Table of contents

The authors apply the well known Wahlquist-Estabrook prolongation technique to the Kuramoto-Sivashinsky equation. The prolongation algebra turns out to be trivial in the sense that it is commutative. This supports nonintegrability of the equation.

The central charge for the integrable higher-spin SU(N) magnets are calculated by solving numerically their associated nested Bethe ansatz equations.

Gives a homomorphism between the braid group B_n and the symmetry group of a face-centred cubic crystal in a (n+1)-dimensional Euclidean space. This representation suggests a continuous family of other realisations of B_n.

A new realisation of the quantum group SU_q(2) is constructed by means of a q-analogue to the Jordan-Schwinger mapping, determining thereby both the complete representation structure and q-analogues to the Wigner and Racah operators. To achieve this realisation, a new elementary object is defined, a q-analogue to the harmonic oscillator. The uncertainty relation for position and momentum in a q-harmonic oscillator is quite unusual.

The correlation functions of the three-dimensional n-vector model are studied near the large-scale d'-dimensional defect in the limit n to infinity . The model is characterised by the fact that the spin lengths close to the defect are changed with respect to their bulk value. The deviations decay as - lambda /r with the distance r from the defect. It is shown that at the critical point the correlation function exhibits nonscaling behaviour if d'=2 and nonuniversal behaviour if d'=1. The deviation of a critical exponent eta is calculated up to order of lambda .

The local magnetisation of the three-dimensional n-vector model is studied near the large-scale d'-dimensional defect in the limit n to infinity . The model is characterised by the fact that the spin lengths close to the defect are changed as S_r²=n(1- lambda /r) with the distance r from the defect. It is shown that near the critical point the local magnetisation exhibits a nonuniversal behaviour if d'=0, 1 and a nonscaling behaviour if d'=2. In the first case the critical exponents beta ' and nu ' are calculated up to the order of lambda and it is shown that these exponents satisfy the usual scaling law relations up to this order.

The concept of approximate symmetry is introduced. The authors describe all nonlinearities F(u) with which the nonlinear wave equation Square Operator u+ lambda u³+ epsilon F(u)=0 with a small parameter epsilon is approximately scale and conformally invariant. Some approximate solutions of wave equations in question are obtained using the approximate symmetry.

Calculates the end-to-end elastic susceptibility for the bond-bending model on the self-avoiding random walk. The numerical result shows clearly that the central force term is irrelevant when the bond-bending term is present. The result also indicates that when the elastic constant of the bond-bending term is set equal to infinity the elastic network is governed by the conductivity exponent of the random resistor network.

A more efficient algorithm for enumerating rigid clusters is presented. Using this algorithm the author has extended the series of site-rigid animals enumerated by Prunet and Blanc (1986) by five more terms and has also calculated, for the first time, the exponent governing the growth of the radius of gyration for rigid animals.

Renormalisation of a model of classical diffusion in a random potential is analysed. It is shown that at two dimensions the one-loop expression for the anomalous dimension of the diffusion coefficient is perturbatively exact leading to the coupling-dependent value 1/ nu =2+g/4 pi of the exponent nu .

The authors study numerically the spreading of damage in the Euclidean travelling salesman problem. There is a critical temperature T_c approximately=0.20-0.25 below which damage does not spread. They search for multifractal behaviour in the moments of the damage probability distribution. Multifractality is found in the frozen phase when the moments are monitored as a function of time. If they are evaluated with the number of cities, multifractal behaviour occurs in the chaotic phase.

A stochastic iterative technique is used to obtain the origin magnetisation distribution of the site-diluted Ising ferromagnet on a rooted Cayley tree with coordination number z=3. This distribution shows marked morphological changes with temperature, crossing over from a broad distribution near the ferromagnetic phase boundary to a highly structured shape at lower temperatures. The large-scale structure of the intermediate- and low-temperature distributions-a superposition of similarly shaped but scaled peaks-can be understood on the basis of the contributions of clusters distinguished by their structure close to the origin.

A general expression of one class of symmetries and their Lie algebra properties for the Burgers equation is presented.

A new strong symmetry, two groups of symmetries and their Lie algebra properties for the higher-order Burgers equations are presented.

Hyperbolic algebras such as E₁₀ are based on Minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root. In string theory terms, finite Lie, affine Kac-Moody and hyperbolic Kac-Moody algebras are generated by tachyon vertex operators, trachyon plus photon vertex operators, and vertex operators for all mass levels, respectively. The 136 possible hyperbolic Dynkin diagrams between the ranks 3 and 10 are classified and exhibited, completing an earlier enumeration by Kac of the 18 rank 7-10 cases. The rank-2 hyperbolic algebras, infinite in number, have been classified by Lepowsky and Moody (1979).

Following Morris (1977) and Luan Dehuai and Wybourne (1981) a simple method is given for the embedding O_n to S_n of ordinary and spin irreps in both n-dependent notation and in an n-independent reduced notation. Basic spin irrep and ordinary irreps are combined using the properties of Q-functions and raising operators in order to give a complete set of branching rules of O_n to S_n for spin irreps. The modification rules for Q-functions given by Morris are redefined to yield a complete and unambiguous set of rules.

Simple algorithms, which are based on the principle of factorisation of an integer, are proposed to generate the polynomial zeros of degree 2 of the 3-j and 6-j coefficients.

The time-dependent Landau-Ginzburg equation of the form eta _t+D Delta eta =P( eta ) where Delta is the three-dimensional Laplacian operator and P( eta ) is an odd-order polynomial up to fifth order, has been used to describe the kinetics of non-conservative order parameters eta at and near criticality. The symmetry reduction method has been applied to solve this equation when P( eta ) is given by P( eta )=a+b eta +c eta ³+d eta ⁵. This can be used to model both first- and second-order phase transitions which take place with (a not=0) or without (a=0) an external field. When either d not=0 or c not=0, two general cases have been found: (i) a=b=0 or a=b=d=0, where the symmetry group involves translations in (3+1)-dimensional spacetime, rotations in three-dimensional space and a dilation; (ii) otherwise, where the symmetry group involves translations in (3+1)-dimensional spacetime and rotations in three-dimensional space. All the reductions to ODE and the corresponding symmetry variables have been derived. The Painleve-type reduced ODE have been solved exactly while the remaining ones can be analysed numerically or approximately. Physical properties of the obtained solutions have been discussed, including their energies.

Higher-order Lagrangian and Hamiltonian systems (time dependent or independent) are interpreted in terms of Lagrangian submanifolds of sympletic higher-order tangent bundles. The relation between both formalisms is given.

The author discusses similarity reductions and Painleve analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations (1972), both of which arise in several physical applications including shallow water waves. Both equations are thought to non-integrable (i.e. not solvable by inverse scattering) since numerical studies show that the interaction of solitary waves in inelastic. In particular, we determine some new similarity reductions of the symmetric regularised long wave equation. These new similarity reductions are not obtainable using the classical Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method which involves no group theoretical techniques. It is shown that every similarity reduction of both the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations obtained using the classical Lie group method reduces the associated partial differential equation to an ordinary differential equation of Painleve type; whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of Painleve type. It is also shown that neither the symmetric regularised long wave equation nor the modified Benjamin-Bona-Mahoney equation possesses the Painleve property for partial differential equations as defined by Weiss et al. (1983).

The self-trapping of a light particle in a fluid is solved exactly in one dimension when the interaction between the light particle and fluid molecules is purely repulsive. It is shown that there is always at least one self-trapped state in the system. The uniqueness of the self-trapped state is proved for the ideal gas. In contrast, multiple trapped states are possible in the hard rod gas.

A dislocation-loop mechanism for the melting of a discotic liquid crystal is presented. With respect to a previous similar work, the contribution of longitudinal edge dislocations is included. It is explicitly shown that a discotic liquid crystal permeated by an equilibrium density of unbound dislocation loops behaves like a nematic liquid crystal in the so-called N+6 phase. The interaction energy between parallel dislocations is calculated.

Transition probabilities in unit time and probability fluxes are compared in studying the elementary quantum processes-the decay of a bound state under the action of time-varying and constant electric fields. It is shown that the difference between these quantities may be considerable, and so the use of transition probabilities W instead of probability fluxes Pi , in calculating the particle fluxes, may lead to serious errors. The difference between Pi and W is due to the fact that in the formulae for probability fluxes the interference is taken into account between the transition amplitudes whereas in the formulae for transition probabilities there are no interference terms. The quantity W represents the rate of change with time of the population of the energy levels relating partly to the virtual ones. For this reason it cannot be directly measured in experiment. Attention is drawn to the concept of the vacuum background that is treated as a physical medium consisting of virtual particles and as a framework in which the real quantum events occur. The vacuum background is shown to be continuously distorted when a perturbation acts on a system. Because of this the viewpoint of an observer on the physical properties of real particles continuously varies with time. This fact is not taken into consideration in the conventional theory of quantum transitions based on using the notion of probability amplitude. As a result, the probability amplitudes lose their physical meaning. All the physical information on quantum dynamics of a system is contained in the mean values of physical quantities. The existence of considerable differences between the quantities W and Pi permits one in principle to make a choice of the correct theory of quantum transitions on the basis of experimental data.

A path integral method is used to calculate the rate of intramolecular electron-vibration transitions taking into account the influence of the environment as the source of random Gaussian forces for vibrational degrees of freedom. The evaluation of the generating function for the rates of multiquantum transitions is reduced to the solution of some integral equations with the kernels dependent on the random force correlation function. For the Markovian Gaussian random process the exact expression for the generating function is obtained. The manifestation of the dynamical modulation effects of the surroundings fluctuations on tunnel and thermostimulated transitions is investigated.

The non-linear Boltzmann equation in the presence of external forces is considered. Exact inhomogeneous homoenergetic solutions are found by means of a nonisotropic generalisation of Nikolskii's transform (1964). In particular, an interesting oscillating behaviour is obtained in the presence of an external time-dependent force. For small values of the vorticity this oscillation becomes a sharp pulsation.

The second virial coefficient for a gas of anyons is computed (i) by discretising the two-particle spectrum through the introduction of a harmonic potential regulator and (ii) by considering the problem in the continuum directly through heat kernel methods. In both cases the result of Arovas et al. (1985) is recovered.

The author applied the generalised master equation (GME) to the analysis of the object function, s(t), which describes the motion of the tunnelling particle in a double-well system coupled to a reservoir consisting of independent bosonic-type elementary excitation. First, using the 'unrelaxed' initial condition and a suitable projection operator we give the exact formal solution of the GME. Then, the memory operator in the GME is explicitly calculated in the weak-tunnelling regime and the function s(t) is obtained. An independent derivation is given which enables us to obtain the exact expansion of the function s(t) by means of a purely algebraic method. This expansion forms a basis on which this GME method and the well known functional-integral approach are compared: the GME-weak-tunnelling approximation is shown to be identical to the commonly used non-interacting-blip approximation ensuing from the functional-integral method. A new approximation is analysed which is weaker than the non-interacting-blip one.

Physical systems with a finite number of connected states are considered. The authors assume the transition rates between the different states are age dependent. The time evolution of the system is described in terms of an age-state probability density. They prove the validity of a general H-theorem, which shows that the age-state probability density evolves towards a time-persistent form. They try to extend the results to hybrid systems, comprising 'jump' and 'non-jump' stochastic processes.

The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.

The authors study continuum percolation and aggregation in binary mixtures of strongly interacting particles. The clustering and connectivity behaviour of the dispersed particles are determined using the Ornstein-Zernike integral equation in the Percus-Yevick (PY) integral equation. Specifically, they consider a binary mixture of spheres in which the interactions between like species are strongly attractive, while the interaction between unlike species is purely repulsive. They model this system through hard-core square-well (SW) potentials, which they approximate by the adhesive hard-sphere model, which yields analytic solutions in the PY theory.

The biaxially next-nearest-neighbour Ising model (BNNNI) is studied using the Monte Carlo procedure. The structure factor is used to identify the possible phases of the system. At intermediate temperatures, evidence for the presence of an additional incommensurate phase with a varying wavevector between the disordered and antiphase states is given.

The author shows that statistical mechanical models, such as the q-state Potts model, whose transfer matrix, may be written in terms of the Hecke algebra H_n(q), respond to block spin scale changes at a certain critical point in a way which may be interpreted as an algebra homomorphism. Part of the structure of the algebra, and hence part of the transfer matrix spectrum of the model, is preserved provided q=4 cos²( pi /r) (r integer). These partial realisations of scale invariance are of a distinct type for each r, characterised in some cases by an endomorphism of the algebra.

Classical homogeneous nucleation theory assumes a constant temperature for droplets of arbitrary size. However, the temperature of a droplet is bound to fluctuate due to both the release of latent heat upon absorption of further molecules, and the collisions with the surrounding gas molecules. The authors examine the effects of such fluctuations upon the nucleation rate and conclude that the corrections could be substantial, in contrast to the results of a previous study.

Important thermodynamic heat engine cycles can be regarded as special cases of a more universal 'generalised' cycle. For specific choices of a continuously variable parameter, this generalised cycle reduces to the Carnot, Otto, Joule-Brayton, Diesel and other known cycles. Of particular interest is the thermal efficiency when characteristic temperatures between the highest and lowest operating temperatures (T₊ and T_-) are chosen to maximise the work output per cycle. This maximum-work efficiency is found to be equal to, or to be well approximated by, the Curzon-Ahlborn efficiency, eta _CA identical to 1-(T_-/T₊)^1/2 for a broad spectrum of cycles and temperatures.

The energy spectra of the one-dimensional Bose gas and Heisenberg chain in an external field (chemical potential in the case of the Bose gas and magnetic field in the case of the Heisenberg chain) are calculated. It is found that, at general values of the external field, the spectra are not of the form expected on the basis of conformal invariance. The conformal structure can be recovered only if some extra conditions are imposed on the size of the system and the external field. It is argued that these additional conditions must be satisfied when taking the continuum limit in order to arrive at a conformally invariant theory.

The authors present a heuristic matching algorithm for the generation of ground states of the short-range +or-J spin glass in two dimensions. It is much faster than previous heuristic algorithms. It achieves near optimal solutions in time O(N) in contrast to the best known exact algorithm which needs a time of O(N⁵2/). From simulations with lattice sizes of up to 210*210 they confirm a phase transition at p=0.105 but they cannot confirm a proposed second transition near p=0.15.

The initial-value problem is studied for two models of gauge field theory in the presence of arbitrary external sources. The non-Abelian SU(2) Yang-Mills theory with a vanishing external charge J₀^a=0 allows for a clear distinction of the gauge field degrees of freedom into dynamical and non-dynamical ones. The case of general external sources J_mu ^a introduces new dynamical quantities-Lagrange multipliers Q^a. Thus a modified Lagrangian has been taken as a starting point. A similar analysis is carried out for an Abelian model of scalar electrodynamics. In all three cases time evolution along classical equations of motion imposes no restriction on external sources and the dynamical degrees of freedom can take almost arbitrary initial values.

The author reviews the equivalence, or the difference, of the two possible force laws in magnetostatics for interacting particles: namely the Ampere and Lorentz force laws. He shows that these two laws are mathematically different and physically correct since they apply to two different systems: the former must be used for a closed system and the latter for an open system.

The asymptotic-energy limit of the density P(S) of spacings between adjacent levels of the two-dimensional harmonic oscillator (TDHO) spectrum is studied. It is shown that in any integer segment (M, M+1), containing approximately M/α levels, of the TDHO spectrum m + αn, P(S) has the form Σ w_i δ(S - S_i) where i takes on at most three values. For large M, P(S) displays strong level repulsion for irrational α, but it does not settle on a stationary form nor does its average over M. This is in marked contrast with the behaviour of generic integrable systems for which the Poisson statistics, P(S)= exp(-S), is known to apply.

Methods of solving the equation for generalised spheroidal eigenfunctions and eigenvalues are considered. An efficient variational method with Jacobi basis functions is worked out. Simple recurrence formulae are derived for analytical calculations of all matrix elements required to solve the variational problem. A fast algorithm for computing generalised spheroidal eigenvalues is proposed.

The author discusses models where particles are added one by one to sites in the one-dimensional lattice. Each site may contain a finite or infinite number of particles. The percolation density depends on the size of the lattice and maximum occupation number of the sites. The author derives site and particle cluster distributions in the case of infinite occupation sites.