Sub-symmetries I. Main properties and applications

TL;DR

This paper introduces the concept of sub-symmetries in differential systems, explores their properties, and demonstrates their usefulness in solving systems, decoupling equations, and generating conservation laws, surpassing traditional symmetries.

Contribution

It defines sub-symmetries, provides an algorithm for finding them, and shows their advantages in solving and analyzing differential systems, including conservation law deformation.

Findings

Sub-symmetries can decouple differential systems effectively.

All lower conservation laws of the nonlinear telegraph system are generated by sub-symmetries.

Sub-symmetries are more powerful than regular symmetries in transforming conservation laws.

Abstract

We introduce a sub-symmetry of a differential system as an infinitesimal transformation of a subset of the system that leaves the subset invariant on the solution set of the entire system. We discuss the geometrical meaning and properties of sub-symmetries, as well as an algorithm for finding sub-symmetries of a system. We show some of the benefits of using sub-symmetries in the search for solutions of a system; in particular , we show how sub-symmetries can be used in decoupling a differential system. We also discuss the role of sub-symmetries in the deformation of known conservation laws of a system into other (often, new) conservation laws and show that, in this regard, a sub-symmetry is a considerably more powerful tool than a regular symmetry. We demonstrate that all lower conservation laws of the nonlinear telegraph system can be generated by sub-symmetries.