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

Abstract

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.

Non-local Poisson structures and applications to the theory of integrable systems

“…are compatible Poisson structures. This was proved in[DSKW10] by direct verification, and deduced from Theorem 5.1 in[TT11].…”

“…Following the idea in [TT11], we will reduce the proof of Theorem 5.1 to the following special case of it: Proof. To simplify notation, in this proof we denote r H by H, and we let R " HK´1 so that R˚" K´1H.…”

“…Following the idea in [TT11], we will reduce the proof of Theorem 5.1 to the following special case of it: H by H, and we let R " HK ´1 so that R ˚" K ´1H . Let H r2s " HK ´1H ´" RH " HR ˚¯, and let t¨λ ¨u2 " t¨λ ¨uH r2s be the non-local λ-bracket on V associated to H r2s P Mat ℓˆℓ VpBq via (4.10).…”

Some new integrable systems of two-component fifth-order equations

“…In recent years studies on fifth-order systems of two-component nonlinear evolution equations have received considerable attention [2,10,8]. Multi-component generalizations of fifth order Kaup-Kupershmidt equation…”

“…By setting v = 0, system (6) reduces to the Sawada-Kotera equation. By setting v = 0 the well known Kupershmidt equation is an obvious reduction of system (8).…”