Gregorio Falqui

On a Camassa-Holm type equation with two dependent variables

We consider a generalization of the Camassa–Holm (CH) equation with two dependent variables, called CH2, introduced in a paper by Liu and Zhang (Liu S-Q and Zhang Y 2005 J. Geom. Phys. 54 427–53). We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie algebra. The Lie algebra involved here is the same algebra as underlies the NLS hierarchy. We study the structural properties of the hierarchy defined by the CH2 equation within the bi-Hamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. Finally we sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.

Separation of variables for Bi-Hamiltonian systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called ωN manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the ωN manifold itself. We apply these results to bi-Hamiltonian systems of the Gelfand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.

Singularity, complexity, and quasi-integrability of rational mappings

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)-integrability, and in particular its links with their singularities (in the 2-plane). Finally, we describe some of their propertiesqua dynamical systems, making contact with Arnolds notion of complexity, and exemplify remarkable behaviours.

Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy

We discuss the Boussinesq system with the stationary time t5 within a general framework of stationary flows of n-Gelfand-Dickey hierarchies. A careful use of the bi-Hamiltonian structure can provide a set of separation coordinates for the corresponding Hamilton-Jacobi equations.

Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Abstract We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton–Jacobi method in the so-called Darboux–Nijenhuis coordinates.

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manins works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal:

A note on fractional KdV hierarchies

[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]

BRS cohomology and topological anomalies

(e.g.

Gaudin Models and Bending Flows: a Geometrical Point of View

[M_{11}, M_{22}]=[M_{21}, M_{12}]

On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds

). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. nOn the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation RTT=TTR and the so--called Cartier-Foata matrices. Also, they enter Talalaevs hep-th/0404153 remarkable formulas: