On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations

TL;DR

This paper explores nonlocal symmetries and conservation laws in evolution equations through potentialisation, introducing higher-degree potential symmetries and demonstrating their application on specific hierarchies.

Contribution

It introduces a systematic approach to identify nonlocal symmetries and conservation laws via potentialisation, including higher-degree symmetries, applied to well-known hierarchies.

Findings

Identification of nonlocal symmetries and conservation laws

Introduction of higher-degree potential symmetries

Application to Burgers' and Calogero-Degasperis-Ibragimov-Shabat hierarchies

Abstract

We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach by the Burgers' hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy.