Abstracts
S. Abenda: Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems
We start from a hyperbolic Dubrovin-Novikov (DN) hydrodynamic type system of dimension n which possesses Riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that, after such a transformation of the independent variables, one of the metrics associated to the initial system be flat. We prove the following statement: let n> 2 in the case of reciprocal transformations of a single independent variable or n> 4 in the case of transformations of both the independent variable; then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws, i.e. the Casimirs, the momentum and the Hamiltonian densities associated to the Hamiltonian operator for the initial metric. Then, we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat. Such characterization has an interesting geometric interpretation in view of previous results by E.V. Ferapontov: the hypersurfaces of two diagonalizable DN systems of dimension n>4 are Lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation.
L. Amico: Some applications of the Gaudin models to condensed matter physics
I will discuss some quantum models ultimately tracing back to the quasiclassical limits of vertex models. I will demonstrate how the rational and trigonometric Gaudin models with periodic or open boundary conditions allow to study pairing force Hamiltonians (the BCS superconductors are an example) exactly. Also spin-boson systems, useful in various fields like atomic and mesoscopic physics and quantum computations, can be seen as related to Gaudin-like Hamiltonian. Such a relation allow to explore important physical regimes non perturbatively.
The first example of a maximally superintegrable system on a N-dimensional space of non-constant curvature is presented. This system can be interpreted as the intrinsic oscillator on such ND space, and the full set of integrals of the motion can be explicitly obtained. Moreover, the intrinsic oscillator on any ND spherically symmetric space can be constructed by making use of the same underlying $sl(2)$-coalgebra symmetry, although for a generic curved space only quasi-maximal superintegrability can be guaranteed. If we relax the spherical symmetry condition by allowing the $h(6)$-coalgebra invariance of the system, many new ND integrable systems can be constructed. In particular, we present a family of integrable perturbations of the ND euclidean oscillator that depend on two arbitrary functions and N free parameters.
F. Calogero: Isochronous dynamical systems
A vector-valued time-dependent function is called isochronous if all its components are periodic in time with the same fixed period T. A dynamical system is called isochronous if its generic solution is isochronous: periodic in all its degrees of freedom with a fixed period T independent of the initial data. It will be shown how essentially any (autonomous) dynamical system can be modified into another (autonomous) dynamical systems which is isochronous with an (arbitrarily !) assigned period T, and which moreover behaves, over time periods very short with respect to T, essentially as the original (unmodified) system---up to a constant time rescaling. This can also be done for a large class of Hamiltonian systems (both the unmodified and the modified one), including the Hamiltonian describing the most general many-body problem (provided it is, overall, translation-invariant). Some implications of this fact for statistical mechanics and thermodynamics will be mentioned. These findings have all been obtained together with F. Leyvraz: some of them are reported in my monograph entitled Isochronous systems (Oxford University Press, February 2008), others are more recent.
G. Carlet: From 2D Toda hierarchy to an infinite-dimensional Frobenius manifold
We define a structure of an infinite-dimensional Frobenius manifold on a space of pairs of holomorphic functions on the inner/outer parts of the unit disk in the complex plane and show that it is naturally associated to the 2D Toda hierarchy. Based on a joint work with B. Dubrovin and L.Ph. Mertens.
A. Doliwa: On tau-function of the quadrilateral lattice
The tau-function of the (commutative) quadrilateral lattice in the class of solutions obtained by the nonlocal $\bar\partial$-dressing method can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.
A. Enciso: Bertrand's theorem revisited and superintegrable Hamiltonian systems
Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this talk we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand's theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems admitting a suitable, globally defined generalization of the Runge--Lenz vector.
G. Falqui: Limits of Gaudin systems: classical and quantum cases
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new ``Gaudin'' algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ``Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case. The talk is based on recent joint works with A. Chervov and L. Rybnykov, and an outhgrowth of previous works with F. Musso.
A. Hone: Hirota-Kimura-Kahan type discretizations of integrable flows in three dimensions
We describe a discretization method for quadratic vector fields which was introduced in the context of integrable systems by Hirota and Kimura, who presented a new integrable map that discretizes the Euler top. This approach to discretization appeared earlier in numerical analysis, in the work of Kahan on numerical integration, and similar schemes were studied by Mickens. Here we consider applying this discretization method to bi-Hamiltonian vector fields in three dimensions, associated to pairs of Lie-Poisson algebras. In the majority of cases, new examples of discrete systems that are algebraically integrable arise. However, in other cases we assert that the resulting maps are non-integrable. This is joint work with Matteo Petrera and Kim Towler.
I. Marshall: Poisson properties of the Schroedinger equation
The Miura map was a crucial piece of the understanding of conservation laws for the KdV equation. It came to be appreciated that the KdV and the mKdV (related to the KdV by the Miura map) could be seen to be completely integrable Hamiltonian systems in the sense of Classical Mechanics. Subsequently this Hamiltonian aspect was developed further and it was found that the KdV equation is just a special case within a large class classified by classical root systems. This analysis uses the tool of reduction, which generalises the notion of conservation of angular momentum in Classical Mechanics to symmetry with respect to the action of any Lie group. This construction is nowadays known as ``Drinfeld-Sokolov reduction''. The result to be explained in my talk is a special case (the simplest one) of an alternative reduction procedure which gives rise to a similar, complementary picture. Factorisation $L=\partial^2+u=(\partial-v)(\partial+v)$ of the Schr\"odinger operator gives rise to the Miura map $v\mapsto u$ which is a Poisson map with respect to the ``Virasoro'' and ``Gardner'' Poisson brackets. It was shown by Wilson that the Miura map can be interpreted via the Galois properties of the differential field extensions $C\langle u\rangle\subset C\langle v\rangle\subset C\langle\psi_1, \psi_2\rangle$, where $\psi_1$ and $\psi_2$ are independent solutions to $L\psi=0$. By identifying the Galois group as a Poisson Lie group, Wilson's result can be shown to arise from Poisson reduction. The same reduction can be applied to the discrete analogue of the Schroedinger operator, giving rise to the Faddeev-Takhtajan Poisson structure on the space of potentials and to a result of Volkov which gives the discrete analogue of the Miura map. This result is quite a simple application of Poisson Lie group reduction and it is expected that it can be understood by nonspecialists.
D. Masoero: Poles of special solutions of the Painlevé first equation
We will show that the poles of an arbitrary solution to the Painlevé first equation are in bijection with a particular class of ramified coverings of the sphere and we will show how to calculate the locations of those poles, through the complex WKB method, in some particular cases.
L. Mertens: 2D Toda Frobenius Manifold
We exhibit the structure of the Frobenius manifold associated to 2D Toda. We will present the construction of the flat coordinates, the potential of the Frobenius Manifold, the three point correlator function and the primary flows of the Principal Hierarchy. The talk is based on our recent paper with G. Carlet and B. Dubrovin.
M. Pedroni: On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system
We show that the bi-Hamiltonian structure of the rational n-particle Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R).
A. Raimondo: Hamiltonian structures for reductions of the Benney chain
It is well known that the Benney moment chain possesses infinitely many semi-Hamiltonian (integrable) reductions, which can be parametrized by families of conformal maps on the complex upper half plane. By following the Ferapontov procedure for semi-Hamiltonian sytems, we show how to construct the Hamiltonian structures of any reduction. All the structures are explicitly written in terms of the associated family of conformal maps; the brackets are usually nonlocal of Ferapontov type. Moreover, we associate to any Benney reduction a class of purely nonlocal Poisson brackets. These -remarkably- turn out to be related with quadratic expansions of the diagonal metrics naturally associated with the system. The latter is a generic fact, which holds for any semi-Hamiltonian system.
We have recently solved the inverse scattering problem and constructed the associated nonlinear Riemann-Hilbert dressing for one-parameter families of vector fields. This theory has been used to study integrable and applicative PDEs arising as commutation of vector fields, like the heavenly, the dispersionless Kadomtsev - Petviashvili and the 2D dispersionless Toda equations. In particular, the following issues have been successfully investigated: i) the Cauchy problem; ii) the longtime behaviour of solutions; iii) distinguished classes of implicit solutions; iv) whether localised initial data break at finite time or not, and, if they do, the geometric and analytic details of such a two-dimensional wave breaking.
C. Scimiterna: On the integrability of the discrete nonlinear Schroedinger equation
In this talk we present an analytic evidence of the nonintegrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a reductive perturbation technique to show an obstruction to its integrability.
P.G. Tempesta: Discrete integrability and zeta functions
The finite operator theory provides a simple and natural language for analyzing integrable systems on a lattice. We show that this approach is closely connected with the theory of formal groups from one side, and with a construction of Riemann-type zeta functions from the other side. Also, a symmetry preserving discretization of continuous models on associative algebras is proposed.