What are symmetries of nonlinear PDEs and what are they themselves?

TL;DR

This paper provides an informal historical overview of the development of the theory of symmetries in nonlinear PDEs, highlighting its foundations, techniques, and applications to attract new researchers to this promising field.

Contribution

It offers an accessible introduction to the modern theory of symmetries of nonlinear PDEs, emphasizing its historical development and potential for future research.

Findings

The theory of symmetries in nonlinear PDEs has solid foundations and wide applications.

Recent developments have expanded the understanding of symmetries in mathematical physics.

The paper aims to attract new researchers to this promising area.

Abstract

The general theory of (nonlinear) partial differential equations originated by S. Lie had a significant development in the past 30-40 years. Now this theory has solid foundations, a proper language, proper techniques and problems, and a wide area of applications to physics, mechanics, to say nothing about traditional mathematics. However, the results of this development are not yet sufficiently known to a wide public. An informal introduction in a historical perspective to this subject presented in this paper aims to give to the reader an idea about this new area of mathematics and, possibly, to attract new researchers to this, in our opinion, very promising area of modern mathematics.