arXiv subjects

Alessandro Arsie

Publications and source records attributed to Alessandro Arsie.

32 records · Page 2Linked to original sources

arXiv2014

Integrable viscous conservation laws

We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven. We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe.

math-ph↗

arXiv2014

On integrable conservation laws

We study normal forms of scalar integrable dispersive (non necessarily Hamiltonian) conservation laws via the Dubrovin-Zhang perturbative scheme. Our computations support the conjecture that such normal forms are parametrised by infinitely many arbitrary functions that can be identified with the coefficients of the quasilinear part of the equation. More in general, we conjecture that two scalar integrable evolutionary PDEs having the same quasilinear part are Miura equivalent. This conjecture is also consistent with the tensorial behaviour of these coefficients under general Miura transformations.

math-ph↗

arXiv2012

Reciprocal $F$-manifolds

We consider the action of a special class of reciprocal transformation on the principal hierarchy associated to a semisimple $F$-manifold with compatible flat structure $(M,\circ,\nabla,e)$. Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an $F$-manifold with compatible flat structure that we call reciprocal $F$-manifold. We also consider the special case of bi-flat $F$-manifolds $(M,\circ,*,\nabla^{(1)},\nabla^{(2)},e,E)$ and we study reciprocal transformations preserving flatness of both the connections $\nabla^{(1)}$ and $\nabla^{(2)}$ and how they act on corresponding solutions of an augmented Darboux-Egorov system.

math-ph↗

arXiv2012

Poisson bracket on 1-forms and evolutionary partial differential equations

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, \mathbb{R}^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of $F$-manifolds with compatible flat connection $(M, \nabla, \circ)$ are Hamiltonian in a generalized sense. Moreover, we show that if a metric $g$ compatible with $\nabla$ is also invariant with respect to $\circ$, then this generalized Hamiltonian set-up reduces to the standard one.

math-ph↗

arXiv2012

From Darboux-Egorov system to bi-flat $F$-manifolds

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 Painlevé VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.

math-ph↗

arXiv2011

$F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains

Given an $F$-manifold with eventual identities we examine what this structure entails from the point of view of integrable PDEs of hydrodynamic type. In particular, we show that in the semisimple case the characterization of eventual identities recently given by David and Strachan is equivalent to the requirement that $E\circ$ has vanishing Nijenhuis torsion. Moreover, after having defined new equivalence relations for connections compatible with respect to the $F$-product $\circ$, namely hydrodynamically almost equivalent and hydrodynamically equivalent connections, we show how these two concepts manifest themselves in several specific situations. In particular, in the case of an $F$-manifold endowed with eventual identity and two almost hydrodynamically equivalent flat connections we are able to derive the recurrence relations for the flows of the associated integrable hierarchy. If the two connections originate from a flat pencil of metrics these reduce to the standard bi-Hamiltonian recursion. Furthermore, using the geometric set-up proposed here we show how the recurrence relations of the principal hierarchy introduced by Dubrovin arise in this general framework and we provide a general cohomological set-up for the conservation laws of the semihamiltonian hierarchy associated to a semisimple $F$-manifold with compatible connection and eventual identity. Therefore, the point of view we propose, not only highlight the conceptual unity of two well-known recursive schemes (principal hierarchy and classical bi-Hamiltonian) but it also provides a far reaching generalization of these recursions that relies on the presence of an eventual identity.

nlin.SI↗

arXiv2011

Inherited structures in deformations of Poisson pencils

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 $ε$. 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 $ε$. 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 $ε$, namely the coefficient of the highest order derivative of the $δ$.

math-ph↗

arXiv2011

On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescaling. Constructing recursively the vector fields $X_{2k}$ one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality the set of linear relations expresses the tangency of the vector fields $X_{2k}$ to the symplectic leaves of $ω_1$ and this tangency condition is equivalent to the exactness of the pencil $ω_λ$. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil $ω_λ$, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of $u$ up to that order.

nlin.SI↗

arXiv2009

Equitable Partitioning Policies for Mobile Robotic Networks

The most widely applied strategy for workload sharing is to equalize the workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents and subregions, and (iii) each agent is responsible for service requests originating within its own subregion. In this paper, we design provably correct, spatially-distributed and adaptive policies that allow a team of agents to achieve a convex and equitable partition of a convex workspace, where each subregion has the same measure. We also consider the issue of achieving convex and equitable partitions where subregions have shapes similar to those of regular polygons. Our approach is related to the classic Lloyd algorithm, and exploits the unique features of power diagrams. We discuss possible applications to routing of vehicles in stochastic and dynamic environments. Simulation results are presented and discussed.

cs.RO↗

arXiv2006

Efficient routing of multiple vehicles with no communications

In this paper we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visited by one of the agents. We propose control strategies that, while making minimal or no assumptions on communications between agents, provide the same level of steady-state performance achieved by the best known decentralized strategies. In other words, we demonstrate that inter-agent communication does not improve the efficiency of such systems, but merely affects the rate of convergence to the steady state. Furthermore, the proposed strategies do not rely on the knowledge of the details of the underlying stochastic process. Finally, we show that our proposed strategies provide an efficient, pure Nash equilibrium in a game theoretic formulation of the problem, in which each agent's objective is to maximize the number of targets it visits. Simulation results are presented and discussed.

math.OC↗

arXiv2003

Stacks of cyclic covers of projective spaces

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer-Severi schemes of rank 1, which are not necessarily uniform, and give a presentation of the Picard group for substacks corresponding to smooth triple cyclic covers.

math.AG↗

arXiv2000

On two simple criteria for recognizing complete intersections in codimension 2

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P^n, n>=3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p<=n-1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P^n, p<=n and the characteristic of the base field is zero. Finally, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu on self-linking and complete intersections in positive characteristic.

math.AG↗

arXiv2000

Maslov class and minimality in Calabi-Yau manifolds

Generalizing the construction of the Maslov class for a Lagrangian embedding in a symplectic vector space, we prove that it is possible to give a consistent definition of this class for any Lagrangian submanifold of a Calabi-Yau manifold. Moreover, we prove that this class can be represented by the contraction of the Kaehler form associated to the Calabi-Yau metric, with the mean curvature vector field of the Lagrangian embedding. Finally, we suggest a possible generalization of the Maslov class for Lagrangian submanifolds of any symplectic manifold, via the mean curvature representation.

math.DG↗

arXiv2000

Special Lagrangian Geometry in irreducible symplectic 4-folds

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are bi-Lagrangian and that they are obtained by complex submanifolds via a sort of "hyperkaehler rotation trick"; thus they retain part of the rigidity of complex submanifolds: indeed all special Lagrangian submanifolds of X turn out to be real analytic.

math.DG↗