A rough analytic relation on partial differential equations

TL;DR

This paper introduces a novel analytic relation for partial differential equations using a comparison method and tropical geometry, enabling asymptotic analysis and classification of PDEs based on automata theory.

Contribution

It presents a new comparison technique and a systematic way to relate different PDEs through tropical geometry and automata, offering insights into their asymptotic behavior.

Findings

Established rough asymptotic estimates for PDE solutions.

Constructed related and unrelated PDE pairs to validate the relations.

Performed numerical comparisons confirming the non-triviality of the relations.

Abstract

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.