Quasi-Lie schemes for PDEs

TL;DR

This paper extends the theory of quasi-Lie systems from ordinary differential equations to partial differential equations, providing new methods for analyzing integrability and superposition rules with applications to models in physics and mathematics.

Contribution

It introduces a generalization of quasi-Lie schemes to PDEs, enabling the construction of quasi-Lie systems and analysis of their integrability and superposition properties.

Findings

Developed a procedure for constructing quasi-Lie systems for PDEs.

Applied the theory to models like Wess-Zumino-Novikov-Witten and Baecklund transformations.

Provided new insights into integrability conditions for complex differential equations.

Abstract

The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining $t$-dependent superposition rules and integrability conditions are analysed. We develop a procedure of constructing quasi-Lie systems through a generalisation to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess-Zumino-Novikov-Witten models, generalised Abel differential equations, Baecklund transformations, as well as other differential equations of physical and mathematical relevance.