Weierstrass integrability of differential equations

The integrability problem consists of finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define Weierstrass integrability and we determine some Weierstrass integrable systems which are Liouvillian integrable. Inside this new class of integrable systems there are non-Liouvillian integrable systems.     

Maximally Superintegrable Gaudin Magnet: A Unified Approach

A classical integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Poisson algebra, while a quantum integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Jordan–Lie algebra of Hermitian operators. We propose a method for obtaining large Abelian subalgebras inside the tensor product of free tensor algebras, and we show that there exist canonical morphisms from these algebras to Poisson algebras and Jordan–Lie algebras of operators. We can thus prove the integrability of some particular Hamiltonian systems simultaneously at both the classical and the quantum level. We propose a particular case of the rational Gaudin magnet as an example.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

Path Integrals in Noncommutative Quantum Mechanics

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.     

New Integrable Two-Dimensional Systems Generated by Deformed Susy Algebras

New solutions of intertwining relations for two-dimensional scalar quantum Hamiltonians by second-order supercharges with Lorentz and degenerate metrics are obtained. The symmetry operators for components of superhamiltonian that lead to integrability of corresponding systems are found. Expressions for the Hamiltonians and the symmetry operators in the classical limit are constructed. A new class of integrable two-dimensional classical systems with integrals of motion of fourth order in momenta is obtained. Bibliography: 23 titles.     

Integration on locally compact noncommutative spaces

We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier traceability.     

Noncommutative integrable systems on <Emphasis Type="Italic">b</Emphasis>-symplectic manifolds

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure.     

Integrability in a discrete world

We review our findings on integrable discrete systems with emphasis on the discrete integrability detector we have proposed under the name of singularity confinement. We have indeed shown, in a host of examples, that it is possible, by studying the structure of the singularities of discrete systems, to identify the integrable ones. A most important result of this approach is the discovery of discrete Painlevé equations of which a lengthy list exists today. These equations, being integrable systems, are characterised by particularly rich properties which are under active investigation. We present here an overview of these properties and stress the similarities and differences that exist between discrete and continuous Painlevé equations.     

Integrability and conservation laws for two systems of hydrodynamic type derived from the Toda and Volterra systems

We prove that two particular systems of hydrodynamic type can be represented as systems of conservation laws, and that they decouple into non-interacting integrable subsystems. The systems of hydrodynamic type in question were previously constructed, via a matrix partial differential equation, from the Lax pairs for the classical Toda and Volterra systems. The decoupling is guaranteed by the vanishing of the Nijenhuis tensor for each system; integrability of the non-interacting subsystems, thus each system as a whole, is proven for low eigenvalue multiplicities.     

Noncommutative Integrability,Moment Map and Geodesic Flows

The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flows onRiemannian manifolds. In particular, we prove that the geodesic flowof the bi-invariant metric on any bi-quotient of a compact Lie group isintegrable in the noncommutative sense by means of polynomial integrals, andtherefore, in the classical commutative sense by means ofC -smooth integrals.     

Quantum integrable systems and differential Galois theory

This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’

The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics.     

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and co-Legendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the existence of a canonical ? ? ? ^n?1 structure on the leaves of the co-Legendrian foliation. Further, the above structure implies the existence of n commuting contact fields preserving a special contact 1-form, thus providing the geometric framework and establishing equivalence with previously known definitions of contact integrability. We also show that contact completely integrable systems are solvable in quadratures. We present an example of contact complete integrability: the billiard system inside an ellipsoid in pseudo-Euclidean space, restricted to the space of oriented null geodesics. We describe a surprising acceleration mechanism for closed light-like billiard trajectories.     

Fractional monodromy of resonant classical and quantum oscillatorsMonodromie fractionnelle des oscillateurs résonnants classiques et quantiques

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.     

Algebraic aspects of integrability for polynomial differential systems

In this article we summarize the results on algebraic aspects of integrability for polynomial differential systems and its application, which include the Darboux, elementary and Liouvelle integrability. Darboux theory of integrability was found by Darboux in 1878, and it becomes extremely useful in study of the center focus problem, of bifurcation, of limit cycle problem and of global dynamics. The importance of Darboux theory of integrability is also presented by the Singer's theorem for planar polynomial differential system. That is, if a polynomial system is Liouville integrable, then it is Darboux integrable, i.e. the system has a Darboux first integral or a Darboux integrating factor.     

Noncommutative dynamics and generalized master equations

We review the basic concepts of quantum probability and stochastics using the universal Itô B*-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Lévy process is defined in terms of the modular B*-Itô algebra, and the corresponding quantum stochastic master equation on the predual space of theW*-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.     

A note on degenerate corank-one singularities of integrable Hamiltonian systems

We prove that, in a neighborhood of a corank-1 singularity of an analytic integrable Hamiltonian system with n degrees of freedom, there is a locally-free analytic symplectic \Bbb T^n-1 {\Bbb T}^{n-1} -action which preserves the moment map, under some mild conditions. This result allows one to classify generic degenerate corank-one singularities of integrable Hamiltonian systems. It can also be applied to the study of (non)integrability of perturbations of integrable systems.