Invertible coupled KdV and coupled Harry Dym hierarchies

TL;DR

This paper explores the conditions for invertibility in coupled KdV and Harry Dym hierarchies, analyzing their nonlocal structures and Poisson properties to deepen understanding of their integrable systems.

Contribution

It identifies the conditions for invertibility and characterizes the nonlocal Poisson structures of coupled KdV and Harry Dym hierarchies.

Findings

Invertible coupled KdV hierarchies have weakly nonlocal Poisson structures.

Coupled Harry Dym hierarchies exhibit third-order nonlocal Poisson structures.

The structure of nonlocal tensor invariants is systematically analyzed.

Abstract

In this paper we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible cKdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.