On a Recently Introduced Fifth-Order Bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators

TL;DR

This paper clarifies that a recently proposed fifth-order bi-Hamiltonian equation is actually a higher symmetry of a known third-order equation, and it explores the relationships and trivial connections among related Hamiltonian operators.

Contribution

It demonstrates that the new fifth-order bi-Hamiltonian equation is not fundamentally new but a higher symmetry of a known third-order equation, and it analyzes the trivial relations among Hamiltonian operators.

Findings

The fifth-order equation is a higher symmetry of a third-order equation.

The new seventh-order Hamiltonian operator is trivially related to known operators.

Nonlocal generalizations of Hamiltonian operators are constructed.

Abstract

We show that a recently introduced fifth-order bi-Hamiltonian equation with a differentially constrained arbitrary function by A. de Sole, V.G. Kac and M. Wakimoto is not a new one but a higher symmetry of a third-order equation. We give an exhaustive list of cases of the arbitrary function in this equation, in each of which the associated equation is inequivalent to the equations in the remaining cases. The equations in each of the cases are linked to equations known in the literature by invertible transformations. It is shown that the new Hamiltonian operator of order seven, using which the introduced equation is obtained, is trivially related to a known pair of fifth-order and third-order compatible Hamiltonian operators. Using the so-called trivial compositions of lower-order Hamiltonian operators, we give nonlocal generalizations of some higher-order Hamiltonian operators.