We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent ... more We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.
We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that ... more We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that some family of eigenfunctions φ satisfies a differential equation in the spectral parameter k of the form For each such V{x) we determine the algebra of all possible operators B and the corresponding functions Θ(x). Table of Contents 0. Introduction 177 1. (adL) m + 1 ((9) = 0 180 2. K(oo) is Finite 187 3. The Rational KdV Potentials 192 4. The Even Case 203 5. The Even Potentials Work Too 213 6. F(oo) = oo is the Airy Case 218 7. Some Illustrative Examples 222
... F. Magri ... This method comes out as a consequence of the change of point of view produced b... more ... F. Magri ... This method comes out as a consequence of the change of point of view produced by the geo-metrical approach, which leads to give preeminence to the Nijenhuis ope-rator A rather than to the integrals which are in involution. ...
ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manif... more ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples.
The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of... more The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.
We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent ... more We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.
The paper describes a new concept of separation of variables with a concrete application to the C... more The paper describes a new concept of separation of variables with a concrete application to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: The first is that the separating coordinates are constructed (not guessed) by solving the Kowalewski separability conditions. The second is that the solutions of the equations of motion are written in terms of theta-functions by means of a generalization of the standard Jacobi inversion problem of algebraic geometry. These two novelties are dealt with in two separate parts of the paper. Part I explains the Kowalewski separability conditions and their implementation to the Clebsch case. It is shown that the new separating coordinates lead to quadratures involving Abelian differentials on two different non-hyperelliptic curves (of genus higher than the dimension of the invariant tori). In Part II these quadratures are interpreted as a new generalization of the standard Abel-Jacobi map, and a procedure of its inversion in terms of theta-functions is worked out. The theta-function solution is different from that found long time ago by F. Kötter, since the theta-functions used in this paper have different period matrix.
This is the second part of a paper describing a new concept of separation of variables applied to... more This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I lead to a new type of the Abel map which contains Abelian integrals on two different algebraic curves. Here we show that this map is from the product of the two curves to the Prym variety of one of them, that it is well defined, although not a bijection. We analyse its properties and formulate a new extention of the Riemann vanishing theorem, which allows to invert the map in terms of theta-functions of higher order. Lastly, we describe how to express the original variables of the Clebsch system in terms of the preimages of the map. This enables one to obtain theta-function solution for the system.
This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with ... more This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with symmetry. The second is to explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.
The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the moti... more The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability conditions obtained by Tullio Levi Civita in 1904.
A systematic approach to the study of nonlinear evolution equations based on the theory of the eq... more A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition.
We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent ... more We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.
We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that ... more We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that some family of eigenfunctions φ satisfies a differential equation in the spectral parameter k of the form For each such V{x) we determine the algebra of all possible operators B and the corresponding functions Θ(x). Table of Contents 0. Introduction 177 1. (adL) m + 1 ((9) = 0 180 2. K(oo) is Finite 187 3. The Rational KdV Potentials 192 4. The Even Case 203 5. The Even Potentials Work Too 213 6. F(oo) = oo is the Airy Case 218 7. Some Illustrative Examples 222
... F. Magri ... This method comes out as a consequence of the change of point of view produced b... more ... F. Magri ... This method comes out as a consequence of the change of point of view produced by the geo-metrical approach, which leads to give preeminence to the Nijenhuis ope-rator A rather than to the integrals which are in involution. ...
ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manif... more ABSTRACT The aim of this paper is to introduce a new category of manifolds, called Haantjes manifolds, and to show their interest by a few selected examples.
The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of... more The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments. The aim of the present paper is to suggest a general theory of Poisson brackets * This work has been sponsored by the Consiglio nazionale delle Ricerche, Gruppo per la Fisica-Matematica.
We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent ... more We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.
The paper describes a new concept of separation of variables with a concrete application to the C... more The paper describes a new concept of separation of variables with a concrete application to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: The first is that the separating coordinates are constructed (not guessed) by solving the Kowalewski separability conditions. The second is that the solutions of the equations of motion are written in terms of theta-functions by means of a generalization of the standard Jacobi inversion problem of algebraic geometry. These two novelties are dealt with in two separate parts of the paper. Part I explains the Kowalewski separability conditions and their implementation to the Clebsch case. It is shown that the new separating coordinates lead to quadratures involving Abelian differentials on two different non-hyperelliptic curves (of genus higher than the dimension of the invariant tori). In Part II these quadratures are interpreted as a new generalization of the standard Abel-Jacobi map, and a procedure of its inversion in terms of theta-functions is worked out. The theta-function solution is different from that found long time ago by F. Kötter, since the theta-functions used in this paper have different period matrix.
This is the second part of a paper describing a new concept of separation of variables applied to... more This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I lead to a new type of the Abel map which contains Abelian integrals on two different algebraic curves. Here we show that this map is from the product of the two curves to the Prym variety of one of them, that it is well defined, although not a bijection. We analyse its properties and formulate a new extention of the Riemann vanishing theorem, which allows to invert the map in terms of theta-functions of higher order. Lastly, we describe how to express the original variables of the Clebsch system in terms of the preimages of the map. This enables one to obtain theta-function solution for the system.
This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with ... more This paper has two purposes. The first is to introduce the definition of Haantjes manifolds with symmetry. The second is to explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.
The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the moti... more The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability conditions obtained by Tullio Levi Civita in 1904.
A systematic approach to the study of nonlinear evolution equations based on the theory of the eq... more A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition.
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