On Hamiltonian Flows on Euler-type Equations

Properties of Hamiltonian symmetry flows on hyperbolic Euler-type equations are analyzed. Their Lagrangian densities are demonstrated to supply the Hamiltonian operators for subalgebras of their Noether symmetries, while substitutions between Euler-type equations define Miura transformations between the symmetry flows; some Miura maps for Liouvillean Euler-type systems are supplied by their integrals. Two examples are considered: the Korteweg–de Vries hierarchy of symmetries of wave equation is connected with multi-component analogues of the modified KdV flows on the 2D Toda lattice (2DTL), and the Boussinesq hierarchy for two-component wave equation is related with the chain of higher modified Boussinesq flows on the 2-component 2DTL. Introduction. In this paper, we consider the problem of constructing pairs of commutative hierarchies of Hamiltonian evolution equations related by Miura-type transformations and identified with Lie subalgebras of the Noether symmetry algebras for Euler–Lagrange-type systems. Two examples are obtained: 1) the Korteweg–de Vries equation (1) st1 = −β sxxx + 3 2 sx, wt1 = −β wxxx + 3wwx, w = sx, β = const, and multi-component modified KdV equations (see [9]) are related with the wave equation and the 2DTL (in particular, associated with Date: September 28, 2004. 2000 Mathematics Subject Classification. 37K10, 37K05.