On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs

Details On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs

2007-11-16

arxiv.org/abs/0711.2599v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages

Scientific paper

10.1007/s11005-008-0234-y

In \cite{LZ2} it is proved that for certain class of perturbations of the hyperbolic equation $u_t=f(u) u_x$, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations possess Hamiltonian structures of certain type, the same quasi-Miura transformations also reduce the Hamiltonian structures to their leading terms. By applying this result, we obtain a criterion of the existence of Hamiltonian structures for a class of scalar evolutionary PDEs and an algorithm to find out the Hamiltonian structures.

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