Dispersive Poisson brackets (and pairs of compatible Poisson brackets, i.e., bi-Hamiltonian structures) on formal loop spaces play a prominent role in the description of integrable hierarchies, especially in the setting of hierarchies of topological type. We will review the general framework and motivation for the study of such objects, including main results like Getzler’s theorem and the notion of central invariants by Liu, Dubrovin and Zhang.
We will then discuss our recent results, which include the proof, using spectral sequences techniques, of the triviality of the bi-Hamiltonian cohomology of semisimple Poisson brackets of DN type. That in turn implies the existence of arbitrary order dispersive deformations, starting from any choice of central invariants, thus solving a conjecture of Dubrovin et al.
Open problems and some generalisations to the multivariable setting will be also discussed. Based on joint works with H. Posthuma, S. Shadrin and with M. Casati, S. Shadrin.
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion'' on the defect point via the space-like and time-like description. We exploit the structural similarity of these equations with the discrete and continuous Backlund transformations. The equations are similar, but not exactly the same to the Backlund transformations. We consider specific examples of integrable models to demonstrate our construction, i.e. the Toda chain and the sine-Gordon model.
The connection between cluster algebras and Poisson structures is by now well-documented. Among the most important examples in which this connection has been utilized are coordinate rings of double Bruhat cells in semisimple Lie groups equipped with (the restriction of) the standard Poisson-Lie structure.
In this talk, based on the joint work with M. Shapiro and A. Vainshtein, I will describe a construction of ageneralized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group GLn and derive from it a generalized cluster structure in GLn compatible with the push-forward of the dual Poisson-Lie bracket.
General reductions of the discrete Hirota equation (aka Hirota-Miwa/discrete KP/octahedron recurrence) to autonomous ordinary difference equations result in certain bilinear recurrence relations of Somos type; these are also referred to as three-term Gale-Robinson recurrences. Each of these bilinear recurrences has the Laurent property and admits a cluster structure associated with a quiver that is periodic under cluster mutations, in the sense of Fordy-Marsh, with a corresponding invariant presymplectic form. Current progress in classifying these reductions is described, in terms of their Lax pairs and underlying symplectic maps that are integrable in the Liouville-Arnold sense.
In this talk we present and study Poisson structures compatible with a class of rational Yang-Baxter maps and integrable lattice equations. By considering periodic initial value problems on the lattice, we derive families of transfer maps which preserve the spectrum of the corresponding monodromy matrix. In particular cases we prove the Liouville integrability of the transfer maps with respect to the introduced Poisson structures.
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Last updated 14th April, 2016