Carlo Morosi

Quasi-bi-Hamiltonian systems and separability

Two quasi-bi-Hamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover, the most general Pfaffian quasi-bi-Hamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

Reduction techniques for infinite dimensional Hamiltonian systems: some ideas and applications

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations.

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.

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.

ON A CLASS OF DYNAMICAL SYSTEMS BOTH QUASI-BI-HAMILTONIAN AND BI-HAMILTONIAN

Abstract It is shown that a class of dynamical systems (encompassing the one recently considered by Calogero [J. Math. Phys. 37 (1996) 1735] is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained through a non-canonical map whose form is directly suggested by the associated Nijenhuis tensor.

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...

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.

On the Constants for Some Sobolev Imbeddings

We consider the imbedding inequalit), I1.(). sr,,,dll IIn,fnu; H (R) is the Sobolev space (or Bessel potential space) ofL type and (integer or fa6tional) order n. We write down upper bounds for the constants Sr,n,a, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r= 2 or) n > d/2, r= oe, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on Sr,n,d for n > d/2, 2 < r < oo; in many cases these are close to the previous upper bounds, as illustrated by a number ofexamples, thus characterizing the sharp constants with little uncertainty.