Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e coefficients $H_i$ involved in the construction are still homogeneous functions of a certain degree $d_i$ but we consider the general case $d_i\ne d_j$. As a consequence the rotation coefficients $\beta_{ij}$ are homogeneous functions of degree $d_i-d_j-1$. It turns out that any semisimple bi-flat $F$ manifold satisfying a natural additional assumption can be obtained in this way. Finally we show that three dimensional semisimple bi-flat $F$-manifolds are parametrized by solutions of the full family of Painlev\'e VI.

Abstract:
Este artigo ressalta que McLuhan, em fragmentos de sua produ o acadêmica, faz uma crítica à escola formata o, na qual o lúdico foi substituído por métodos educacionais pragmáticos-tecnicistas que visam, antes de formar uma pessoa original, formatar um sujeito competitivo para o mercado de trabalho; um indivíduo que compete pela sua rapidez produtiva e n o pela sua atividade criativa, transformandose assim numa pe a de fácil substitui o na engrenagem de produ o. Destaca que, na vis o do autor, a escola deve desenvolver e afinar os sentidos e as percep es do indivíduo, de forma que a educa o seja uma atividade contínua, na perspectiva de que o homem compreenda que menos de ganhar a vida o mais importante é renovar a vida. Aponta que a crítica do autor à estandardiza o do ensino é, também, uma crítica aos modelos educacionais impostos pelas formas econ micas e políticas dominantes na aldeia global, sugerindo que os cidad os do futuro ser o recompensados pela sua diversidade e n o mais pela forma o e pontos de vista semelhantes.

Abstract:
Given a semi-Hamiltonian system, we construct an $F$-manifold with a connection satisfying a suitable compatibility condition with the product. We exemplify this procedure in the case of the so-called $\epsilon$-system. The corresponding connection turns out to be flat, and the flat coordinates give rise to additional chains of commuting flows

Abstract:
We discuss, in the framework of Dubrovin-Zhang's perturbative approach to integrable evolutionary PDEs in 1+1 dimensions, the role of a special class of Poisson pencils, called exact Poisson pencils. In particular we show that, in the semisimple case, exactness of the pencil is equivalent to the constancy of the so-called "central invariants" of the theory that were introduced by Dubrovin, Liu and Zhang.

Abstract:
Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of $F$-manifolds due to Manin [22], we consider a special class of $F$-manifolds, called bi-flat $F$-manifolds. A bi-flat $F$-manifold is given by the following data $(M, \nabla_1,\nabla_2,\circ,*,e,E)$, where $(M, \circ)$ is an $F$-manifold, $e$ is the identity of the product $\circ$, $\nabla_1$ is a flat connection compatible with $\circ$ and satisfying $\nabla_1 e=0$, while $E$ is an eventual identity giving rise to the dual product *, and $\nabla_2$ is a flat connection compatible with * and satisfying $\nabla_2 E=0$. Moreover, the two connections $\nabla_1$ and $\nabla_2$ are required to be hydrodynamically almost equivalent in the sense specified in [2]. First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat $F$-manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric. Although any Frobenius manifold possesses automatically the structure of a bi-flat $F$-manifold, we show that the latter is a strictly larger class. In particular we study in some detail bi-flat $F$-manifolds in dimensions n=2, 3. For instance, we show that in dimension 3 bi-flat $F$-manifolds are parametrized by solutions of a two parameters Painlev\'e VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.

Abstract:
In this paper we study some properties of bi-Hamiltonian deformations of Poisson pencils of hydrodynamic type. More specifically, we are interested in determining those structures of the fully deformed pencils that are inherited through the interaction between structural properties of the dispersionless pencils (in particular exactness or homogeneity) and suitable finiteness conditions on the central invariants (like polynomiality). This approach enables us to gain some information about each term of the deformation to all orders in $\epsilon$. Concretely, we show that deformations of exact Poisson pencils of hydrodynamic type with polynomial central invariants can be put, via a Miura transformation, in a special form, that provides us with a tool to map a fully deformed Poisson pencil with polynomial central invariants of a given degree to a fully deformed Poisson pencil with constant central invariants to all orders in $\epsilon$. In particular, this construction is applied to the so called $r$-KdV-CH hierarchy that encompasses all known examples with non-constant central invariants. As far as homogeneous Poisson pencils of hydrodynamic type is concerned, we prove that they can also be put in a special form, if the central invariants are homogeneous polynomials. Through this we can compute the homogeneity degree about the tensorial component appearing in each order in $\epsilon$, namely the coefficient of the highest order derivative of the $\delta$.