Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented. Received: 19 March 2001 / Accepted: 11 July 2001     

Strominger–Yau–Zaslow Geometry,Affine Spheres and Painlevé III

We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of anti–self–dual Yang–Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin–Yau–Zaslow construction to give an explicit expression for a six–real dimensional semi–flat Calabi–Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3^rd Painlevé transcendents.     

Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus

We realize affine Weyl group symmetries on the Schr?dinger equations for the quantum Painlevé equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schr?dinger equations for the quantum Painlevé equations. In other words, since the Schr?dinger equations for the quantum Painlevé equations are equivalent to the Knizhnik?CZamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik?CZamolodchikov equations.     

Similarity Reductions and Painlevé Property of Coupled KdV Equations

Using Ablowitz-Ramani Segur algorithm, the coupled KdV systems are reclassified under the Painlevé integrable sense while the similarity reductions of the model are obtained by using the Clarkson and Kruskal＇s direct method. Some new types of Painlevé integrable models including a model with different dispersion relations for two layer fluids are found.     

Painlevé Integrability of Nonlinear Schrödinger Equations with bothSpace- and Time-Dependent Coefficients

We investigate the Painlevé integrability of nonautonomous nonlinearSchrödinger (NLS) equations with both space- and time-dependent dispersion, nonlinearity, and external potentials. The Painlevé analysis is carried out without using the Kruskal's simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painlevé analyses and/or become unknown new equations.     

Painlevé Test and Higher Order Differential Equations

Abstract

Starting from the second Painlevé equation, we obtain Painlevé type equations of higher order by using the singular point analysis.     

Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials

We study double integral representations of Christoffel–Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its–Izergin–Korepin–Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlevé IV differential equation for each family.     

Painlevé test of the McKean and Carleman models

The Painlevé test of the system of nonlinear partial differential first-order equations u₁+u_k=k₁v²+k₂u²+k₃uv, v₁–v_x=–k₁v²–k₂u²–k₃uv is performed. The system includes the Carleman and McKean models which are caricatures of the Boltzmann equation. For k ₁=k ₂=0 the system describes the interaction of two waves u and v. The results of the Painlevé test are discussed in connection with whether or not the system is integrable. We also study in detail the constraint on (whose vanishing defines a noncharacteristic hypersurface S) which arises at the resonance.     

Rational Solutions of the Master Symmetries¶of the KdV Equation

In this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for KdV. The result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schr?dinger operator. By bispectral potentials we mean that the corresponding Schr?dinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable. This complements certain results of Airault–McKean–Moser [4], Duistermaat–Grünbaum [10], and Magri–Zubelli [40]. As a consequence of bispectrality, the rational solutions of the master symmetries turn out to be solutions of a (generalized) string equation. Received: 28 January 1999 / Accepted: 22 October 1999     

Painlevé analysis and integrability of the trapped ionic system

Integrability in the Painlevé sense of the trapped ionic system in the quadrupole field with superpositions of rotationally symmetric hexapole and octopole fields is studied. Five integrable cases of the system are reported. First Integrals of the planar motion are founded. Confirming three-dimensional integrability of the equations of motion, the third explicit integrals of motion are constructed directly for each case. We carried out a numerical study to observe the regularity and chaotic regions via the Poincaré surface of sections, and corroborate the analytical results.     

Continuous and Discrete Painlevé Equations Arising from the Gap Probability Distribution of the Finite n Gaussian Unitary Ensembles

In this paper we study the gap probability problem in the Gaussian unitary ensembles of \(n\) by \(n\) matrices : The probability that the interval \(J := (-a,a)\) is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke, and Forrester and Witte on this subject, it has been shown that two Painlevé type differential equations arise in this context. The first is the Jimbo–Miwa–Okomoto \(\sigma \) -form and the second is a particular Painlevé IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted as \(\sigma _n(a)\) , \(R_n(a)\) and \(r_n(a)\) . We show that each one satisfies a second order Painlevé type differential equation as well as a discrete Painlevé type equation. In particular, in addition to providing an elementary derivation of the aforementioned \(\sigma \) -form and Painlevé IV we are able to show that the quantity \(r_n(a)\) satisfies a particular case of Chazy’s second degree second order differential equation. For the discrete equations we show that the quantity \(r_n(a)\) satisfies a particular form of the modified discrete Painlevé II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities \(R_n(a)\) and \(\sigma _n(a)\) .     

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000     

and as Vertex Operator Extensions¶of Dual Affine Algebras

We discover a realisation of the affine Lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with “dual” levels (and an auxiliary level-1 algebra). The duality relation between the levels is . We construct the representation of on a sum of tensor products of , , and modules and decompose it into a direct sum over the spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs. Conversely, we show how the representations are constructed from representations. Received: 29 July 1999 / Accepted: 6 February 2000     

The Painlevé Property for Quasihomogenous Systems and a Many-Body Problem in the Plane

We investigate a many-body problem in the plane introduced by Calogero and intensively studied by Calogero, Françoise and Sommacal. An ad hoc complexification transforms the many-body problem to a system of second order autonomous complex equations depending on some complex constants that describe the two-body interactions. We investigate the sets of two-body interaction constants that make the complexified equation have the Painlevé Property, this is, its solutions are given by single-valued meromorphic functions. In this case the original system has only periodic isochronous solutions. We exhibit a family of settings where the system displays this property and show that it is not present in the three- and four-body problems that do not fall within our class. For this, we introduce a necessary condition for the presence of the Painlevé Property in some quasihomogenous systems.     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

Painlevé analysis and integrability of the damped anharmonic oscillator equation

The Painlevé analysis is applied to the anharmonic oscillator equation . The following three integrable cases are identified: (i)C=0,d ²=25A/6,A>0,B arbitrary, (ii)d ²=9A/2,B=0,A>0,C arbitrary and (iii)d ²=−9A/4,C=2B ²/(9A),A<0,C<0,B arbitrary. The first two integrable choices are already reported in the literature. For the third integrable case the general solution is found involving elliptic function with exponential amplitude and argument.