arXiv subjects

A. Lesfari

Publications and source records attributed to A. Lesfari.

13 recordsLinked to original sources

arXiv2019

On Poincaré lemma or Volterra theorem about differential forms and cohomology groups

The Poincaré lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim of this paper is to present some direct proofs of this lemma and explore some of its numerous consequences. Some connections with Cech-De Rham-Dolbeault cohomologies, $\overline{\partial}$-Poincaré lemma or Dolbeault-Grothendieck lemma are given.

math.GM↗

arXiv2019

Symplectic manifolds and Hamiltonian dynamical systems

This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete integrability of these systems, with detailed proofs. Several explicit calculations for which references are not immediately available are given. These results are exemplified by applications to some Hamiltonian dynamical systems.

math.SG↗

arXiv2019

Isospectral deformations, the spectrum of Jacobi matrices, infinite continued fraction and difference operators. Application to dynamics on infinite dimensional systems

This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory, cohomology groups and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. Another part will be dedicated to the study of infinite continued fraction, orthogonal polynomials, the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Some connections with Cauchy-Stieltjes transform of a suitable measure and Abelian integrals are given. Finally the notion of algebraically completely integrable systems is explained, techniques to solve such systems are presented and some interesting cases appear as coverings of such dynamical systems. These results are exemplified by several problems of dynamical systems of relevance in mathematical physics.

math.DS↗

arXiv2016

Fonctions méromorphes et fonctions thêta sur les surfaces de Riemann

Theta functions play a major role in many current researches and are powerful tools for studying integrable systems. The purpose of this paper is to provide a short and quick exposition of some aspects of meromorphic theta functions for compact Riemann surfaces. The study of theta functions will be done via an analytical approach using meromorphic functions in the framework of Mumford. Some interesting examples will be given : the classical Kirchhoff equations in the cases of Clebsch and Lyapunov-Steklov, the Landau-Lifshitz equation and the sine-Gordon equation.

math.CV↗

arXiv2014

The Hénon-Heiles system as part of an integrable system in five unknowns with three constants of motions

In this paper we construct a new completely integrable system. This system is an instance of a master system of differential equations in $5$ unknowns having $3$ quartics constants of motion.We find via the Painlevé analysis the principal balances of the hamiltonian field defined by the hamiltonian. Consequently, the system in question is algebraically integrable. A careful analysis of this system reveals an intimate rational relationship with a special case of the well known Hénon-Heiles system. The latter admits asymptotic solutions with fractional powers in $t$ and depending on $3$ free parameters. As a consequence, this system is algebraically completely integrable in the generalized sense.

math.AG↗

arXiv2009

Surfaces de Riemann compactes, courbes algébriques complexes et leurs Jacobiennes

Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holed torus (a sphere with $g$ handles). This paper is an introduction to the theory of compact Riemann surfaces and algebraic curves. It presents the basic ideas and properties as an expository essay, explores some of their numerous consequences and gives a concise account of the elementary aspects of different viewpoints in curve theory. We discuss and prove most intuitively some geometric-topological aspects of the algebraic functions and the associated Riemann surfaces. Abelian and normalized differentials, Riemann's bilinear relations and the period matrix for $X$ are defined and some consequences drawn. The space of holomorphic 1-forms on $X$ has dimension $g$ as a complex vector space. Fundamental results on divisors on compact Riemann surfaces are stated and proved. The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. We present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves. Abel's theorem classifies divisors by their images in the jacobian. The Jacobi inversion problem askes whether we can find a divisor that is the preimage for an arbitrary point in the jacobian. In the first appendix, we introduced intuitively and explicitly elliptic and hyperelliptic Riemann surfaces. In the second appendix, we study some results of resultant and discriminant as needed in the paper.

math.AG↗

arXiv2007

Etude Des Solutions Meromorphes D'Equations Differentielles

In this paper we shall study differential equations in the complex domain. The method of indeterminate coefficients and the majorant method lead to a proof of the existence and uniqueness of meromorphic solution of differential equations. We discuss their connection with the concept of algebraic integrability systems.

math.CA↗

arXiv2007

Fonctions Et Integrales Elliptiques

This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum, the Euler rigid body motion and some others integrable hamiltonian systems.

math.CV↗

arXiv2007

The Yang Mills system and cyclic covering of abelian varieties

In this paper, we consider a dynamical system related to the Yang-Mills system for a field with gauge group SU(2). We solve this system in terms of genus two hyperelliptic functions and we show that it is algebraic completely integrable in the generalized sense.

math-ph↗

arXiv2007

Affine parts of abelian surfaces as complete intersection of three quartics

We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a genus two hyperelliptic curve. This system is algebraic completely integrable and it can be integrated in genus two hyperelliptic functions.

nlin.SI↗

arXiv2007

Le théorème de Riemann-Roch et ses applications

The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this article is to present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves.

math.CV↗

arXiv2007

Prym varieties and applications

The classical definition of Prym varieties deals with the unramified covers of curves. The aim of the present paper is to give explicit algebraic descriptions of the Prym varieties associated to ramified double covers of algebraic curves. We make a careful study of the connection with the concept of algebraic completely integrable systems and we apply the methods to some problems of Mathematical Physics.

math.AG↗

arXiv2007

Integrable systems and complex geometry

In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety.

math.DS↗