On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs

TL;DR

This paper demonstrates that quasi-Miura transformations can simplify Hamiltonian structures in certain evolutionary PDEs, providing a criterion and algorithm for identifying Hamiltonian structures in these equations.

Contribution

It shows that under specific conditions, quasi-Miura transformations preserve Hamiltonian structures, offering a new method to identify and analyze Hamiltonian properties in PDEs.

Findings

Quasi-Miura transformations reduce Hamiltonian structures to their leading terms.

A criterion for the existence of Hamiltonian structures in scalar evolutionary PDEs is established.

An algorithm for finding Hamiltonian structures in these PDEs is developed.

Abstract

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.