Table of contents

A generalised definition for invariance of partial differential equations is proposed. Exact solutions of the equations with broken symmetry are obtained.

Using variable transformations and certain theorem proofs the asymptotic behaviour and the proper analytical solution of the Korteweg-de Vries-Burgers equation is found.

The author obtains relativistic brachistochrones for both a uniform electric and uniform gravitational field, applying the same method as that used by Bernoulli (1696) in order to solve the classic brachistochrone problem.

The author obtains the Green function for a non-relativistic charged particle moving in the field of a Dirac monopole. Several properties of the Green function are derived.

A two-parameter model of Hermitian matrices in zero-dimensional space is solved in the large-N limit. A very interesting phase diagram and a spontaneous magnetisation are exhibited.

Based on the Landauer formula for DC conductivity, a computer experiment is performed on a chain of 10⁷ atoms to show the energy-dependent metal-insulator transition in the Aubry (1978) model. The self-similarity of conducting channels in the conducting phase is also demonstrated.

The authors study irreversible reaction-diffusion models of the form pX from k₁ from mX to k₂ to qX, sX from k₃ from nX, m<n. Competition between k₁, k₂ gives a continuous transition of d>d_c= 2(n-m+1)/(n-1) which they study for d<d_c using the renormalisation group in conjunction with then epsilon expansion and exact techniques. For m=1 and any n the transition remains continuous. They present epsilon expansions for the case n=3. By contrast, for m=2 and any n the system shows a fluctuation-induced first-order transition.

The author introduces and investigates a susceptibility-like quantity which characterises the sensibility of neural models. It can be calculated without the replicas in the Hopfield model and may be useful in the investigation of highly nonlinear learning rules.

The authors investigate the growth of viscous fingers (VF) and diffusion-limited aggregates (DLA) in a disordered porous medium. They propose that the DLA model and VF are not equivalent in a porous medium with topological disorder as represented by a percolation system and suggest new universality classes for VF and DLA in such media. They also present evidence that, for a wide class of porous media with geometrical disorder (i.e. with pores of random sizes), the DLA model and VF are not equivalent, even though they have often been characterised by approximately the same fractal dimension. Moreover, VF appear to be sensitive to the local properties of the medium such as the pore size distribution. Therefore, the DLA model cannot be used to simulate VF in disordered porous media and to predict the efficiency of the displacement process.

Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.

Using a modified version of a Monte Carlo algorithm proposed by Sterling and Greensite (1983) the authors obtain the exponents theta =1.51+or-0.25 and nu =0.502+or-0.024 for planar self-avoiding surfaces with a fixed boundary in three dimensions, consistent with the conjectured exact values for branched polymers. They show how the modifications are needed to obtain a viable distribution of surfaces.

The authors study the distribution function f(L) for the length of single-connected links for a percolation backbone in 3D. They argue that f(L) approximately L^{- mu} exp(-(L/G)^gamma ), where mu =1.25+or-0.05, G=9.5+or-1 and gamma =1.0+or-0.1. They also derive the functional dependence of the distribution of single-connected links in terms of the droplet model of the backbone.

The explicit construction of an orthonormal basis for states of good spin, isospin and SU(4) Wigner supermultiplet symmetry is given in a Bargmann representation space. A complete set of quantum labels is provided by a Sp(3, R) contains/implies U(3) complementary symmetry.

A non-linear ordinary differential equation is linearisable if it possesses SL(3, R) symmetry. The conditions under which the Abelian two-dimensional subalgebras of sl(3, R) are sufficient for linearisation are established.

The authors have obtained the quasi-periodic solutions to the discrete non-linear Schrodinger equation by a variant of a method due to Date and Tanaka (1976). It is shown explicitly that the non-linear field variables at the different lattice points can be determined in a recursive fashion in terms of combinations of Riemann's theta functions depending on lattice position and time.

The Burgers' equation is used to model sedimentation of colloidal suspensions in terms of solitary waves. The importance of diffusive terms is highlighted. Analytic expressions are given for the distribution of dispersed material and the wavefront velocities are calculated as functions of time. An explanation of observed sedimentation behaviour is provided along with a discussion of the important features of real sedimenting systems.

The authors write down the recursion operator which defines the class of integrable nonlinear evolution equations (NEE) associated with a spectral problem (SP) for matrices of rank 2 depending quadratically on the spectral parameter. Then they show the existence of four different transformations which transform the given SP into the well known one of Zakharov and Shabat (ZS) (1972) by composing these transformations they recover, among others, the so-called elementary Backlund transformations for the ZS SP. Under some restrictions, one is able to prove the complete equivalence of the given glass of NEE to the one associated with the ZS SP.

A new method is proposed for the analysis of the magnetic polarisability of small apertures of arbitrary shape. The method is based on an integral representation for the reciprocal distance between two points previously obtained by the author. A general formula is derived for the coefficients of magnetic polarisability of small apertures. Specific formulae are obtained for the apertures shaped as a polygon, a triangle, a rectangle, a rhombus, a circular sector and a circular segment. All the formulae are checked against the solutions known in the literature and their accuracy is confirmed.

An operator analogous to the almost tangent structure on a tangent bundle is defined on jet bundles of a fibred manifold. This operator is used to construct a Cartan form. The construction is unique for first-order Lagrangians and is also unique when restricted to higher-order mechanics.

Recently, the Birkhoff-Gustavson normal form (BGNF) approach has been used in both classical and semiclassical calculations. The authors consider the limitations associated with this approach using as an illustration the anharmonic oscillator. At the classical level, they emphasise the need to properly take into account the divergence properties of the BGNF series. By reviewing an earlier work they demonstrate that this divergence problem may, in some cases, be resolved. However, after considering several standard quantisation rules, they find that it is not possible to obtain the correct quantisation of a system via the BGNF. As a result, they maintain that care should be used when working with the BGNF.

Two groups (old and new) of symmetries and their Lie algebra properties for the KdV and cylindrical KdV equations are unified and extended to the general KdV equation.

Two groups of symmetries and their Lie algebra properties for the modified KdV equation are extended to the general modified KdV equation and the Miura transformation between the general KdV equation and the general modified equation is also established.

For pt.I see ibid., vol.19, p.3755 (1986). Different definitions of the notion of the multisoliton are investigated. It is demonstrated that, even in more complicated cases (like the CDGSK equation), multisolitons can be completely characterised in terms of the discrete spectrum of the strong symmetry and that degeneracy of this spectrum leads to resonance solitons (yielding singular solitons in case of the KdV equation and non-singular ones in other cases). Furthermore a method is described which allows us to compute strong symmetries out of auto-Backlund transformations for nonlinear systems.

On the basis of the approach of Eriksen and Kolsrud (1960), the author derives a canonical transformation which diagonalises the Dirac Hamiltonian for a particle interacting with an external electromagnetic field. It is shown that an explicitly gauge invariant electromagnetic interaction can be obtained without recourse to the 1/m expansion. When expanded in 1/m, the result coincides with the Foldy-Wouthuysen interaction.

The authors compare three problems of quantum mechanics (or more generally of wave mechanics) which reduce to the same problem in classical mechanics and which can also be treated semiclassically by the Einstein, Brillouin and Keller (1958) semiquantisation rules. The free particle motion in an ellipsoidal oblate or prolate cavity (a deformed nucleus) is compared to that in a plane elliptical billiard box. Separation of variables is performed in appropriate coordinate systems. The presence of a separatrix in phase space is exhibited, which is connected to a potential barrier that is different for each problem. The uniform approximation is used to calculate WKB phase rules appropriate to each symmetry. An important difference results between the prolate and the oblate systems.

The thermofield representation of a thermal state by a pure-state wavefunction in a doubled Hilbert space is generalised to arbitrary and pure states. The authors employ a Liouville space formalism to investigate the connection between these generalised thermofield wavefunctions and a generalised thermofield state vector in Liouville space which is valid for all cases of the quantum density operator. The system dynamics in the Schrodinger and Heisenberg pictures are discussed.

An infinite number of exact solutions for the ground and excited states of an s-wave hydrogenic atom with the perturbation V(r)= gr+ lambda r² are constructed, subject to special relations between g, lambda and the nuclear charge Z. A powerful method of Stieltjes moments is employed to generate rapidly converging upper and lower bounds to the ground-state energy E₀(Z, g, lambda ) for arbitrary g, Z and positive lambda . The accuracies of these bounds are relatively insensitive to the strength of the perturbation. The moment method also generates the exact s-wave ground-state solutions mentioned above.

The author studies the quantum scattering of geodesic curves on the hyperboloid via established techniques in geometric quantisation. The wave operators are shown to be the dual Radon transform, and the S matrix thus calculated is the same as the one from Coulomb scattering. The procedure used can be extended to treat any non-compact symmetric spaces since they involve only geometric and group theoretic arguments.

The author gives probabilistic solutions to the equations describing non-relativistic quantum 'electrodynamical' systems. These solutions involve, besides the usual diffusion processes, also birth and death processes corresponding to the 'photon number' variables. Some inequalities are stated and in particular bounds to the ground-state energy of systems composed of a non-relativistic particle interacting with a field are established. The result is general and it is applied as an example to the polaron problem.

The scaling behaviour of the magnetic hard-square lattice gas is examined in the vicinity of the multicritical line located by an exact solution. A simple scaling form is proposed for the singular part of the free energy and corresponding scaling forms are derived for the sublattice density difference, the magnetisation and the densities. These are all shown to be consistent with the exact results when restricted to the exact solution manifold. In the case of the densities this is achieved by proving appropriate elliptic function identities.

The author studies some conformal variance properties of the polymer and percolation problems in two dimensions. By analysing the transfer matrix spectrum of these models at criticality, their series of thermal and magnetic exponents are identified. The results for percolation agree with the recent conjectures of Dotsenko and Fateev (1984) while some of the results for polymers are different. In the case of polymers, these series are interpreted as a new set of geometrical exponents. In each case the question of corrections to scaling is discussed.

The 2D ANNNI model is studied using a new iterative method, the method of ring recurrence. The authors show that the method produces an iterative scheme which enables the phase diagram and behaviour of the wavevector to be obtained using low-order matrices, with minimal computational effort. They find a one-dimensional attractor associated with the incommensurate phase.

The authors study the behaviour of spiral self-avoiding walks on the triangular lattice. The spiral constraint simply says that no step in an anticlockwise direction may be taken. Imposing the additional constraint that steps may not deviate from the straight ahead direction by +pi/3 defines model I, previously solved by Joyce and Brak (1985) as well as by Lin (1985). If deviations of +2pi/3 are forbidden, this model is referred to as model II, while model III allows deviations of both +pi/3 and +2pi/3. One finds for both model II and model III that the number of n-step self-avoiding spirals is s_n approximately c exp(2pi square root n)log(n/12)/n^13/4 where c= phi²/768gamma⁵, phi(model II) approximately= 0.009, phi(model III) approximately= 0.16 and gamma = 1-12(log 2/pi)². The confluent logarithm is an additional feature not present in the simpler case of the square lattice and model I triangular spiral self-avoiding walks. Use of two new results in the theory of partitions are made.

An exact solution to the problem of diffusion on a quasiperiodic chain is provided by a decimation method involving scaling by an irrational length factor.

The authors show that the arguments of Doktorov and Sakovich (1985) against the validity of the Painleve test are unfounded, being based on an incomplete singularity analysis.

An algorithm is presented to enumerate the polynomial zeros of degree 1 of the Racah (6j) coefficients. This algorithm is based on a Diophantine equation linear in two variables, obtained from a condition satisfied by the parameters in the parametrical formula of Brudno (1985) for the polynomial zeros of degree 1 of the 6j coefficient.

In a recent paper Wood (1985) conjectured that the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is at z=-0.119392 . . .. A careful analysis of the available series data implies that this conjecture is almost certainly false. The author finds the correct value to be z=-0.1193388809+or-0.000000001, with the corresponding exponent indistinguishably close to 5/6 (corresponding to a zero of the singular part of the function).