Table of contents

The perturbed Kadomtsev-Petviashvili equation is analysed for media with a split dispersion law. For arbitrary perturbations the method of asymptotic description of the equation solutions is constructed. Applications of this method are given for problems in hydrodynamics and plasma physics.

The author considers the consequences of Q-reducibility of a point group for the structure of a corresponding space group. First he shows that the space group is a subgroup of a direct product of two Euclidean groups of lower dimensions. He introduces the so-called separation diagram from which follows that (i) the space group is a subdirect product of space groups of lower dimensions, (ii) its point group is a subdirect product of point groups of lower-dimensional groups, (iii) the space group has two complementary normal translation subgroups of lower dimensions and the corresponding factor groups are complementary subperiodic groups. The latter conclusion enables us to classify space groups into finer classes than the point classes. The most important fact is that the space groups in higher dimensions can be constructed from Q-irreducible groups of lower dimensions unless they are themselves Q-irreducible.

The authors have analysed the types of non-linear coupled Klein-Gordon equations possessing non-trivial Lie-Backlund vector fields. The Dodd-Bullough equation occurs as a special case of their system. These systems have the special feature of having the lowest symmetry generator depending upon the fifth derivative of the fields.

The asphericity, (A), of a random walk is defined and calculated for non-self-avoiding walks. The definition of (A) generalises to self-avoiding walks, percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects.

A stochastic numerical scheme is used to simulate radial motion of a fluid interface through a two-dimensional porous medium. The point at which the interface moves is chosen with a probability proportional to the local pressure gradient. The interface can intersect itself, enclosing islands of the original fluid. The number of islands of area s is found to scale approximately as s^{- alpha} , where alpha lies in the range 2.07+or-0.7.

The authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which, differing from previous phenomenological or small-cell real space renormalisation groups, is sensitive to the existence of an infinite cluster along any lattice direction. This enables them to explore new aspects of the phase diagram for the problem, which were not accessible through existing treatments; in particular, the crossover exponent between one- and two-dimensional behaviour is shown to be exactly one for directed percolation. Numerical estimates are in agreement with known inequalities.

It is shown that solid-on-solid models for which steps between neighbouring sites are limited to at most n adatoms exhibit self-dual points. At these points the amplitude of the height-height correlation function is exactly known. This extends a recent results of den Nijs (ibid., vol.18, p.L549, 1985) for the case n=1. Some doubts are raised to his assertion that correction to scaling terms due to the sine-Gordon operators should be absent at self-dual points.

An Ising chain with Hamiltonian H=-J Sigma _i<j epsilon _ijS_iS_j/ mod i-j mod ^{(1+ sigma )/2} is considered where the epsilon _ij are independent random variables. According to Kotliar et al. (1983) this model has a phrase transition with non-mean-field exponents when ¹/₃< sigma <1 and mean-field exponents beyond the upper critical value of sigma =¹/₃. By means of an epsilon expansion about the lower critical value of sigma =1, the model is shown to be replica symmetric as epsilon to 0 and it is speculated that this result holds for ¹/₃< sigma <1.

The Painleve property for partial differential equations (PDES) proposed by Weiss et al. (1983) is studied for a system of PDES, namely the reduced Maxwell-Bloch (RMB) equations. The RMB equations describe the propagation of short optical pulses through dielectric materials with a resonant non-degenerate transition. The author demonstrates that the RMB system passes the Painleve test, and constructs a Backlund transformation and solutions of the RMB equations.

The general equation Del ² Phi =-4 pi f is studied where f not=0 all over a three-dimensional Euclidean space and has no symmetries. The necessary and sufficient conditions for the existence of a solution vanishing at infinity are derived. Some specific differential equations of the above form appearing in general relativity are explicitly solved. Several properties of the solutions are established.

A Monte Carlo method for quantum spin-1 systems is presented. The internal energy and longitudinal correlation functions are computed in the antiferromagnetic region. Some exponents and the phase diagram are derived with good accuracy.

The author shows how to obtain the quasiclassical evolution of a class of field theories called ultralocal fields. Coherent states that follow the 'classical' orbit as defined by Klauder's weak correspondence principle and restricted action principles are explicitly shown to approximate to the quantum evolution as h(cross) to O.

The authors study the quantum mechanical evolution generated by Hamiltonians which are quadratic polynomials in q and p with time-dependent coefficients. This evolution is determined by a unitary implementation of the phase flow of the corresponding classical Hamiltonian. In the case of quadratic Hamiltonians which are periodic in time, the Floquet operator is shown to have either a pure point spectrum or a purely transient absolutely continuous spectrum. Thus, the motion is non-stochastic. In a simple model of a quadratic Hamiltonian with random time dependence, the quantum mechanical motion is shown to be non-stochastic almost surely.

It has been claimed that the polaron exhibits a transition from a delocalised to a localised ground state as the coupling to the phonon field is increased. The statistical mechanics analogy of this transition is the roughening transition. The author argues that no such roughening can occur for the polaron. He points out the possibility of a pinning transition in a suitable, fixed pinning potential.

The author discusses a one-dimensional nonlinear one-step process defined by a master equation. This equation describes a random walk in a symmetric potential which may be attractive, flat or repulsive. The eigenvalues of the master operator and its symmetric counterpart are found.

A new numerical method for estimating asymptotic properties is proposed. A generalisation of the Romberg algorithm leads to a systematic elimination of higher-order analytical terms in finite-size calculations and determines both the asymptotic value and the leading convergence exponent in a self-consistent way.

The phase diagram and the critical properties of the Hamiltonian version of the two-dimensional n-component cubic model are investigated. Judging from the results of simple limits, mean-field calculation and RG transformations, the phase diagram of the one-dimensional quantum system is similar to that of the two-dimensional classical model. Several RG transformations were used to investigate the critical properties using different cell sizes in the transformation. The coincidence of critical and tricritical fixed points and the presence of a marginal operation showed the formation of the Ashkin-Teller fixed line and the breaking of the universality. The cubic transition is found to be first order for n>2.

For part I see ibid., vol.19, p.563-74, 1986. The phase transition behaviour of the Hamiltonian version of the two-dimensional n-component cubic model is studied along the cubic transition line. 1/n expansion and anisotropy expansion around the 2n-state Potts point are used to determine the phase transition line and the latent heat. The latent heat depends on the value of the coupling whereas the crossover value of n, where the transition changes from second to first order, does not (it is n_c=2). The latent heat has an essential singularity at n=2 along the cubic transition line.

An elementary derivation of the complete asymptotic expansion for the number S_n of spiral self-avoiding walks with n steps on a triangular lattice is given.

The authors present an effective free-energy functional formalism which allows systematic epsilon -expansion methods to be used deep inside the critical region. Applications are provided to a d-dimensional layered geometry with periodic boundary conditions, and calculations are presented for the shift of the critical temperature and its associated exponent lambda . The theory shows how the (d-1)-dimensional physics emerges with power law dependences on the interlayer separation.

Using Beleznay's modified Romberg algorithm the author re-analyses the transfer matrix data for the two-dimensional problems of Derrida and Stauffer (1985). He gets for percolation: p_c=0.59273(6) for the threshold of the square site problem, nu =1.3330(7) for the correlation length exponent and omega =1.9(3) for the correction exponent, in the case of lattice animals x_c=0.246152(4) (square lattice), x_c=0.19294(1) (triangular lattice critical fugacity) and nu =0.64062(10) (radius exponent).

A plausible argument using the concepts of dynamic renormalisation group is used to derive the dynamical exponent z of the one-dimensional q=2, 3 and 4 Potts and four-state clock models. The exact result (z=2) for the Ising model with Glauber dynamics is recovered and in addition the authors find that the q=3, 4 Potts and four-state clock models belong to the same universality class. Moreover they find that z=3.05+or-0.1 for the (q=2) Potts model with Kawasaki dynamics in agreement with the z=4- eta prediction of Halperin, Hohenberg and Ma (1974). A simple qualitative derivation is also given.