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La recherche en mathématiques n'est pas une activité pour amateur. Il s'agit de s'y consacrer à fond et cela est très absorbant.

           

Pierre van Moerbeke

 

 

     

Publications dans des journaux internationaux indexés, avec comité de lecture, ...

 

Nota : A l’exception de certains articles que l’on peut télécharger via cette page, plusieurs autres articles qui sont publiés sont soumis à un copyright par les revues correspondantes. Par respect à ces revues, on ne peut donc les télécharger via cette page. Cependant, pour avoir un exemplaire et pour un usage strictement personnel, contacter l’auteur pour avoir une copie.

 

• Lesfari, A.: Geometric study of a family of integrable systems, Int. Electron. J. Geom., No. 1, 78–92, Vol. 11 (2018).

Abstract: The aim of this paper is to study a family of integrable hamiltonian systems from a different angle, assemble different geometric methods and several views.

• Lesfari, A.: Normally generated line bundle and Laurent series solutions of nonlinear differential equations,  J. Dyn. Syst. Geom. Theor., Vol. 16, No. 2, 151-171 (2018).

Abstract: The aim of this paper is to demonstrate the rich interaction between the properties of dynamical systems, the geometry of its asymptotic solutions, and the theory of Abelian varieties. We are going to illustrate as well that the nature of many methods for finding solutions of some dynamical systems is determined by the Laurent series for solutions of nonlinear differential equations. We apply the methods to the Kowalewski’top a solid body rotating about a fixed point, the Kirchhoff’s equations of motion of a solid in an ideal fluid and the Ramani-Dorizzi-Grammaticos (RDG) series of integrable potentials.

• Lesfari, A.: Adjunction formula, Poincaré residue and holomorphic differentials on Riemann surfaces,  Methods Funct. Anal. Topology, Vol. 24, no.1, pp. 41–52 (2018).

Abstract: There is still a big gap between knowing that a Riemann surface of genus g has g holomorphic differential forms and being able to find them explicitly. The aim of this paper is to show how to construct holomorphic differential forms on compact Riemann surfaces. As known the dimension of the space H^1(D;C) of holomorphic differentials of a compact Riemann surface D is equal to its genus dimH^1(D;C)=g(D)=g. When the Riemann surface is concretely described, we show that one can usually present the basis of holomorphic differentials explicitly. We apply the method to the case of relatively complicated Riemann surfaces.

• Lesfari, A.: Integrable systems, spectral curves and representation theory, Gen. Lett. Math., Vol. 3, No. 1, pp.1-24 (2017).

Abstract: The aim of this paper is to present an overview of the active area via the spectral linearization method for solving integrable systems. New examples of integrable systems, which have been discovered, are based on the so called Lax representation of the equations of motion. Through the Adler-Kostant-Symes construction, however, we can produce Hamiltonian systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form. We outline an algebraic-geometric interpretation of the flows of these systems, which are shown to describe linear motion on a complex torus. These methods are exemplified by several problems of integrable systems of relevance in mathematical physics.

• Lesfari, A.: Meromorphic functions and theta functions on Riemann surfaces, Res. Rep. Math., Vol. 1, Issue 1, pp. 1-9 (2017).

Abstract: 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 important 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.

• Lesfari, A.: Etude géométrique et topologique du flot géodésique sur le groupe des rotations, Surv. Math. Appl., 11, 107-134 (2016).

Abstract: The aim of this paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps so(4) → so(4)’≡so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from R to C, and then the regular invariant manifolds are complex instead of real tori ; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem.

• Lesfari, A.: Some fundamental properties of complex geometry. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 57, 207-220 (2016).

Abstract: This paper gives sufficient conditions, which guarantee that a complex n-dimensional manifold is analytically isomorphic to a n-dimensional complex torus and a Kähler manifold. We discuss the relation with Hodge theory and an immediate consequence is that a complex manifold will complete to abelian variety by adjoining some divisors. Several important examples are given.

• Lesfari, A.: Symplectic structures on coadjoint orbits of Lie groups, J. Advances Maths, Vol. 2, No 4, pp. 29-31 (2016).

Abstract: This article is devoted to the explicit determination of symplectic structures on coadjoint orbits of Lie groups. It will first be devoted to the study of adjoint and coadjoint orbits of a Lie group with an application in the case of special orthogonal group SO (n). We will see how to determine explicitly a symplectic structure on the orbit of the coadjoint representation of a Lie group. Particular attention is given to the groups SO(3) and SO(4).

• Lesfari, A.: Géométrie et intégrabilité algébrique. Rendiconti di Matematica, Serie VII, Volume 36, Roma, 27-76 (2015).

Résumé : Cet article est consacré à l’étude des systèmes dynamiques non-linéaires algébriquement complètement intégrables. Cela veut dire que l’on demande que les invariants du système différentiel soient polynomiaux (dans des coordonnées adéquates) et que de plus les variétés complexes obtenues en égalant ces invariants polynomiaux à des constantes génériques forment la partie affine d’un tore complexe algébrique (variété abélienne) de telle façon que les flots complexes engendrés par les invariants soient linéaires sur ces tores complexes. D’autres systèmes intégrables (appelés systèmes généralement algébriquement complètement intégrables) apparaissent comme des revêtements de systèmes algébriquement complètement intégrables. Ces systèmes concernent des situations où les exposants de Kowalewski (c’est-à-dire les valeurs propres de l’opérateur linéaire intervenant dans la recherche des solutions sous forme de séries de Laurent) sont des fractions. On verra que le problème est lié à la notion d’équivalence rationnelle entre systèmes complètement intégrables, aux surfaces de type général ainsi qu’à la classification des singularités. Ensuite, on expose dans les sections suivantes plusieurs systèmes importants : le système différentiel de Hénon-Heiles, la toupie de Goryachev-Chaplygin, le réseau de Toda, la toupie de Kowalewski, le flot géodésique sur le groupe SO(n), la toupie de Lagrange, le réseau périodique de Kac-van Moerbeke, les systèmes de Toda périodiques généralisés, le système de Gross-Neveu et le potentiel de Kolossof.

   • Lesfari, A.: Champ de Yang-Mills avec groupe de jauge SU(2) (Yang-Mills field with SU(2) as gauge group). Mathematical Reports. Vol. 17(67), No. 1, 133-153 (2015).

Abstract : In this paper we consider a hamiltonian system which, in a special case and under the gauge group SU(2), can be considered as some reduction of the Yang-Mills field equations. We study this system from a different angle. We show that this system is completely integrable and we realize explicitly, using the Lax spectral curve technique that the fows generated by the constants of the motion are straight lines on the Jacobi variety of a genus 2 hyperelliptic curve. We show that at some special values of the parameters, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation. We show that the complex affine variety defined by putting the invariants of the system equal to generic constants completes into an abelian surface and the system is algebraic completely integrable. We give a direct proof that the abelian surface obtained in this paper is dual to Prym variety of an hyperelliptic curve of genus 3.

   • Lesfari, A.: Systèmes dynamiques algébriquement complètement intégrables et géométrie. Analele Universitatii de Vest din TimisoaraSeria Matematica-Informatica, LIII, 1, 109– 136 (2015).

Abstract : In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems: Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.

  • Lesfari, A.: The Hénon-Heiles system as part of an integrable system. Journal of Advanced Research in Dynamical and Control Systems, Vol. 6, Issue 3, pp. 24-31 (2014).

Abstract : 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.

• Lesfari, A.: Rotation d'un corps solide autour d'un point fixe. Rend. Sem. Mat. Univ. Pol. Torino, Vol.72, 1-2, 255-284 (2014).

Abstract: The aim of this paper is to present an overview and to make a careful study of different integrable cases for the equations of a rigid body rotation around a fixed point. We study this interesting problem from a different angle using different methods: the Kowalewski-Painlevé analysis, isospectral deformation method and others techniques. 

   • Lesfari, A.: Moser's lemma and the Darboux theorem, Univ. J. Appl. Maths, 2(1): 36-39 (2014).

Abstract : The main Darboux theorem is fundamental for the theory of symplectic manifolds, which states that locally all symplectic manifolds of the same dimension are isomorphic.  Every point in a symplectic manifold has a neighborhood with Darboux coordinates. In this paper we will give a proof of the classical Moser's lemma. Using it, we give a direct proof of the Darboux theorem.

    • Lesfari, A.: Interversion des dérivées partielles. Quadrature, Paris, No. 91, pp. 41-43 (2014).

Résumé : Cet article est consacré à l'étude du problème de l'interversion de l'ordre de dérivation et aux différentes conditions suffisantes pour que la permutabilité des dérivées partielles soit licite. Nous verrons que l'on peut affaiblir les hypothèses du théorème de Schwarz.

 Lesfari, A.: Algebraic integrable systems, abelian varieties and Kummer surfaces. C. Adv. Maths Res.Volume 1, Issue 1, 1-11 (2014).

Abstract : In the present paper, we discuss an interesting interaction between complex algebraic geometry and dynamical systems. We construct a new integrable system in five unknowns having three quartics invariants. This system is interesting because is the first known algebraic completely integrable system in C^5 it can be integrated in genus 2 hyperelliptic functions and it establishes some correspondences for old and new integrable systems. The paper is supported by two appendices which contain some basis concepts concerning abelian varieties and hamiltonian systems.

• Lesfari, A.: Etude des équations stationnaire de Schrödinger, intégrale de Gelfand-Levitan et de Korteweg-de-Vries. Solitons et méthode de la diffusion inverse. Aequat. Math., Springer, Vol. 85, 243-272 (2013).

Abstract: The Korteweg-de Vries (KdV) equation is a universal mathematical model for the description of weakly nonlinear long wave propagation in dispersive media. It is a most remarkable nonlinear partial differential equation in 1+1 dimensions whose solutions can be exactly specified; it has a soliton like solution or solitary wave of sech^2 form. Various physical systems of dispersive waves admit solutions in the form of generalized solitary waves. The study of this equation is the archetype of an integrable system and is one of the most fundamental equation of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. This article will cover in detail: the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly.

• Lesfari, A.: Etude du théorème de plongement de Kodaira. Analele Universitatii de Vest din TimisoaraSeria Matematica-Informatica, LI, 1, 75– 90 (2013).

Abstract: The archetype of embedding theorems is the Kodaira Embedding theorem for compact complex manifolds. The aim of the present paper is to give a direct proof of this theorem and discuss the relation with Hodge theory and Kähler manifolds.

• Lesfari, A. : Application d’Abel-Jacobi, Espace des modules des surfaces de Riemann et le Problème de Schottky. Mathematica (Cluj), tome 54(77), No 1, 64-77 (2012).

Abstract: In this paper we discuss some interesting problems. 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. The Schottky problem is the problem of characterizing jacobian varieties among all abelian varieties. We study the map X (Riemann surface) to its jacobian variety Jac(X) from a moduli point of view. The problem consists in finding an analytical characterization of the complex tori that arise as jacobians inside the Siegel upper half space.

• Lesfari, A. : Algèbres de Lie affines et opérateurs pseudo-différentiels d’ordre infini. Maths report, Vol.14 (64), No.1, 43-69 (2012).

Abstract: In this paper, we make a careful study of some connection between pseudo-differential operators, Kadomtsev-Petviashvili (KP) hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the Korteweg-de Vries (K-dV) and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given. The study of the KP and KdV hierarchies, the use of tau functions related to infinite  dimensional Grassmannians, vertex operators and the Hirota's bilinear formalism led to obtaining remarkable properties concerning these algebras as for example the existence of an infinite family of first integrals functionally independent and in involution. The paper is supported by an appendix which contain some informations about coadjoint orbits in Kac-Moody algebras and a proof of the Adler-Kostant-Symes theorem.

• Lesfari, A.: Fonctions différentiables. Quadrature, Paris, No. 84, pp.45-47 (2012).

Résumé : Dans cet article, nous prouvons que si l'une des dérivées partielles premières d'une fonction à n variables existe en un point a et si les n-1 autres dérivées partielles premières existent dans un voisinage de a et sont continues en a, alors cette fonction  est différentiable en a. Dans le cas particulier d'une fonction à deux variables la condition suffisante de différentiabilité de cette fonction consiste à utiliser la continuité d'une des dérivées partielles mais pas les deux!

• Lesfari, A. : Algebraic integrability : the Adler-van Moerbeke approach. Regul. Chaotic Dyn. Vol. 16, Nos. 3–4, pp. 187–209 (2011).

Abstract: In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski-Painlev'{e} analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method's which will be used is devoted to illustrate how to decide about the algebraic completely integrable hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the H'{e}non-Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group SO(n) for a left invariant metric.

• Lesfari, A.: Etude des théorèmes d'annulation de Kodaira-Nakano, de Lefschetz sur les sections hyperplanes et de Lefschetz pour les classes de type (1,1), Analele Universitatii de Vest din TimisoaraSeria Matematica-Informatica, XLIX, 2, 37– 60 (2011).

Abstract: Many problems and results of contemporary complex geometry involve vanishing theorems for holomorphic vector bundles. In this paper, we prove some vanishing theorems and explore some of their numerous consequences. It is in the book of Griffiths and Harris that we went to draw most of the ideas discussed here. In order to fully state and prove these results, we will have to develop some background material. The paper begins with the study of divisors and line bundles on complex manifolds. Standard properties about global sections of line bundles, adjunction formulas, the Poincaré residue, Kodaira-Serre duality and the relationship with the Riemann-Roch theorem are proved, and some results are discussed. We describe a proof of the Kodaira vanishing theorem due to Akizuki and Nakano. The Kodaira-Nakano vanishing theorem is a statement that certain cohomology groups of positive line bundles vanish. Using Kodaira vanishing theorem, we can prove the Lefschetz hyperplane theorem. Finally we prove the Lefschetz theorem on (1, 1)-classes. The paper contains numerous interesting examples.

• Lesfari, A. : Systèmes hamiltoniens complètement intégrables. Aequat. Math., Springer, Vol. 82, 165-200 (2011).

Abstract: This article is dedicated to one of the greatest mathematicians of our time: VI Arnold, who died suddenly Thursday, June 3, 2010 in France. Integrable hamiltonian systems are nonlinear ordinary differential equations described by a hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential geometric fact: a compact and connected n-dimensional manifold on which there exist $n$ vector fields which commute and are independent at every point is diffeomorphic to an n-dimensional real torus and each vector field will define a linear flow there. We make a careful study of the connection with the concept of completely integrable systems and we apply the methods to several problems.

• Lesfari, A. : Équations couplées non-linéaires de Schrödinger.  Afr. Diaspora J. Math. 10, No. 2, 96-108 (2010). 

Abstract: In this paper, we give a complete description of the invariant surfaces of the system governing the motion of the coupled nonlinear Schrödinger equations and their completion into abelian surfaces. We derive the associated Riemann surface on the basis of Painlevé-type analysis in the form of a genus 3 Riemann surface Γ, which is a double ramified covering of an elliptic curve Γ0 and a two sheeted genus two hyperelliptic Riemann surface C. We show that the affine surface Vc obtained by setting the two quartics invariants of the problem equal to generic constants, is the affine part of an abelian surface Vc. The latter can be identified as the dual of the Prym variety Prym(Γ / Γ0) on which the problem linearizes, that is to say their solutions can be expressed in terms of abelian integrals. Also, we discuss a connection between Vc and the jacobian variety Jac(C) of the genus 2 hyperelliptic Riemann surface C. 

• Lesfari, A. : Théorie spectrale et problèmes non linéaires. Surv. Math. Appl., 5, 151-190 (2010).

Abstract: We present a Lie algebra theoretical schema leading to integrable systems, based on the Kostant-Kirillov coadjoint action. Many problems on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem and the van Moerbeke-Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety of the spectral curve. All the complex flows generated by the constants of the motion are straight line motions on these Jacobi varieties. We study the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view and their relation with the Kac-Moody extension of some algebras. We will present the Griffith's linearization method for solving integrable systems without reference to Kac-Moody algebras. We solve a number of interesting integrable systems of relevance in mathematical physics.

• Lesfari, A.: Fonctions thêta et critère de projectivité des tores complexes. Imhotep, Vol.8, No. 1, 13-48 (2010). 

Abstract: This paper is devoted to some properties of the theory of thêta functions on Riemann surfaces and explores some of its numerous consequences, a subject of renewed interest in recent years. We show how the meromorphic functions on the Riemann surfaces of an arbitrary genus can be constructed explicitly in terms of the multi-dimensional theta functions. We discuss the important role of the zeros of theta function and the Jacobi inversion problem which askes whether we can find a divisor that is the preimage for an arbitrary point in the jacobian. The Lefschetz theorem on projective embeddings over the complex numbers is of utmost importance in the complex geometric theory of compact manifolds. We present an analytic proof of this theorem. We explain how positive line bundles on abelian varieties can be explicitly described in terms of multipliers and how their sections can be described by theta functions.

• Lesfari, A.: Integrable systems and complex geometry. Lobachevskii Journal of Mathematics, Vol. 30, No.4, 292-326 (2009).

Abstract: 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.

• Lesfari, A.: Fonctions et Intégrales elliptiques. Surv. Math. Appl., 3, 27-65 (2008).

Abstract: 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.

• Lesfari, A.: Affine parts of abelian surfaces as complete intersection of three quartics. Int. Electron. J. Geom., Volume 1, No. 2, 15-32 (2008).

Abstract: 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

• Lesfari, A.: Prym varieties and applications. J. Geom. Phys., 58, 1063-1079 (2008).

Abstract: 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 such as the Hénon-Heiles system, the Kowalewski rigid body motion and Kirchhoff's equations of motion of a solid in an ideal fluid.

• Lesfari, A.: Cyclic coverings of abelian varieties and the generalized Yang Mills system for a field with gauge groupe SU(2). Int. J. Geom. Methods Mod. Phys., Vol.5, N.6, 947-961 (2008).

Abstract: 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. 

• Lesfari, A. : Integrables hamiltonian systems and the isospectral deformation method. Int. J. of Appl. Math. And Mech.3(4) :35-55 (2007).

Abstract: Integrable hamiltonian systems are nonlinear ordinary differential equations described by a hamiltonian function and possessing sufficiently many independent constants of motion in involution. The regular compact level manifolds defined by the intersection of the constants of motion are diffeomorphic to a real torus on which the motion is quasi-periodic as a consequence of the following purely differential geometric fact : a compact and connected n-dimensional manifold on which there exist n  vector fields which commute and are independent at every point is diffeomorphic to an n-dimensional real torus and each vector field will define a linear flow there. New examples of completely integrable hamiltonian systems, which have recently 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. We make a careful study of the connection with the concept of completely integrable systems and we apply the methods to several problems.

• Lesfari, A. : Étude des solutions méromorphes d’équations différentielles. Rend. Sem. Mat. Univ. Pol. Torino, Vol.65, 4, 451-464 (2007).

Abstract: 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.

• Lesfari, A.: Abelian varieties, surfaces of general type and integrable systems. Beiträge Algebra Geom, Vol.48, N.1, 95-114 (2007).

Abstract: In recent years, there has been much effort given for finding integrable hamiltonian systems. However, there is still no general method for testing the integrability of a given dynamical system. In this paper, we shall be concerned with finite dimensional algebraic completely integrable systems. A dynamical system is algebraic completely integrable if it can be linearized on an abelian variety (i.e., a complex algebraic torus. The invariants (often called first integrals or constants) of the motion are polynomials and the phase space coordinates (or some algebraic functions of these) restricted to a complex invariant variety defined by putting these invariants equals to generic constants, are meromorphic functions on an abelian variety. Moreover, in the coordinates of this abelian variety, the flows (run with complex time) generated by the constants of the motion are straight lines. However, besides the fact that many hamiltonian completely integrable systems posses this structure, another motivation for its study which sounds more modern is: algebraic completely integrable systems come up systematically whenever you study the isospectral deformation of some linear operator containing a rational indeterminate. Indeed a theorem of Adler-Kostant-Symes applied to Kac-Moody algebras provides such systems which, by a theorem of van Moerbeke-Mumford, are algebraic completely integrable. Therefore there are hidden symmetries which have a group theoretical foundation. Also someinteresting integrable systems appear as coverings of algebraic completely integrable systems. The invariant varieties are coverings of abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The concept of algebraic complete integrability is quite effective in small dimensions and has the advantage to lead to global results, unlike the existing criteria for real analytic integrability, which, at this stage are perturbation results. In fact, the overwhelming majority of dynamical systems, hamiltonian or not, are non-integrable and possess regimes of chaotic behavior in phase space. In the present paper, we discuss an interesting interaction between complex algebraic geometry and dynamical systems. We construct a new integrable system in five unknowns having three quartics invariants. This 5-dimensional system is algebraic completely integrable and it establishes some correspondences for old and new integrable systems. The paper is organized as follows: In section 2, we construct a new and interesting integrable system of differential equations in five unknowns having three quartics invariants. We make a careful study of the algebraic geometric aspect of the complex affine variety A defined by putting these invariants equal to generic constants. We find via the Painlevé analysis the principal balances of the hamiltonian field defined by the hamiltonian. To be more precise, we show that the system in question possesses Laurent series, which depend on 4 free parameters. These meromorphic solutions restricted to the surface A are parameterized by two copies of the same Riemann surface C of genus 7, that intersect in two points at which they are tangent to each other. The affine variety A is embedded into P^15 and completes into an abelian variety A by adjoining a divisor. The latter has geometric genus 17 and is very ample. The flow in question evolves on this abelian variety and is tangent to each copie of the Riemann surface C at the points of tangency between them. Consequently, the system in question is algebraically integrable. In section 3, we show that the above system  include in particular, another system in four unknowns which is intimately related to the potential obtained by Ramani, Dorizzi and Grammaticos. We show that this system admits Laurent solutions in Ït, depending on 3 free parameters. These pole solutions restricted to the invariant surface B are parameterized by two copies of the same Riemann surface G  of genus 16. We show that the invariant variety B can be completed as a cyclic double cover B of the abelian variety A, ramified along the above divisor. Moreover, B is smooth except at the point lying over a singularity (of type A_3) and the resolution B of B is a surface of general type. Consequently, the system in question is algebraically completely integrable in the generalized sense. The paper is supported by two appendices which contain some basis concepts concerning abelian varieties and hamiltonian systems. The methods which will be used are primarily analytical but heavily inspired by algebraic geometrical methods. Abelian varieties and cyclic coverings of abelian varieties, very heavily studied by algebraic geometers, enjoy certain algebraic properties which can then be translated into differential equations and their Laurent solutions. Among the results presented in this paper, there is an explicit calculation of invariants for hamiltonian systems which cut out an open set in an abelian variety or cyclic coverings of abelian varieties, and various Riemann surfaces related to these systems are given explicitly. The integrable dynamical systems presented here are interesting problems, particular to experts of abelian varieties who may want to see explicit examples of a correspondence for varieties defined by different Riemann surfaces.

• Lesfari, A. and Elachab, A.: The Yang-Mills equations and the intersection of quartics in projective 4-space CP4. Int. J. Geom. Methods Mod. Phys., Vol.3, N.2, 1-8 (2006).

Abstract: In this paper, we discuss an interesting interaction between complex algebraic geometry and dynamics: the integrability of the Yang-Mills system for a field with gauge group SU(2) and the intersection of quartics in projective 4-space CP^4. Using Enriques classification of algebraic surface and dynamics, we show that these two quartics intesect in the affine part of an abelian surface and it follows that the system of differential equations is algebraically completely integrable.

• Lesfari, A. and Elachab, A.: A connection between geometry and dynamical systems. J. Dyn. Syst. Geom. Theor., Vol 3, Number 1, 25-44 (2005).

Abstract: This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential. We study this system from a different angle: a)  We show, using a Lax-type representation of the Hamilton's equations of motion, that the system is linearized in the jacobian variety of a smooth genus 2 hyperelliptic curve. b)  We find via Kowalewski-Painlevé analysis the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. c)  We also describe an explicit embedding of the abelian variety which completes the generic invariant surface, into projective space. d)  We give a direct proof that the abelian variety obtained in this paper is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. e) We show that at some special values of the parameters l1 and l2  we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation.

• Lesfari, A. : Analyse des singularités de quelques systèmes intégrables, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Abstract: Dans cette note, nous construisons un nouveau système intégrable de cinq variables ayant trois invariants quartiques. Ce système est algébriquement complètement intégrable.

• Lesfari, A. and Elachab, A.: On the integrability of the generalized Yang-Mills system. Appl. Math. (Warsaw), 31, 3, 345-351 (2004).

Abstract: In this paper we consider a hamiltonian system which, in a special case and under the gauge group SU(2), can be considered as some reduction of the Yang-Mills field equations. We realize explicitly, using the Lax spectral curve technique and the van Moerbeke-Mumford method, that the flows generated by the constants of the motion as straight lines on the Jacobi variety of a genus two Riemann surface.

• Lesfari, A. et Elachab, A. : Etude géométrique d’une famille de systèmes intégrables. Math. Pannon., 15/2, 275-282 (2004).

Abstract: In this paper, we consider a hierarchy of hamiltonian systems. Usually, this system is nonintegrable, but we give two integrable cases in the sens of Liouville. In the first case, we show that the system is linearized in the jacobian variety of a smooth hyperelliptic Riemann surface. For the second case, we describe a connection with the system of two coupled nonlinear Schrödinger equations. We use this connection for deriving a Lax representation and a spectral curve for the system. The linearized flow can be realized on an elliptic curve.

• Lesfari, A.: The complex geometry of an integrable system. Arch. Math. (Brno), T. 39, 257-270 (2003).

Abstract: A finite dimensional algebraic completely integrable system is considered. We show that the intersection of levels of integrals completes into an abelian surface (a two dimensional complex algebraic torus) of polarization (2,8) and that the flow of the system can be linearized on it; that is to say their solutions can be expressed in terms of abelian integrals.

• Lesfari, A. : Le système différentiel de Hénon-Heiles et les variétés Prym. Pacific J. Math., Vol. 212, No 1, 125-132 (2003).

Résumé: On montre que la fibre définie par l'intersection des invariants du système différentiel de Hénon-Heiles se complète en un tore complexe algébrique par l'adjonction d'une surface de Riemann lisse hyperelliptique de genre 3 ; laquelle est un revêtement double ramifié le long d'une courbe elliptique. Une description géométrique de ce tore complexe via la théorie des variétés abéliennes est faite.

• Lesfari, A. : Le théorème d’Arnold-Liouville et ses conséquences. Elem. Math., Vol. 58, I.1, 6-20 (2003).

Résumé: Les variétés de niveau communes des intégrales premières définies par les groupes à un paramètre de difféomorphismes d’un système dynamique, sont invariantes du flot. La solution d’un problème non-linéaire se ramène actuellement, à l’étude de son flot et de ces variétés invariantes. Le théorème d’Arnold-Liouville  joue un rôle crucial dans l’étude de ces problèmes. Il permet, entre autres, d’étudier la situation topologique suivante : si les variétés invariantes sont compactes et connexes, alors elles sont difféomorphes aux tores réels sur lesquels le flot de phase détermine un mouvement quasi-périodique. Les équations du problème sont intégrables par quadratures et le théorème en question montre un comportement très régulier des solutions. Le but ici est de donner une nouvelle démonstration de ce théorème aussi directe que possible, d’étudier explicitement ses liens avec la théorie des systèmes hamiltoniens intégrables et enfin l’appliquer à des situations concrètes.

• Lesfari, A.: The Hénon-Heiles system via the Kowalewski-Painlevé analysis. Intern. J. Theo. Phys, Group  theory & Nonlin.Opt., Vol. 9, N°4, 305-330 (2002).

Abstract: This work deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the Hénon-Heiles system. We study this system via the Kowalewski-Painlevé analysis. We find the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. We also describe an explicit embedding of the abelian surface which completes the generic invariant surface into projective space. We give a direct proof that the abelian surface obtained is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. We also discuss the Kummer surface associated with this abelian surface, it will be constructed explicitly.

• Lesfari, A.: A new class of integrable systems. Arch. Math., 77, 347-353 (2001).

Résumé: De nouveaux systèmes intégrables au sens de Liouville ont été découverts. Ils généralisent plusieurs systèmes connus. Certains d’entre eux ont pu être résolu par des méthodes élémentaires sur la variété Jacobienne d’une courbe lisse hyperelliptique de genre deux. En outre, un lien avec certaines équations aux dérivées partielles non-linéaires a été explicitement établi.

• Lesfari, A.: The generalized Hénon-Heiles system, Abelian surfaces and algebraic complete integrability. Reports on Math.Phys., Vol. 47, 9-20 (2001).

Résumé: La complète intégrabilité algébrique du système différentiel de Hénon-Heiles. Il s’agit d’un système non-linéaire défini sur une variété symplectique de dimension quatre et sert entre autres à modéliser le mouvement d’une étoile dans une galaxie cylindrique. En général ce système n’est pas intégrable. Mais pour des valeurs particulières des paramètres intervenants dans le problème, la linéarisation a été obtenue sur la duale d’une variété de Prym d’une surface de Riemann de genre trois. Des résultats nouveaux ont été établis en utilisant des techniques de géométrie complexe contemporaine.

• Lesfari, A.: The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch’s case. NoDEA, Nonlinear diff.Equ. Appl., Vol. 8, N°1, 1-13  (2001).

Résumé: La géométrie complexe des équations de Kirchhoff dans le cas de Clebsch. Ces équations peuvent être associées aux flots géodésiques sur les groupes de Lie pour une métrique invariante à gauche. Il s’agit du flot géodésique sur le groupe des déplacements de l’espace euclidien à trois dimensions. Les solutions de ces équations ont été exprimées en termes d’intégrales hyperelliptiques sur les surfaces de polarisation (1,2). Les tores invariants, vus dans le complexe, ne sont pas principalement polarisés, mais sont isogènes à des surfaces principalement polarisées. Une étude géométrique et topologique complète de cette situation a été faite du point de vue de la caractérisation des surfaces abéliennes.

• Lesfari, A.: Completely integrable systems : Jacobi’s heritage. J. Geom. Phys., 31, 265-286 (1999). 

 Abstract: During the last few decades, algebraic geometry has become a tool for solving differential equations and spectral questions of mechanics and mathematical physics. This paper deals with the study of the integrable systems from the point of view of algebraic geometry, inverse spectral problems and mechanics from the point of view of Lie groups. The first section is preliminary giving a little background. In section 2, we study a Lie algebra theoretical method leading to completely integrable systems, based on the Kostant-Kirillov coadjoint action. The third section is devoted to illustrate how to decide about the algebraic complete integrability 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). Adler-van Moerbeke's method is a very useful tool not only to discover among families of Hamiltonian systems those which are a.c.i., but also to characterize and describe the algebraic nature of the invariant tori (periods, etc...) for the a.c.i. systems. Some integrable systems, such as Korteweg-de Vries equation, Toda lattice, Euler rigid body motion, Kowalewski's top, Manakov's geodesic flow on SO(4),…are treated.

• Lesfari, A.: Geodesic flow on SO(4), Kac-Moody Lie algebra and singularities in the complex t- plane. Publ. Mat., Barc., Vol. 43, 261-279 (1999).

Abstract: The article studies geometrically the Euler-Arnold equations associated to geodesic flow on SO(4) for a left invariant diagonal metric. Such metric were first introduced by Manakov and extensively studied by Mishchenko-Fomenko and Dikii. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke and Haine. In this problem there are four invariants of the motion defining in an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coadjoint action. This method allows us to linearizes the problem on a two-dimensional Prym variety of a genus 3 Riemann surface. In section 2, the method consists of requiring that the general solutions have the Painlevé property, i.e., have no movable singularities other than poles. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler-Arnold equations occurs on a Prym variety of another genus 3 Riemann surface. In the last section the Riemann surfaces are compared explicitly.

• Lesfari, A. : Une méthode de compactification de variétés liées aux systèmes dynamiques. Les Cahiers de la recherche, Rectorat-Université Hassan II  Aïn Chock, Casablanca, Vol. I, N° 1, 147-157 (1999).

• Lesfari, A.: On affine surface that can be completed by a smooth curve. Result. Math. 35, 107-118 (1999).

Abstract: We consider a non linear system of differential equations with four invariants: two quadrics, a cubic and a quartic. Using Enriques-Kodaira classification of algebraic surfaces, we show that the affine surface obtained by setting these invariants equal to constants is the affine part of an abelian surface. This affine surface is completed by gluing to it a one genus 9 curve consisting of two isomorphic genus 3 curves intersecting transversely in 4 points.

• Lesfari, A.: Abelian surfaces and Kowalewski’s top. Ann. Scient. Ec. Norm. Sup., Paris, 4° série, t. 21, 193-223 (1988).

Abstract: This paper deals with a geometric and systematic approach to the integration of Kowalewski’s rigid body motion. It is well known that this motion is completely integrable and Kowalewski integrates the problem in terms of hyperelliptic quadratures after a complicated and mysterious change of variables. The classical approach to solving integrable systems was based on solving the Hamilton-Jacobi equation by separation of variables, after an appropriate change of coordinates; for every problem finding this transformation required a great deal of ingenuity. Up to now Kowalewski’s method has been neither understood, nor improved nor extended to other cases except for some modest amelioration contributed by Kötter and Kolossoff. This paper presents a new and systematic method to integrate the problem, and leads to a detailed geometric description of the invariant surfaces (tori) on which the motion evolves.

• Lesfari, A. : Une approche systématique à la résolution des systèmes  intégrables. Proceeding du Colloque (2ème Ecole de Géométrie-Analyse). Conférences organisées en l’honneur du Professeur A. Lichnerowicz, EHTP, Casablanca, p. 83 (15-19 Juin 1987).

• Lesfari, A. : Une approche systématique à la résolution du corps solide de Kowalewski. C. R. Acad. Sc., Paris, t.302, série I, 347-350 (1986).

Résumé: On utilise la méthode des développements asymptotiques de Laurent, en vue de linéariser le flot de la toupie de Kowalewski et obtient la linéarisation sur la duale d’une variété de Prym d’une courbe de genre 3.

• Lesfari, A.: Abelian surfaces and Kowalewski’s top. Séminaire de Mathématique/N.S., Vol.3,  81,  SC/MAPA - Institut de mathématique pure et appliquée, UCL,  1986.

• Lesfari, A. : La complète algébrique du corps solide de Kowalewski. Séminaire de  Mathématique /N.S., Vol.1, 72,  SC/MAPA -Institut de mathématique pure et appliquée, UCL,  1985.

    • Lesfari, A. : Equation de Korteweg-de Vries. M1982/LES/1A, SC/MAPA - Institut de mathématique pure et appliquée, UCL, Louvain-la-Neuve, 1982.

 

 

 

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