HAMILTONIAN STRUCTURES FOR THE GENERALIZED DISPERSIONLESS KdV HIERARCHY

Abstract

We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and third Hamiltonian structures are calculated directly from the r-matrix approach. Since the third structure is not related recursively with the first two ones the generalized dispersionless KdV hierarchy can be characterized as a truly tri-Hamiltonian system.

Properties of Moyal–Lax representation

“…While the Lax representations for various dispersionless integrable models are known [15]- [17], the determination of the Hamiltonian structures (at least the second) from such a Lax function has remained an open question. The Moyal-Lax representation provides a solution to this problem in a natural way.…”

“…First, let us consider the KdV hierarchy, which in the dispersionless limit, goes over to the Riemann hierarchy [15]. With…”

“…These are indeed the correct Hamiltonian structures of the Riemann equation and while the first structure was already constructed from the Lax function, the construction of the second structure, from the Lax description, was not known so far [15].…”

Dispersionless fermionic KdV

“…Then, a generalization of the Gelfand-Dikii definition of the first Hamiltonian structure [10] {sTr LQ, sTr LV } = sTr L {Q, V }…”

On the Integrability of a Class of Monge–ampère Equations

“…In Sec. 3, using results from [8,11,12], we obtain the dispersionless Lax representation of (1) (Proposition 3.1) and write the two sets of local conserved charges densities for the Monge-Ampère equation. In Sec.…”