Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Geometric Numerical Integration of Differential Equations'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with symplectic or multisymplectic integration; volume-preserving integration; symmetry-preserving integration; integrators that preserve first integrals, Lyapunov functions, or dissipation; exponential integrators; integrators for highly oscillatory systems; Lie-group integrators, etc. Papers on geometric integration of both ODEs and PDEs will be considered, as well as application to molecular-scale integration, celestial mechanics, particle accelerators, fluid flows, population models, epidemiological models and/or any other areas of science. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field.

The Editorial Board has invited G R W Quispel and R I McLachlan to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are the following:

• The subject of the paper should relate to geometric numerical integration in the sense described above.

• Contributions will be refereed and processed according to the usual procedure of the journal.

• Papers should be original; reviews of a work published elsewhere will not be accepted.

The guidelines for the preparation of contributions are as follows:

• The DEADLINE for submission of contributions is 1 September 2005. This deadline will allow the special issue to appear in late 2005 or early 2006.

• There is a strict page limit of 16 printed pages (approximately 9600 words) per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. Further advice on publishing your work in Journal of Physics A: Mathematical and General may 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 e-mail to jphysa@iop.org, quoting `JPhysA Special Issue—Geometric Integration'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the web site 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 the electronic code on floppy disk if available and quoting `JPhysA Special Issue—Geometric Integration'.

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

This 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.

Directed polymers (strings) and semiflexible polymers (filaments) are one-dimensional objects governed by tension and bending energy, respectively. They undergo unbinding transitions in the presence of a short-range attractive potential. Using transfer matrix methods we establish a duality mapping for filaments and strings between the restricted partition sums in the absence and the presence of a short-range attraction. This allows us to obtain exact results for the critical exponents related to the unbinding transition, the transition point and transition order.

A famous inverse problem posed by M Kac 'Can one hear the shape of a drum?' is concerned with isospectrality of drums or planer billiards, and the first counter example was constructed by Gordon, Webb and Wolpert (1992 Invent. Math.110 1). Here we present pieces of numerical evidence showing that 'One can distinguish isospectral drums by measuring the scattering poles of exterior Neumann problems'. This is based on the observation that the Fredholm determinant appearing in the boundary element method admits a factorization into interior and exterior parts.

A physical realization of scattering by -symmetric potentials is provided: we show that the Maxwell equations, for an electromagnetic wave travelling along a planar slab waveguide filled with gain and absorbing media in contiguous regions, can be approximated in a parameter range by a Schrödinger equation with a -symmetric scattering potential.

We investigate propagation of a slow-light soliton in Λ-type media such as atomic vapours and Bose–Einstein condensates. We show that the group velocity of the soliton monotonically decreases with the intensity of the controlling laser field, which decays exponentially after the laser is switched off. The shock wave of the vanishing controlling field overtakes the slow soliton and stops it, while the optical information is recorded in the medium in the form of spatially localized polarization. In the strongly nonlinear regime we find an explicit exact solution describing the whole process.

We present an integral formula for the density matrix of a finite segment of the infinitely long spin- XXZ chain. This formula is valid for any temperature and any longitudinal magnetic field.

The ubiquitous Lipkin model is investigated for an interaction parameter beyond the traditional critical point. It is argued that a phase transition occurs higher up in the spectrum for such larger interaction, where, using appropriate scaling of the energies, the position of the phase transition becomes independent of the particle number. The phase transition is related to near singularities in the complex interaction plane, the exceptional points. Consideration of finite temperature yields the well-known physical features associated with phase transitions.

Equilibrium solutions for a sample of ferroelectric smectic C (SmC*) liquid crystal in the 'bookshelf' geometry under the influence of a tilted electric field will be presented. A linear stability criterion is identified and used to confirm stability for typical materials possessing either positive or negative dielectric anisotropy. The theoretical response times for perturbations to the equilibrium solutions are calculated numerically and found to be consistent with estimates for response times in ferroelectric smectic C liquid crystals reported elsewhere in the literature for non-tilted fields.

In a previous paper (2004 J. Phys. A: Math. Gen.37 9651–68) we have given the Fuchsian linear differential equation satisfied by χ⁽³⁾, the 'three-particle' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the (n − 1)-dimensional integrals representing the n-particle contribution to the isotropic square lattice Ising model susceptibility χ. The integration rules are straightforward due to remarkable formulae we derived for these variables. We obtain without any numerical approximation χ⁽³⁾ as a fully integrated series in the variable w = s/2/(1 + s²), where s = sh(2K), with K = J/kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

We study the dynamics of two-component Bose–Einstein condensates in periodic potentials in one dimension. Elliptic potentials which have the sinusoidal optical potential as a special case are considered. We construct exact nonstationary solutions to the mean-field equations of motion. Among the solutions are two types of temporally periodic solutions—in one type there are condensate oscillations between neighbouring potential wells, while in the other the condensates oscillate from side to side within the wells. Our numerical studies of the stability of these solutions suggest the existence of one-parameter families of stable nonstationary solutions.

We realize the Pauli Hamiltonians (with constant magnetic field ν > 0) on a simply connected Riemann surface M_κ of constant scalar curvature as second-order differential operators acting on differential 1-forms of M_κ. We also study the asymptotic behaviour of some aspects of their L²-spectral properties when the Euclidean limit is taken. More exactly, we show that the L²-eigenprojector kernels on the plane (i.e., κ = 0) corresponding to the Landau levels 8νl; l = 0, 1, ..., can be recovered from the L²-eigenprojector kernels of of the curved Riemann surfaces M_κ, κ ≠ 0, in the limit .

We study quantum systems whose scattering modes are governed by unitary representation of the algebra. The S-matrices of the systems under consideration are defined from intertwining relations for the Weyl equivalent representations of the group SL(2, C) or its Lie algebra.

This paper is the last of a series of three articles presenting a classification of Vornoi and Delone tilings determined by point sets Σ(Ω) ('quasicrystals'), built by the standard projection of the root lattice of type A₄ to a two-dimensional plane spanned by the roots of the Coxeter group H₂ (dihedral group of order 10). The acceptance window Ω for Σ(Ω) in the present paper is a regular decagon of any radius 0 < r < ∞. There are 14 distinct VT sets of Voronoi tiles and 6 sets DT of Delone tiles, up to a uniform scaling by the factor and . The number of Voronoi tiles in different quasicrystal tilings varies between 3 and 12. Similarly, the number of Delone tiles is varying between 4 and 6. There are 7 VT sets of the 'generic' type and 7 of the 'singular' type. The latter occur for seven precise values of the radius of the acceptance window. Quasicrystals with acceptance windows with radii in between these values have constant VT sets, only the relative densities and arrangement of the tiles in the tilings change. Similarly, we distinguish singular and generic sets DT of Delone tiles.

We present an algorithm to reduce the coloured box–ball system, a one-dimensional integrable cellular automaton described by motions of several colours (kinds) of balls, into a simpler monochrome system. This algorithm extracts the colour degree of freedom of the automaton as a word which turns out to be a conserved quantity of this dynamical system. It is based on the theory of crystal basis and in particular on the tensor products of sl_n crystals of symmetric and anti-symmetric tensor representations.

A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of . This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: we show that the generalized Lamé potential possesses four algebraic sectors, and describe a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.

We consider the Fermi quantization of the classical damped harmonic oscillator (dho). In past work on the subject, authors double the phase space of the dho in order to close the system at each moment in time. For an infinite-dimensional phase space, this method requires one to construct a representation of the CAR algebra for each time. We show that the unitary dilation of the contraction semigroup governing the dynamics of the system is a logical extension of the doubling procedure, and it allows one to avoid the mathematical difficulties encountered with the previous method.

A mechanism of spin decoherence caused by spacetime curvature in general relativity is discussed. The spin state of a particle is shown to decohere only if the particle moves in a curved spacetime. In particular, when a particle is near the event horizon of a black hole, an extremely rapid spin decoherence occurs for an observer who is static in a Killing time, however slow the particle's motion.

The Newtonian potential is computed exactly in a theory that is fundamentally non-commutative in the space–time coordinates. When the dispersion for the distribution of the source is minimal (i.e. it is equal to the non-commutative parameter θ), the behaviour for large and small distances is analysed.

We study the principal bifurcation curve of a third-order equation which describes the nonlinear evolution of several systems with a long-wavelength instability. We show that the main bifurcation branch can be derived from a variational principle. This allows us to obtain a close estimate of the complete branch. In particular, when the bifurcation is subcritical, the large amplitude stable branch can be found in a simple manner.