Local and nonlocal solvable structures in ODEs reduction

TL;DR

This paper explores the use of solvable structures, including nonlocal ones, for integrating scalar ODEs, especially when local symmetries are scarce, and introduces the concept of nonlocal solvable structures.

Contribution

It introduces the notion of nonlocal solvable structures and discusses their adaptation to local and nonlocal symmetry algebras of any order.

Findings

Solvable structures can be used to integrate ODEs by quadratures.

Any ODE admits solvable structures under regularity assumptions.

Introduction of nonlocal solvable structures for broader applicability.

Abstract

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure.