Laplace maps and constraints for a class of third order partial differential operators

TL;DR

This paper investigates generalized Laplace maps for a specific class of third order partial differential operators, identifying necessary constraints on invariants for their existence and related systems.

Contribution

It introduces conditions on invariants that enable the construction of Laplace maps for third order PDEs of a particular form.

Findings

Derived constraints on invariants for Laplace map existence

Established links between third order PDEs and first order systems

Provided a framework for analyzing third order operators

Abstract

We explore the existence of a class of generalised Laplace maps for third order partial differential operators of the form \[\partial_1\partial_2\partial_3+a_1\partial_2\partial_3+a_2\partial_1\partial_3+a_3\partial_1\partial_2+a_{12}\partial_3+a_{23}\partial_1+a_{13}\partial_2+a_{123}\] and related first order $3\time 3$ systems and show that they require the satisfaction of constraints on the invariants for such operators.