Symmetry approach to integrability and non-associative algebraic structures

TL;DR

This paper explores the symmetry-based classification of integrable PDEs and their connection to algebraic structures, including associative and non-associative algebras, with applications to polynomial and vector systems.

Contribution

It introduces a novel approach linking integrability of PDEs to algebraic structures, expanding understanding of polynomial and vector integrable systems.

Findings

Classification of scalar integrable PDEs via symmetry methods

Association of polynomial integrable systems with algebraic structures

Geometric description of polynomial integrable systems

Abstract

The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand side are considered. Such systems are associated with associative algebras and, more generally, with non-associative algebraic structures, whose structural constants are defined by the coefficients of the system under consideration. We explore which structures correspond to polynomial integrable models. Integrable systems polynomial in derivatives are described in geometric terms. The third part we consider integrable vector izotropic and anizotropic systems.