
considered when their defining equations are transformed to the same automata
at infinity. We have a systematic way to construct related pairs of different
partial differential equations, and also construct some unrelated pairs
concretely. These verify that the new relations are non trivial. We also make
numerical calculations and compare the results for both related and unrelated
pairs of PDEs.
