Livio Pizzocchero

On the euler equation: Bi-Hamiltonian structure and integrals in involution

We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the ‘physical’ phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.

Mesoscale averaging of nucleation and growth models

The aim of this paper is to derive a general theory for the averaging of heterogeneous processes with stochastic nucleation and deterministic growth. We start by generalizing the classical Johnson--Mehl--Avrami--Kolmogorov theory based on the causal cone to heterogeneous growth situations. Moreover, we relate the computation of the causal cone to a Hopf--Lax formula for Hamilton--Jacobi equations describing the growth of grains. As an outcome of the approach we obtain formulae for the expected values of geometric densities describing the growth processes; in particular we generalize the standard Avrami--Kolmogorov relations for the degree of crystallinity. By relating the computation of expected values to mesoscale averaging, we obtain a suitable description of the process at the mesoscale. We show how the variance of these mesoscale averages can be estimated in terms of quotients of the typical length on the microscale and on the mesoscale. Moreover, we discuss the efficient computation of the mesoscale ...

On the continuous limit of integrable lattices. I. The Kac-Moerbeke system and KdV theory

KdV theory is constructed systematically through the continuous limit of the Kac-Moerbeke system. The infinitely many commuting vector fields, the conserved functionals, the Lax pairs and the biHamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the Kac-Moerbeke system. The combinatorial aspects of this recombination method are treated in detail.

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.

A FUNCTIONAL REPRESENTATION FOR NON-COMMUTATIVE C*-ALGEBRAS

We obtain a representation of non commutative C*-algebras as function algebras on the pure state space, with a convenient product. This construction is an extension of the classical functional representation of commutative C*-algebras.

R‐matrix theory, formal Casimirs and the periodic Toda lattice

The nonunitary r‐matrix theory and the associated bi‐ and triHamiltonian schemes are considered. The language of Poisson pencils and of their formal Casimirs is applied in this framework to characterize the biHamiltonian chains of integrals of motion, pointing out the role of the Schur polynomials in these constructions. This formalism is subsequently applied to the periodic Toda lattice. Some different algebraic settings and Lax formulations proposed in the literature for this system are analyzed in detail, and their full equivalence is exploited. In particular, the equivalence between the loop algebra approach and the method of differential‐difference operators is illustrated; moreover, two alternative Lax formulations are considered, and appropriate reduction algorithms are found in both cases, allowing us to derive the multiHamiltonian formalism from r‐matrix theory. The systems of integrals for the periodic Toda lattice known after Flaschka and Henon, and their functional relations, are recovered thr...

On approximate solutions of semilinear evolution equations II : generalizations, and applications to Navier-Stokes equations

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Td of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential dec...

ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II. VOLTERRA SYSTEMS AND SP(N) THEORIES

A connection is suggested between the zero-spacing limit of a generalized N-fields Volterra (VN) lattice and the KdV-type theory which is associated, in the Drinfeld–Sokolov classification, to the simple Lie algebra sp(N). As a preliminary step, the results of the previous paper [1] are suitably reformulated and identified as the realization for N=1 of the general scheme proposed here. Subsequently, the case N=2 is analyzed in full detail; the infinitely many commuting vector fields of the V2 system (with their Hamiltonian structure and Lax formulation) are shown to give in the continuous limit the homologous sp(2) KdV objects, through conveniently specified operations of field rescaling and recombination. Finally, the case of arbitrary N is attacked, showing how to obtain the sp(N) Lax operator from the continuous limit of the VN system.

A fully supersymmetric AKNS theory

We construct a fully supersymmetric biHamiltonian theory in four superfields, admitting zero curvature and Lax formulation. This theory is an extension of the classical AKNS, which can be recovered as a reduction. Other supersymmetric theories are obtained as reductions of the susy AKNS, namely a nonlinear Schrödinger, a modified KdV and the Manin-Radul KdV. The susy nonlinear Schrödinger hierarchy is related to the one of Roelofs and Kersten; we determine its biHamiltonian and Lax formulation. Finally, we show that the susy KdVs mentioned before are related through a susy Miura map.

QUANTUM PHASE SPACE FORMULATION OF SCHRÖDINGER MECHANICS

We develop a geometrical approach to Schrodinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kahler structure possessed by the set of the pure states of a quantum system. The Kahler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kahler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrodinger quantum mechanics.