On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields

TL;DR

This paper explores the connections between different classes of integrable nonlinear PDEs, focusing on a specific five-dimensional scalar PDE related to commuting vector fields and its derivation from hierarchies of $Ch$-integrable PDEs.

Contribution

It introduces a new five-dimensional scalar PDE linked to commuting vector fields and derives its matrix generalization from hierarchies of $Ch$-integrable PDEs, expanding understanding of integrable systems.

Findings

Identifies the origin of a five-dimensional scalar PDE from $Ch$-integrable hierarchies.

Derives a matrix generalization of the scalar PDE.

Establishes relations among different classes of integrable PDEs.

Abstract

It was shown recently that Frobenius reduction of the matrix fields reveals interesting relations among the nonlinear Partial Differential Equations (PDEs) integrable by the Inverse Spectral Transform Method ($S$-integrable PDEs), linearizable by the Hoph-Cole substitution ($C$-integrable PDEs) and integrable by the method of characteristics ($Ch$-integrable PDEs). However, only two classes of $S$-integrable PDEs have been involved: soliton equations like Korteweg-de Vries, Nonlinear Shr\"odinger, Kadomtsev-Petviashvili and Davey-Stewartson equations, and $GL(N,\CC)$ Self-dual type PDEs, like Yang-Mills equation. In this paper we consider the simple five-dimensional nonlinear PDE from another class of $S$-integrable PDEs, namely, scalar nonlinear PDE which is commutativity condition of the pair of vector fields. We show its origin from the (1+1)-dimensional hierarchy of $Ch$-integrable PDEs after certain composition of Frobenius type and differential reductions imposed on the matrix fields. Matrix generalization of the above scalar nonlinear PDE will be derived as well.