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.

The solutions of Maxwell equations with electric and magnetic fields existing only in a finite part of space are given, due to massless charges moving with speed of light. Under proper conditions, there is no interaction between charges and the field.

The bounded positive- and negative-parity spectrum E^(±)_n of the symmetric, one-dimensional, finite quantum square-well potential is computed exactly, explicitly and analytically in the form E^(±)_n = f^(±)(n), where f^(±) are known functions.

Starting from the continuous time random walk (CTRW) scheme with the space-dependent waiting-time probability density function (PDF) we obtain the time-fractional diffusion equation with varying in space fractional order of time derivative. As an example, we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non-trivial drift and subdiffusion whose laws are changed in the course of time.

In this letter, we use the newly available explicit multi-soliton and multi-cuspon solutions of the Camassa–Holm equation to study the interactions of a soliton and a cuspon, two cuspons and two solitons. Some interesting phenomena are found, e.g., a larger soliton can 'eat up' a smaller cuspon during the collison and it is the other way round if the amplitude of the cuspon is larger. It is also found that a soliton and a cuspon can emerge from an almost zero mass. The interaction of two cuspons is also investigated in detail for the first time.

A previously reported simple method for calculating complex matrix eigenvalues is modified to incorporate the traditional HEG approach for the case of even parity potentials. Two examples of resonance calculations are given. Our matrix and perturbation results agree with each other, but are not in full accord with previously published results for one of the test potentials. New results are given for the resonances of the inverted Gaussian potential.

We study a restricted class of self-avoiding walks (SAWs) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] × [0, L] on the square lattice . The number of distinct walks is known to grow as . We estimate λ = 1.744 550 ± 0.000 005 as well as obtaining strict upper and lower bounds, 1.628 < λ < 1.782. We give exact results for the number of SAWs of length 2L + 2K for K = 0, 1, 2 and asymptotic results for K = o(L^1/3). We also consider the model in which a weight or fugacity x is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For x < 1/μ the average length of a SAW grows as L, while for x > 1/μ it grows as L². Here μ is the growth constant of unconstrained SAWs in . For x = 1/μ we provide numerical evidence, but no proof, that the average walk length grows as L^4/3. Another problem we study is that of SAWs, as described above, that pass through the central vertex of the square. We estimate the proportion of such walks as a fraction of the total, and find it to be just below 80% of the total number of SAWs. We also consider Hamiltonian walks under the same restriction. They are known to grow as on the same L × L lattice. We give precise estimates for τ as well as upper and lower bounds, and prove that τ < λ.

Using a form factor approach, we define and compute the character of the fusion product of rectangular representations of . This character decomposes into a sum of characters of irreducible representations, but with q-dependent coefficients. We identify these coefficients as (generalized) Kostka polynomials. Using this result, we obtain a formula for the characters of arbitrary integrable highest weight representations of in terms of the fermionic characters of the rectangular highest weight representations.

In the present work, we establish a simple relation between the Dirac equation with a scalar and an electromagnetic potential in a two-dimensional case and a pair of decoupled Vekua equations. In general, these Vekua equations are bicomplex. However, we show that the whole theory of pseudoanalytic functions without modifications can be applied to these equations under a certain nonrestrictive condition. As an example we formulate the similarity principle which is the central reason why a pseudoanalytic function and as a consequence a spinor field depending on two space variables share many of the properties of analytic functions. One of the surprising consequences of the established relation with pseudoanalytic functions consists in the following result. Consider the Dirac equation with a scalar potential depending on one variable with fixed energy and mass. In general, this equation cannot be solved explicitly even if one looks for wavefunctions of one variable. Nevertheless, for such Dirac equation, we obtain an algorithmically simple procedure for constructing in explicit form a complete system of exact solutions (depending on two variables). These solutions generalize the system of powers 1, z, z², ... in complex analysis and are called formal powers. With their aid any regular solution of the Dirac equation can be represented by its Taylor series in formal powers.

The simple Lie algebra usp(4) ≃ so(5) is a ten-dimensional algebra that contains the angular momentum algebra su(2). In nuclear structure physics, the algebra so(5) describes beta-rigid collective modes in the Bohr–Mottelson and interacting boson models. The so(5) dual space consists of density matrices which are defined by the expectations of so(5) generators. A coadjoint orbit is a common level surface in the dual space of the two so(5) Casimirs. This paper develops mean field theory on any coadjoint orbit of so(5) densities. When the densities of a set of quantum states lie on one orbit, the system is said to have a weak dynamical symmetry. A Lax pair determines the dynamics of so(5) densities on each coadjoint orbit. Analytic solutions are reported for rotating so(5) densities in equilibrium for a particular energy function.

We construct a noncommutative extension of the U(N) principal chiral model with Wess–Zumino term and obtain an infinite set of local and non-local conserved quantities for the model using the iterative procedure of Brezin et al (1979 Phys. Lett. B 82 442). We also present the equivalent description as a Lax formalism of the model. We expand the fields perturbatively and derive zeroth- and first-order equations of motion, zero-curvature condition, iteration method, Lax formalism, local and non-local conserved quantities.

We deduce a class of non-Markovian completely positive master equations which describe a system in a composite bipartite environment, consisting of a Markovian reservoir and additional stationary unobserved degrees of freedom that modulate the dissipative coupling. The entanglement-induced memory effects can persist for arbitrary long times and affect the relaxation to equilibrium, as well as induce corrections to the quantum-regression theorem. By considering the extra degrees of freedom as a discrete manifold of energy levels, strong non-exponential behaviour can arise, such as for example power law and stretched exponential decays.

A classical theorem of Stone and von Neumann states that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner–Moyal transform, we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase-space Schrödinger equation.

Within a special multi-coin quantum walk scheme, we analyse the effect of the entanglement of the initial coin state. For states with a special entanglement structure, it is shown that this entanglement can be measured with the mean value of the walk, which depends on the i-concurrence of the initial coin state. Further, the entanglement evolution is investigated and it is shown that the symmetry of the probability distribution is reflected by the symmetry of the entanglement distribution.

By using the Lewis–Riesenfeld quantum invariant theory and properly choosing Hermitian invariant operator, a closed solution of the Schrödinger equation is derived for two forced quantum oscillators with mixing of two modes, and the quantum fluctuations in the output fields are evaluated. For the initial two-mode squeezed number or squeezed coherent state, in some particular conditions, the time evolution of the oscillators can not only preserve the initial two-mode squeezing, but also produce squeezing in the individual modes; and exhibit a periodical squeezing behaviour. For the initial two-mode number state or coherent state, there is no squeezing in the individual and mutual quadrature phases of the two-mode fields. Furthermore, regardless of which state above being initially considered, the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters. In particular, for the initial two-mode coherent state, the variances of the output fields are also independent of other parameters in the Hamiltonian, and always preserve their initial values 1/4.

We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the Poincaré disc under a perpendicular uniform magnetic field. The corresponding coherent state transforms enable us to connect, by an integral transform, spaces of bound states of the particle with the space of square integrable functions on the real line. The established connection provides us with a new way to obtain hyperbolic Landau states.

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V(x) = 1/x², where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

An algebraic approach is employed to formulate N = 2 supersymmetry transformations in the context of integrable systems based on loop superalgebras , with homogeneous gradation. We work with extended integrable hierarchies, which contain supersymmetric AKNS and Lund–Regge sectors. We derive the one-soliton solution for p = 1 which solves positive and negative evolution equations of the N = 2 supersymmetric model.

We present numerical results for chains of SU(2) BPS monopoles constructed from Nahm data. The long chain limit reveals an asymmetric behaviour transverse to the periodic direction, with the asymmetry becoming more pronounced at shorter separations. This analysis is motivated by a search for semiclassical finite temperature instantons in the 3D SU(2) Georgi–Glashow model, but it appears that in the periodic limit the instanton chains either have logarithmically divergent action or wash themselves out.

Reductions of the self-dual Yang–Mills (SDYM) system for ∗-bracket Lie algebra to the Husain–Park (HP) heavenly equation and to sl(N, C) SDYM equation are given. An example of a sequence of su(N) chiral fields (N ⩾ 2) tending for N → ∞ to a curved heavenly space is found.

We construct a supersymmetrized version of the model to the radiation damping introduced by the present authors (Mendes, Neves, Oliveira and Takakura 2005 Preprint hep-th/0503135). We discuss its symmetries and the corresponding conserved Noether charges. It is shown that this supersymmetric version provides a supersymmetric generalization of the Galilei algebra of the model. We have shown that the supersymmetric action can be split into dynamically independent external and internal sectors.

In this paper, the radiation force per length resulting from a plane standing wave incident on an infinitely long cylindrical shell is computed. The cases of elastic and viscoelastic shells immersed in ideal (non-viscous) fluids are considered with particular emphasis on their thickness and the content of their interior hollow spaces. Numerical calculations of the radiation force function Y_st are performed. The fluid-loading effect on the radiation force function curves is analysed as well. The results show several features quite different when the interior hollow space is changed from air to water. Moreover, the theory developed here is more general since it includes the results on cylinders.