Ideals of partial differential equations

TL;DR

This paper introduces an algebraic framework using commutative algebra, algebraic geometry, and Gr"obner bases to analyze the compatibility and structure of partial differential equations, exemplified through the sinh-Gordon equation.

Contribution

It presents a novel algebraic approach to assess PDE compatibility, providing conditions for passivity and demonstrating the generation of manifolds in jet space.

Findings

Established algebraic criteria for PDE passivity

Connected passivity with manifold structures in jet space

Constructed passive systems related to the sinh-Gordon equation

Abstract

We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gr\"obner bases to clarify crucial notions concerning compatibility such as passivity (involution) and reducibility. One obtains sufficient conditions for a differential system to be passive and prove that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with the sinh-Gordon equation are given.