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Register a new userTri-Hamiltonian Structure of the Ablowitz-Ladik Hierarchy
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We construct a local tri-Hamiltonian structure of the Ablowitz-Ladik hierarchy, and compute the central invariants of the associated bihamiltonian structures. We show that the central invariants of one of the bihamiltonian structures are equal to 1/24, and the dispersionless limit of this bihamiltonian structure coincides with the one that is defined on the jet space of the Frobenius manifold associated with the Gromov-Witten invariants of local CP1. This result provides support for the validity of Brinis conjecture on the relation of these Gromov-Witten invariants with the Ablowitz-Ladik hierarchy.
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We construct the Baxters operator and the corresponding Baxters equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Backlund transformations. For comparison we find the same result by using the well-known Bethe ansatz technique. General results about integrable models governed by the same r-matrix algebra will be given. The Baxters equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Backlund transformations gives a trace formula representing the classical analogue of the Baxters equation. An explicit q-integral representation of the Baxters operator is discussed.
Complete integrability and multisoliton solutions are discussed for a multicomponent Ablowitz-Ladik system with branched dispersion relation. It is also shown that starting from a diagonal (in two-dimensions) completely integrable Ablowitz-Ladik equation, one can obtain all the results using a periodic reduction.
The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field $tau$ such that $A_2=L_tau A_1$. We use this result in order to find the Lagrangian representation when $A_2$ is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in $3$ components.
Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential-geometric constraints. Complete classification results in the 2-component and 3-component cases are obtained.
We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture.
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