The Hamiltonian structures associated with a generalized Lax operator

Abstract

Articles you may be interested inDegenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equationsThe Hamiltonian structures of the superKP hierarchy associated with an even parity superLax operator It is shown that with every Lax operator, which is a pseudodifferential operator of nonzero leading order, is associated a KP hierarchy. For each such operator, we construct the second Gelfand-Dikii bracket associated with the Lax equation and show that it defines a Hamiltonian structure. When the leading order is positive the corresponding compatible first Hamiltonian structure, which turns out, in general, to be different from the naive first Gelfand-Dikii bracket is derived. The corresponding Hamiltonian structures for the constrained Lax operator, where the next to leading-order term vanishes or has a constant coefficient, is discussed.

Gelfand-Dikii Brackets for Nonstandard Lax Equations

“…(2)), it can be easily checked that the presence or absence of the term ∂ −n q −1 in the dual makes no difference in the definitions of F Q n (L n ) and the first Hamiltonian structure. However, the structure of the second Hamiltonian structure requires that q −1 be constrained for the Lax equation to be Hamiltonian with respect to the second Hamiltonian structure [4]. Alternately, one can modify the second Hamiltonian structure in such a case…”

“…Most integrable systems [1,2] can be understood in terms of pseudo-differential operators as follows [3,4]. Consider, for simplicity, the formal pseudo-differential operator of the form…”

“…The conserved quantities (Hamiltonians) for the system can be seen to correspond to (for more details see ref. 4)…”

Dispersionless fermionic KdV

“…The Lax description, for the bosonic KdV equation, of course, gives the Hamiltonian structures (at least the first one) naturally through a generalization of the Gelfand-Dikii bracket [10]. A similar analysis fails in this case, in spite of a careful treatment (Dirac analysis) of the constrained nature of the Lax function [21][22]. The derivation of the Hamiltonian structure for the Kupershmidt equation from the Lax description, therefore, remains an open question.…”

Properties of Moyal–Lax representation

“…where k, l are integers, prove that different flows commute and can define Hamiltonian structures in a straightforward manner [11]. Let us illustrate these ideas through some examples.…”