Table of contents

Pairs of coupled nonlinear evolution equations are considered. Factorisation of the L-operator that relates the equations in such pairs may be realised by the operators of creation and annihilation of solitons, and yields the unitary scattering operator for the processes of collision of arbitrary wave disturbances with solitons as well as new forms of the Backlund transformation.

The spectrum for a Gaussian orthogonal ensemble of random matrices augmented by a pairing interaction, which has been given by Edwards and Jones (1976) and by Jones, Kosterlitz and Thouless (1978), is derived in other ways. Several extensions are given and relationships to other problems of current interest discussed. Stress is laid on the importance of the dominance of binary Hamiltonian correlations in the moments which define the density and relevant correlation functions.

The technique of differential form and exterior calculus has been applied to the analysis of the prolongation structure of the nonlinear equations of self-induced transparency. The process of prolongation yields a completely different scheme of inverse scattering for these equations.

Reflection of waves from an asymmetric stratified medium is studied, whose refractive index obeys a power-law on both sides of a resonance point. Reflection is enhanced due to the asymmetry of the medium but the effect becomes weaker with increasing order of the discontinuity.

Contrary to the implications of a recent paper by Allcock and Hall (1978), spin ³/₂ may be coupled consistently to electromagnetism if gravity is also included; likewise, the gauge variance of the free spin-³/₂ stress tensor is no hindrance to consistent gravitational coupling.

A simple experiment is proposed, with the aid of which it is suggested that the Earth's absolute velocity may be measured.

The quantum open system weakly coupled to thermal reservoirs at different temperatures and under the influence of slowly varying external conditions is studied. The famous Carnot inequality for the efficiency of any heat engine is obtained.

Ultrafine smoke particles stick together to form chain-like aggregates. It is found that the particle density has long-range correlations of the same form in iron, zinc of silicon dioxide aggregates. The correlation data suggest a power-law spatial dependence giving a Hausdorff dimension between 1.7 and 1.9. The authors discuss the consistency of these results with a model based on percolation. They also compare their results with a random-walk model, which has a nominal Hausdorff dimension of 2.

Exact values of the numbers of connected clusters of n sites, each site having valence no larger than v, are presented for the triangular lattice for v=2,3,4 and 5 for small values of n. Assuming a plausible asymptotic form for the dependence of these numbers on n and v the authors show, using series analysis techniques, that the exponent characterising the dominant singularity in the generating function has the same value for all v>or=3 but a different value for v=2.

It is proved that the interface between two coexisting pure phases in the classical lattice gas is not localised by a vanishing phase-separating field whenever the free interface is diffuse. This result is obtained for nearest neighbour, attractive interactions.

The structure of the interface between phases in the spherical model of a ferromagnet is solved exactly.

The authors derive a semiclassical formula for the Wigner function W(q,p,t) describing the evolution in the two-dimensional phase space qp of a nonstationary quantum state psi (q,t) for a system with one degree of freedom. The initial state psi (q,0) corresponds to a family of classical orbits represented by a curve C₀ in qp. Under the classical motion C₀ evolves into a curve C_t; it is shown that the region where W is large hugs C_t in an adiabatic fashion, and that W has semiclassical oscillations depending only on the geometry of C_t and neighbouring curves.

A common assumption that quantisation is simply a representation of the Lie algebra of classical observables by a Lie algebra of self-adjoint operators in Hilbert space is shown to be generally invalid.

The relativistic wave equations considered in this study are of the form (-i Gamma . delta +m) psi (x)=0, and describe a unique mass m and spin s with 2(2s=1) independent components. Furthermore, their Gamma _mu matrices form a reducible, but not necessarily decomposable, set over the representation space of the Lorentz group. It is shown that such reducible equations are dynamically equivalent to simpler irreducible equations: (-i beta . delta +m) phi (x)=0 where the beta _mu can be constructed from the Gamma _mu of the original equation. The beta _mu form the irreducible 'core' of the original equation. The external field interactions for reducible wave equations in general are also studied. It will be shown that if the external fields do not introduce any new independent components into the equation and if the interactions are made up of the Gamma -matrices contracted over the external potentials, then such interactions can be studied as the same type of interactions of the irreducible beta -equation. Thus the dynamical equivalence of the free Gamma and beta equations extends to most interactions of interest. The results of the study show that the number of theories available which can lead to different physical predictions is significantly restricted.

It is known that while spin-1 (Proca) theory with minimal electromagnetic interaction is marred by the Corben-Schwinger anomaly, the inclusion of specific types of 'anomalous' interactions lead to other difficulties such as non-causality of propagation, and the energy spectrum (in homogeneous magnetic fields of sufficient strength) becoming partially non-real. The authors investigate whether, by making the anomalous terms (introduced into the Duffin-Kemmer-Petiau equation) sufficiently general, it is possible to escape these inconsistencies. The answer, unfortunately, turns out to be in the negative. They comment briefly on other formulations for spin-1.

The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha ₁ lambda ²+O( lambda ²) with alpha ₁ approximately=1.5.

The authors consider a field theory in which the fields from a (2l+1)-dimensional (complex) irreducible representation of SO(3). This theory contains an additional internal discrete symmetry of the 3-state Potts model. There exists only one quadratic and one cubic invariant. There are many quartic invariants, but they are irrelevant below six dimensions as far as the infra-red behaviour is concerned. It is shown that as l to infinity the model becomes soluble. In contrast to the infinite N limit here it is the wavefunction, rather than the coupling constant, that is renormalised in a nontrivial way, as well as phi ². The asymptotic behaviour of the infinite l limit is analysed explicitly, and compared with renormalisation group results.

The laser field is treated as a classical external field and represented as a monochromatic plane-wave field of circular polarisation. The differential cross section for the scattering of unpolarised electrons in this field is calculated and numerically evaluated for the non-relativistic energy region. Because of the complicated kinematics of this process an experimental situation is considered where the incoming electrons have opposite and equal momentum. It is shown that the off resonance cross section in a strong field can differ considerably from the free case in regions accessible to experimentation for various laser and electron parameters. Resonances are found to exist in physical regions. Renormalisation is performed. The resonance cross section, the width and the spacing of the resonances are evaluated. Under specific circumstances the resonance cross section is found to exceed the off-resonance cross section by several orders of magnitude. The experimental conditions for the resonance case, however, seem to be harder to meet than in the off resonance case.

The diffraction of long waves in a rotating channel of constant depth, consisting of an infinite and a semi-infinite parallel barrier of zero thickness, is investigated. The excitation is due to a Kelvin wave travelling towards the open end from inside the channel. An exact solution is given for the region inside the semi-infinite channel, consisting of a 'reflected' Kelvin wave and the various travelling modes, which become attenuated waves when the width of the channel is less than a certain value. MOreover it is shown that there is a resonance effect for some critical values of the channel width. In the remaining region an asymptotic expression for the far field is obtained. This expression involves a Kelvin wave travelling along the infinite barrier and also an outgoing cylindrical wave (Poincare wave) coming from the edge of the semi-infinite barrier of the channel.

The critical isotherm of the Ising model for the four-dimensional simple hypercubic lattice is studied using a high-field expansion through mu ¹⁵. The series for the magnetisation is analysed for singularities of the asymptotic form E(1- mu )^1/3 mod ln(1- mu ) mod ^p, predicted by renormalisation group theory. Good convergence is obtained for values of p around ¹/₃ (the renormalisation group prediction) and we estimate p=0.30+or-0.05. Assuming p=¹/₃, the critical amplitude E is estimated to be 0.896+or-0.02.