Hamiltonian Flows on Euler-Type Equations

Abstract

We analyze properties of Hamiltonian symmetry flows on hyperbolic Euler-Liouville-type equations ε ′_EL. We obtain the description of their Noether symmetries assigned to the integrals of these equations. The integrals provide Miura transformations from ε′_EL to the multicomponent wave equations ε. Using these substitutions, we generate an infinite-Hamiltonian commutative subalgebra $\mathfrak{A}$ of local Noether symmetry flows on ε proliferated by weakly nonlocal recursion operators. We demonstrate that the correspondence between the Magri schemes for $\mathfrak{A}$ and for the induced “modified” Hamiltonian flows $\mathfrak{B}$ ⊂ sym ε ′_EL is such that these properties are transferred to $\mathfrak{B}$ and the recursions for ε′ _EL are factored. We consider two examples associated with the two-dimensional Toda lattice.