Table of contents

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `One hundred years of Painlevé VI, the Fuchs–Painlevé equation'. The motivation behind this special issue is to celebrate the centenary of the discovery of this famous differential equation. The Editorial Board has invited P A Clarkson, N Joshi, M Mazzocco, F W Nijhoff and M Noumi to serve as Guest Editors for the issue.

The nonlinear ordinary differential equation, which is nowadays known as the Painlevé VI (PVI) equation, is one of the most important differential equations in mathematical physics. It was discovered 100 years ago by Richard Fuchs (son of the famous mathematician Lazarus Fuchs) and reported for the first time in Comptes Rendus de l'Academie des Sciences Paris141 555–8 (1905). Gambier, in his seminal paper of 1906, included this equation as the top equation in the list of what are now known as the six Painlevé transcendental equations. The Painlevé list emerged from the work on the classification of all ordinary second-order differential equations whose general solution are `uniform', in the sense that there are no movable (i.e. as a function of the initial data) singularities (meaning branch points) worse than poles. The latter is known as the Painlevé property.

As the top equation in the Painlevé list of transcendental equations, the importance of PVI can be appreciated by recognizing that this is a universal differential equation, which is the most general (in terms of number of free parameters) of the known second order ODEs defining nonlinear special functions. As such, parallels can be drawn between the role played by PVI transcendents in the nonlinear case and the hypergeometric functions at the linear level. In fact, the monograph From Gauss to Painlevé by K Iwasaki, H Kimura, S Shimomura and M Yoshida (Vieweg, 1991), draws very clearly the line stretching over more than 150 years of special function theory in which PVI is placed as the key equation. In recent years these lines have been extended into the discrete domain, i.e. the field of nonlinear ordinary difference equations, and discrete analogues of PVI have opened entirely new fields of investigation.

The aim of the special issue, dedicated specifically to the PVI equations and its avatars rather than to general Painlevé theory, is to consolidate the state-of-the-art knowledge of the properties of this equation and to highlight modern developments (including generalizations of PVI, such as the Garnier system, as well as discrete versions of the equation). The issue should be a repository of high-quality original research papers as well as some invited topical reviews.

Scope of the special issue

The special issue is dedicated to the study of the Painlevé VI equation, its solutions and properties, and to its generalizations—either in the direction of higher-order differential equations associated with PVI (and related Garnier and Schlesinger systems), or in the direction of difference analogues of the equation. The special issue will welcome contributions that go into the analysis (including asymptotic theory) of Painlevé VI transcendents, the corresponding monodromy theory, the representation theory aspects, the underlying algebraic geometry of the solution manifolds, associated combinatorics and random matrix theory, as well as q-difference and discrete versions of the equation, and last but not least applications in physics.

Papers dealing primarily with Painlevé equations other than PVI, or with general Painlevé theory, are not encouraged as these would deflect the contents of the special issue from its specific celebrational motivation.

Editorial policy

All contributions to the special issue will be refereed in accordance with the refereeing policy of the journal. The Guest Editors will reserve the right to judge whether a contribution fits the scope of the topic of the special issue.

Guidelines for preparation of contributions

• We aim to publish the special issue in the first half of 2006, in order not to lose the connection with the celebrational year 2005, marking the 100-year anniversary of the discovery of PVI. To realize this, the deadline for contributed papers will be 31 January 2006.

• There is a page limit of 16 printed pages (approximately 9600 words) per contribution. For submitted papers exceeding this length the Guest Editors reserve the right to request a reduction in length. Further advice on document preparation can be found at www.iop.org/Journals/jphysa

• Contributions to the special issue should if possible be submitted electronically by web upload at www.iop.org/Journals/jphysa, or by email to jphysa@iop.org, quoting `J. Phys. A Special Issue: Painlevé VI'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the website for further information on electronic submissions.

• Authors unable to submit electronically may send hard-copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing electronic code on floppy disk if available and quoting `J. Phys. A Special Issue: Painlevé VI'.

• All contributions should be accompanied by a read-me file or covering letter giving the postal and email address for correspondence. The Publishing Office should be notified of any subsequent change of address.

The special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

We present high-precision Monte Carlo data for the phase diagram of a two-species driven diffusive system, reminiscent of traffic across a narrow bridge. Earlier studies reported two phases with broken symmetry; the existence of one of these has been the subject of some debate. We show that the disputed phase disappears for sufficiently large systems and/or sufficiently low bulk mobility.

We theoretically analyse the defect modes generated by equispaced twist defects in 1D helical (cholesteric-like) structures within their frequency gap which is such that only the first two of the four eigenwaves 1^±, 2^± are exponentially attenuated. n₀ identical defects generate n₀ different defect modes, each one represented by a linear combination of the four eigenwaves. The components 1⁺ and 1⁻ are by far the dominant ones and they are localized near the defect planes. We give exact analytic expressions for the elements of the transfer and scattering matrices of the defect planes, for the functions defining the defect mode when n₀ = 1, and for the defect frequencies when n₀ = 1, 2, 3. In the particular case n₀ = 2 and twist angle θ = π/2, the difference between the two defect wavelengths λ_d2, λ_d1 depends exponentially on the distance z₁ between the defect planes, going to zero for z₁ → ∞ and becoming as large as the entire frequency gap for z₁ → 0.

Consider a homogeneous fluid membrane described by the Helfrich–Canham energy, quadratic in the mean curvature of the membrane surface. The shape equation that determines equilibrium configurations is fourth order in derivatives and cubic in the mean curvature. We introduce a Hamiltonian formulation of this equation which dismantles it into a set of coupled first-order equations. This involves interpreting the Helfrich–Canham energy as an action; equilibrium surfaces are generated by the evolution of space curves. Two features complicate the implementation of a Hamiltonian framework. (i) The action involves second derivatives. This requires treating the velocity as a phase-space variable and the introduction of its conjugate momentum. The canonical Hamiltonian is constructed on this phase space. (ii) The action possesses a local symmetry—reparametrization invariance. The two labels we use to parametrize points on the surface are themselves physically irrelevant. This symmetry implies primary constraints, one for each label, that need to be implemented within the Hamiltonian. The two Lagrange multipliers associated with these constraints are identified as the components of the acceleration tangential to the surface. The conservation of the primary constraints implies two secondary constraints, fixing the tangential components of the momentum conjugate to the position. Hamilton's equations are derived and the appropriate initial conditions on the phase-space variables are identified. Finally, it is shown how the shape equation can be reconstructed from these equations.

We present a model of polydisperse granular mixture with a power-law size distribution subjected to Brownian forces. The driven inelastic mixture which has n components can be regarded as 'granular gas'. The stationary state where the mixture reaches is the result of the balance between the dissipation and random forces which inject energies. We have studied the nonequilibrium properties of the system by means of Monte Carlo method. When the typical relaxation time τ of the driving Brownian process is greater than the mean collision time τ_c, the velocity probability distribution strongly deviates from the Gaussian one and the system has a strong spatial clustering. We define the partial granular temperature and the global granular temperature of the multi-component mixture. The partial granular temperature is the average kinetic energy per particle of the same component and the global one is the statistic average value per partial granular temperature of the mixture. One of the most useful properties of the temperature, i.e., the independence from the thermal substance, is lost. We introduce a fractal dimension D as a measurement of the inhomogeneity of the size distribution of the mixture. Our simulations represent that the deviation and clusterization become more and more pronounced with the increasing value of the fractal dimension D. The global granular temperature decreases, and dissipated energy per particle increases as D augments.

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a three-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology—illustrating the onset in a deterministic context of irregular motions—is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

We study the degree of order and chaos in the anisotropic Kepler problem. The Lyapunov characteristic number (LCN) is constant for all orbits in the chaotic domain with a fixed mass ratio µ. It increases from zero as the mass ratio µ increases from µ = 1, reaches a maximum for µ = 4 and then decreases slowly to zero for a very large µ. We found the characteristics of many families of the periodic orbits of multiplicities up to 17. Several families have a stable part for µ close to µ = 1 (namely, µ < 1.748). We conjecture that there are stable orbits of high multiplicities for larger values of µ. The stable orbits are surrounded by islands of stability consisting of closed invariant curves. The rotation number along these invariant curves decreases from the periodic orbits outwards. Higher-order islands are enclosed in every main island around a low-order periodic orbit. In the case µ = 1.1, we find a large number of islands of multiplicities from 3 up to 17. Then we calculate the asymptotic curves from unstable periodic orbits. These curves go through the collision orbits but continue beyond these points. The intersections of the stable and unstable asymptotic curves are homoclinic and heteroclinic points. The corresponding orbits are the main characteristics of chaos.

Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace–Beltrami operators for pairs of flat tori in with n ⩾ 4. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the case of isospectral flat tori in four dimensions—where a four-parameter family of isospectral pairs is known—we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count. Thus one can count the shape of a drum (if it is designed as a flat torus in four dimensions).

The permutation associated with the decimal expression of the binary reflected Gray code with N bits is considered. Its cycle structure is studied. Considered as a set of points, its self-similarity is pointed out. As a fractal, it is shown to be the attractor of an IFS. For large values of N the set is examined from the point of view of time series analysis.

The Toda lattice hierarchy with self-consistent sources and their Lax representation are derived. We construct a forward Darboux transformation (FDT) with arbitrary functions of time and a generalized forward Darboux transformation (GFDT) for Toda lattice with self-consistent sources (TLSCS), which can serve as a non-auto-Bäcklund transformation between TLSCS with different degrees of sources. With the help of such DT, we can construct many types of solutions to TLSCS, such as rational solution, solitons, positons, negetons, and soliton–positons, soliton–negatons, positon–negatons etc, and study properties and interactions of these solutions.

In this paper, we derive a formula for the ratio of the ζ-determinants of the Laplacian with Neumann and Dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the ζ-determinant of the Dirichlet to Neumann map.

We study Hamiltonians with point interactions in spaces of vector-valued functions. Using some information from the theory of quantum graphs, we describe a class of the operators which can be reduced to the direct sum of several one-dimensional problems. It shown that such a reduction is closely connected with the invariance under channel permutations. Examples are provided by some 'model' interactions, in particular, the so-called δ, δ' and the Kirchhoff couplings.

Two families of series with terms involving spherical spinors (spinor spherical harmonics) for a spin one-half particle, and are summed to closed forms involving the Legendre function of the first kind and its derivatives with respect to an argument and/or an index. In particular, it is shown that the closed forms of the series Γ_±(0; n, n') are remarkably simple: and with I denoting the 2 × 2 unit matrix and σ standing for the vector of Pauli matrices. Integral eigenvalue equations solved by the spherical spinors, with either Γ₊(κ; n, n') or as kernels, are provided. It is pointed out that Γ₊(κ; n, n') is the Green function of the operator −σ ⋅ Λ − (κ + 1)I, (κ ≠ ±1, ± 2, ...), while is the generalized (reduced) Green function of the operator −σ ⋅ Λ − (k + 1)I, (k = ±1, ± 2, ...), with Λ = −ir × ∇.

Starting with a given generalized boson algebra known as the bosonized version of the quantum super-Hopf algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra . With both the Hopf algebras being finitely generated, we produce a closed form expression of the universal matrix that caps the duality and generalizes the familiar exponential map relating a Lie algebra with its corresponding group. Subsequently, using an inverse Mellin transform approach, the coherent states of single-node systems subject to the symmetry are found to be complete with a positive-definite integration measure. The nonclassical coalgebraic structure of the algebra is found to generate naturally entangled coherent states in bipartite composite systems.

We investigate the structure of SO(3)-invariant quantum systems which are composed of two particles with spins j₁ and j₂. The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P_J onto the eigenspaces of the total angular momentum J, and by certain invariant operators Q_K which are built out of spherical tensor operators of rank K. It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6-j symbols. The operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the Q_K-representation. These results are employed to obtain a complete characterization of spin systems with j₁ = 1 and arbitrary j₂ ⩾ 1. We prove that the Peres–Horodecki criterion of positive partial transposition (PPT) is necessary and sufficient for separability if j₂ is an integer, while for half-integer spins j₂ there always exist entangled PPT states (bound entanglement). We construct an optimal entanglement witness for the case of half-integer spins and design a protocol for the detection of entangled PPT states through measurements of the total angular momentum.

We construct a class of 3 ⊗ 3 entangled edge states with positive partial transposes using indecomposable positive linear maps. This class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes.

We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R²ⁿ, but its infinite-dimensional analogue. Here the phase space Ω = H × H, where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f:Ω → R. The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion ≈h. The quantum statistical model is obtained as the lim_h→0 of the classical one. All quantum states including so-called 'pure states' (wavefunctions) are simply Gaussian fluctuations of the 'vacuum field', ω = 0 ∊ Ω, having dispersions of the Planck magnitude.

A family of N-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the 3-tangle and some of the four-, five- and N-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form with L = 2^N−n ⩾ l = 2ⁿ. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L, l) of l-planes in C^L that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the Plücker embedding. The invariants are neatly expressed in terms of the Plücker coordinates of the Grassmannians.

We analyse the dynamics of single- and two-particle states in Ising-type networks. The mutual entanglement is quantified using the concept of concurrence. We derive explicit expressions for the concurrence for single- and two-particle initial states in arbitrary passive networks and specify the result for Ising-type networks. We show how to design a network to prepare a prescribed pattern of entanglement for one excitation and study the maximum attainable entanglement for passive optical networks in general. The effect of network randomization on the average entanglement is also studied.

We consider a two-dimensional quantum control system evolving under an entropy-increasing irreversible dynamics in the semigroup form. Considering a phenomenological approach to the dynamics, we show that the accessibility property of the system depends on whether its evolution is assumed to be positive or completely positive. In particular, we characterize the family of maps having different accessibilities and show the impact of that property on observable quantities by means of a simple physical model.

A recently developed algebraic approach for constructing coherent states for solvable potentials is used to obtain the displacement operator coherent state of the Pöschl–Teller potential. We establish the connection between this and the annihilation operator coherent state and compare their properties. We study the details of the revival structure arising from different time scales underlying the quadratic energy spectrum of this system.

We discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical self-coupling, and for low temperatures, it has been argued that the configuration space for vortex dynamics in each topological class of the Abelian Higgs model approximately truncates to a finite-dimensional moduli space with a Kähler structure. For the case where the vortices live on a 2-sphere, we explain how localization formulae on the moduli spaces can be used to compute exactly the partition function of the vortex gas interacting with a background potential. The coefficients of this analytic function provide geometrical data about the Kähler structures, the simplest of which being their symplectic volume (computed previously by Manton using an alternative argument). We use the partition function to deduce simple results on the thermodynamics of the vortex system; in particular, the average height on the sphere is computed and provides an interesting effective picture of the ground state.

A recent method to obtain short-time propagators for finding path-integral solutions of Fokker–Planck equations is applied here to numerically solve the non-linear kinetic Fokker–Planck equation in plasma physics. Furthermore, we extend the use of this method to solve non-homogeneous equations. Cylindrical geometry in velocity space is used and two-species plasma is considered with no linearization of the exact conservative collisional operator. Numerical singularities in the diffusion tensor determinant are avoided by the splitting of the collisional operator into two parts, each one leading to different multiplicative integral operators which describe electron–electron and electron–ion interactions separately. The accurate advancing path-integral numerical formalism preserves conservative physical properties making this procedure a promising alternative to the classical linearized collisional operators used in kinetic theory. Here, we show the feasibility of the method by giving a new calculation of Spitzer's transport coefficients.