arXiv subjects
Publications and source records attributed to Paolo Lorenzoni.
We investigate the role of Hertling-Manin condition on the structure constants of an associative commutative algebra in the theory of integrable systems of hydrodynamic type. In such a framework we introduce the notion of F-manifold with compatible connection generalizing a structure introduced by Manin.
We study purely nonlocal Hamiltonian structures for systems of hydrodynamic type. In the case of a semi-Hamiltonian system, we show that such structures are related to quadratic expansions of the diagonal metrics naturally associated with the system.
We study the Harry Dym hierarchy of nonlinear evolution equations from the bi-Hamiltonian view point. This is done by using the concept of an S-hierarchy. It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian flows in two fields that projects onto the scalar Harry Dym hierarchy. We also show that the conserved densities of the matrix Harry Dym equation can be found by means of a Riccati-type equation.
We show how to construct the Hamiltonian structures of any reduction of the Benney chain (dKP) starting from the family of conformal maps associated to it.
In this paper we consider a class of semihamiltonian systems characterized by the existence of a special conservation law. The density and the current of this conservation law satisfy a second order system of PDEs which has a natural interpretation in the theory of flat bifferential ideals. The class of systems we consider contains important well-known examples of semihamiltonian systems. Other examples, like genus 1 Whitham modulation equations for KdV, are related to this class by a reciprocal trasformation.
We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a 3-component extension of the CH equation.
We show the relevance of the dispersive analogue of the shock waves in the FPU dynamics. In particular we give strict numerical evidences that metastable states emerging from low frequency initial excitations, are indeed constituted by dispersive shock waves travelling through the chain. Relevant characteristics of the metastable states, such as their frequency extension and their time scale of formation, are correctly obtained within this framework, using the underlying continuum model, the KdV equation.
We explain how to use the theory of bidifferential ideals to construct integrable hierarchies of hydrodynamic type.
We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.
In this paper we study the sine-Gordon and the Liouville hierarchies in laboratory coordinates from a bi-Hamiltonian point of view. Besides the well-known local structure these hierarchies possess a second compatible non-local Poisson structure.
In this paper we study the deformations of bihamiltonian PDEs of hydrodynamic type with one dependent variable. The reason we study such deformations is that the deformed systems maintain an infinite number of commuting integrals of motion up to a certain order in the deformation parameter. This fact suggests that these systems could have, at least for small times, multi-solitons solutions. Our numerical experiments confirm this hypothesis.