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.
Abstract. Two-component second and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New symmetry integrable systems with their master symmetries are obtained. Some third order systems are observed to possess conservation laws. Bi-Poisson structures of systems possessing conservation laws are given.
Keywords:Integrable system Symmetry Symplectic operator Hamiltonian operator Recursion operator Dedicated to professor Atalay Karasu on his 60th birthday.
a b s t r a c tThe symplectic-Hamiltonian formulation and recursion operator of the fifth-order Mikhailov-Novikov-Wang system are given.Ó 2014 Elsevier Inc. All rights reserved.The Hamiltonian and symplectic structures are well known to be of paramount importance, as the whole conserved quantity hierarchy and symmetry hierarchy for the system in question are then readily generated by the repeated application of these operators to the seed symmetry. Complete integrability of an equation is established by proving a compatible bi-Hamiltonian, (or bi-symplectic or Hamiltonian-symplectic operators) defining the Magri scheme of infinite symmetry hierarchy u t i ¼ F i ½u all of which is in conservation law formwith respect to two compatible Hamiltonian (skew-adjoint, Jacobi identity satisfying) operators J and K by the conserved density q i ½u which are in involution. Recently, Mikhailov et al. [3] applied the symmetry analysis to a class of fifth-order nonlinear equations, and found two new cases that pass the symmetry test:
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