A new application of k-symplectic Lie systems

TL;DR

This paper explores a novel application of $k$-symplectic structures to analyze $k$-symplectic Lie systems, particularly in the context of diffusion equations, expanding their use beyond classical field theories.

Contribution

It introduces the concept of $k$-symplectic Lie systems and investigates their properties, including superposition rules, in the analysis of differential equations.

Findings

Identified superposition rules for $k$-symplectic Lie systems

Developed a new example related to diffusion equations

Extended the application of $k$-symplectic structures beyond field theories

Abstract

The $k$-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the $k$-symplectic structures to investigate a type of systems of first-order ordinary differential equations, the $k$-symplectic Lie systems. In particular, we analyse the properties, e.g. the superposition rules, of a new example of $k$-symplectic Lie system which occurs in the analysis of diffusion equations.