Partial Euler operators and the efficient inversion of Div

TL;DR

This paper introduces partial Euler operators and scalings to efficiently invert divergence operators, facilitating the computation of conservation law components in complex PDE systems.

Contribution

It presents a novel, efficient method for inverting divergence operators using partial Euler operators and scalings, improving computational approaches for conservation laws.

Findings

Developed a line integral formula for inversion of total derivatives.

Provided a concise procedure for inverting total divergences.

Enhanced computational efficiency in finding conservation law components.

Abstract

The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.