Super Integrability Research Articles

Superintegrability of matrix Student's distribution

For ordinary matrix models, the eigenvalue probability density decays rapidly as one goes to infinity, in other words, has “short tails”. This ensures that all the multiple trace correlators (multipoint moments) are convergent and well-defined. Still, many critical phenomena are associated with an enhanced probability of seemingly rare effects, and one expects that they are better described by the “long tail” models. In absence of the exponential fall-off, the integrals for high moments diverge, and this could imply a loss of (super)integrability properties pertinent to matrix and eigenvalue models and, presumably, to non-perturbative (exact) treatment of more general quantum systems. In this paper, we explain that this danger to modern understanding could be exaggerated. We consider a simple family of long-tail matrix models, which preserve the crucial feature of superintegrability: exact factorized expressions for a full set of basic averages. It turns out that superintegrability can survive after an appropriate (natural and obvious) analytical continuation even in the presence of divergencies, which opens new perspectives for the study of the long-tail matrix models.

Superintegrable geodesic flows versus Zoll metrics

Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. (1,3), up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S2 are Zoll. We will generalize these results to the case (1,n) for any n≥2. Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S2, as it is already the case for the (1,2) model constructed by Koenigs.

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

We present a family of superintegrable (SI) sytems living on a riemannian surface of revolution and which exhibits one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2 one recovers a metric due to Koenigs. The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called simple case (see definition 2). Some globally defined examples are worked out which live either in H2 or in R2.

The Binary Nonlinearization of the Super Integrable System and Its Self-Consistent Sources

Abstract The super integrable system and its super Hamiltonian structure are established based on a loop super Lie algebra and super-trace identity in this paper. Then the super integrable system with self-consistent sources and conservation laws of the super integrable system are constructed. Furthermore, an explicit Bargmann symmetry constraint and the binary nonlinearization of Lax pairs for the super integrable system are established. Under the symmetry constraint,the n $n$ -th flow of the super integrable system is decomposed into two super finite-dimensional integrable Hamilton systems over the supersymmetric manifold. The integrals of motion required for Liouville integrability are explicitly given.

Bargmann Symmetry Constraint and Binary Nonlinearization of a Super Integrable Hierarchy

Bargmann Symmetry Constraint and Binary Nonlinearization of a Super Integrable Hierarchy

Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians

We construct one-soliton solutions for the nonlinear Schr¨odinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev´e transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlev´e transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose–Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.

Two families of superintegrable and isospectral potentials in two dimensions

As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky–Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators, and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Supersymmetric Yang-Mills equations and supertwistors

This paper is a survey of results relating the supertwistor correspondence on N-extended super-Minkowski space M 4|4 N to supersymmetric Yang-Mills (SSYM) theory. A theorem of Manin relating bundles on the (3- N)th infinitesimal neighborhood L (3− N) 5|2 N of super null-line space L 5 2 N → P 3 N × P 3 N ∗ to solutions of the SSYM equations is analyzed in terms of component fields, interpolating between the N = 0 and N = 3 results studied previously. Using an inductive approach based on the degree of odd homogeneity and a particular gauge condition (the D -gauge), the graded Frobenius equations for covariant constancy along super null-lines are solved. The resulting solution space is shown to define a bundle over L 5|2 N which extends to L (3− N) 5|2 N when the SSYM equations are satisfied. Conversely, the inverse transform determines super connections that are integrable along super null-lines in M 4|4 N . These superconnections determine a supermultiplet which solves the SSYM equations when the bundle over L 5|2 N extends to L (3− N) 5|2 N . A clarification is given concerning the relation between supersymmetry transformations of the component fields and Lie derivations of superconnections on M 4|4 N satisfying super null-line integrability conditions and the D -gauge conditions. Our approach is aimed at bridging the gap between the abstract sheaf-theoretic formulation preferred by mathematicians and the coordinate formulation familiar to physicists.

Super Integrable Systems and its Infinite Conserved Currents

The super extended eigenvalue problem is considered. We deduce a class of completely integrable equations. Its reduced forms are given and their infinite conserved currents have been discussed.