Table of contents

A method of constructing (2+1)-dimensional non-linear integrable equations and their solutions by means of the non-local delta problem is developed. A 'basic set' of equations is obtained by using different normalisations of the non-local delta problem and the Lagrangian of the set is found. Other integrable equations, which are degenerate cases of the basic set, are also Lagrangian.

Complete integrability is proved for a system of ODE describing the double three-wave interaction between five waves, when the coupling constant in one triplet is twice as large as in the other. Known results from the Painleve analysis for this case are combined with the Yoshida-Kovalevskaya approach (1983) to gain insight about the degree of the missing first integral. This integral of degree six is then obtained via a direct search based on irreducible forms, elementary building blocks for polynomial first integrals in involution with the Manley-Rowe invariants.

The model of a randomly stirred fluid introduced by De Dominicis and Martin (1979) is used to study the convection of a passive scalar, namely temperature. Mode coupling theory of dynamic critical phenomena is used to calculate the turbulent Prandtl number, which is a universal amplitude ratio. The result agrees with that obtained from the dynamic renormalisation group.

A four-mode Lorenz model for the rotating Rayleigh-Benard system is constructed which reproduces the results of a linear stability analysis of the hydrodynamic equations. An extension of this model predicts that the Kuppers-Lortz instability occurs at a critical Taylor number of T_c=80 pi ⁴ and a critical angle theta _c given by tan theta _c/4= square root 5.

From the exact enumeration on a triangular lattice of trail configurations at the tricritical regime as the fugacity for intersections is increased, the authors find phi _t=0.68 (8) for the crossover exponent (and hence alpha _t=0.5 (2) for the specific heat exponent). It indicates the collapse transition of trails to differ from that of polymer chains at the theta point in two dimensions.

The authors derive an exact expression for the distribution function for the (algebraic) area enclosed by a plane closed random walk, using a continuous model from the start. Their result has a different analytical form but is numerically close to an expression for the same quantity of a discrete model, derived by Brereton and Butler (1987).

The wetting transition of a two-dimensional system with short-range forces is studied in the presence of disorder, fully correlated in the direction of the substrate. In a solid-on-solid description the row-to-row transfer matrix is of the same type as in random one-dimensional systems. Therefore the substrate bound state is still localised when it becomes degenerate with a bulk state. This results in an unusual first-order wetting transition. The corresponding wetting temperature varies from sample to sample.

The author presents two proofs of the Atiyah-Singer index theorem (1969) for the Dirac operator for even-dimensional orientable manifolds without boundary with totally antisymmetric torsion with vanishing curl (H torsion) which, for example, may be provided by the field strength of the antisymmetric tensor occurring in the supergravity multiplet of superstring-inspired ten-dimensional theories, or by the structure constants of the group manifold in Kaluza-Klein theories. One of the methods, which is considered as formal, utilises a modified version of the conventional heat kernel regularisation. The other which is shown to be connected to the first one through a Legendre transformation uses the supersymmetric path integral approach. Of course, as far as the index is concerned, the H parts form a globally defined exact form and hence the author's result is consistent with the invariance of the index under the inclusion of torsion. The obvious advantage of the author's approach is the simplicity of the calculations in contrast to conventional regularisation methods which are practically impossible to handle beyond four dimensions, despite the trivial effects of the torsion on the (physical) results.

Combining the extended Painleve conjecture with Yoshida's singularity and stability analyses (1986) it is shown that, for two-dimensional homogeneous potentials of degree 2m, integrability restricts Kowalevskaya exponents and integrability coefficients to discrete sets of values. This result is made use of in the analysis of integrability of symmetric potentials with m=2, 3 and 4. Direct construction of additional first integrals is successful only in special cases which can be transformed to known integrable ones.

Classical scattering with singularities of Cantor-set type can be observed if unstable localised orbits exist whose homoclinic structures are transported to infinity by the Hamiltonian flow. An electron moving in the field of a magnetic dipole is a simple example of physical relevance to demonstrate this transport mechanism. In the asymptotic plane spanned by impact parameter and incoming direction the deflection function is singular for initial conditions leading to captured orbits. Using the method of Poincare sections, the authors find a correspondence between this set of singularities and the stable manifolds of localised orbits. The scattering data which are measured in the asymptotic region of free motion provide information about chaotic motion in a finite part of the position space.

The author gives a thermodynamical description of the spectrum of generalised dimensions for invariant sets recently proposed by Tel (1986). The author's approach allows a definition of an uncountable set of generalised spectra, depending on the equilibrium measure supported by the set, and the author indicates that the dimensions are independent of the measurable partition of the set.

The quasiclassical generalised Moyal quantisation procedure is applied to the problem of quantum canonical invariance. Simple new formulae for the Moyal bracket corresponding to various correspondence rules are given, and are used to calculate the 'additional potential terms' arising from a point canonical transformation made on a free Hamiltonian. The status of canonical transformations in other equivalent quantum phase space formulations is also reviewed.

The author presents a method to construct potentials for Schrodinger equations with some prescribed features, for which all eigenfunctions and the time-dependent propagator can be explicitly calculated. The prescribed features can be formulated by choosing arbitrarily the N lowest eigenvalues. Alternatively one can prescribe some qualitative behaviour for the potential, like the number and relative depths of wells and barriers. The results can also be applied to the construction of Fokker-Planck models with prescribed properties and explicitly calculable transition probability density. The method is based on ideas of supersymmetric quantum mechanics and the theory of solitons, that can be traced back to the work of Darboux (1894) and Crum (1955).

The classical path method is used to derive the WKB energy levels and wavefunctions for the periodic potential. The method generalises the instanton approach and can accommodate excited-state energy bands in addition to the ground-state band.

A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

The generalisation of the quantum inverse scattering method is used to derive the Bethe ansatz equation of a system for a multicomponent nonlinear Schrodinger models with n species (or colours) of bosons or fermions. The matrix periodic boundary conditions for bosons and fermions are obtained and the relationship between them discussed. The eigenstates for the infinite conserved quantities of the system are constructed. Lastly, the author generalises these results to the non-linear Schrodinger model with the mirror-image delta potentials proposed by Schulz (1987).

For pt.I see ibid., vol.21, p.2391, (1988). The Bethe ansatz equations for a mixture of species of fermions and bosons in one dimension, interacting with a repulsive delta potential, are obtained. It is shown that the matrix periodic boundary conditions of the system can be obtained from those of a fermion or boson system. Finally, the extension of these results to the nonlinear Schrodinger model with a supermatrix is made.

This paper is a collection of examples showing various effects on configurational properties of a random walk under the constraint that the ends of the random walk be fixed. Some formalism is developed for the solution of such problems, allowing the authors to express results in terms of the characteristic function of the unconstrained random walk. Several applications of this formalism are made in the paper. Typical of these is the result that fixing the ends of a random walk with an underlying stable-law density forces the steps of the resulting walk to have a number of finite moments. Other results show a range of differences to be expected between properties of constrained and unconstrained random walks.

Finite-size methods and the theory of conformal invariance are applied to the (1+1)DO(2) or X-Y planar model. The critical point is found to be x_c approximately=2.0, the value sigma =0.501+or-0.005 is obtained for the index governing the critical divergence of the correlation length and the conformal anomaly is confirmed to be c=1. The Kosterlitz-Thouless features of a low-temperature phase with zero spontaneous magnetisation but algebraic decay of correlations, together with a standard high-temperature phase, are observed. The critical exponent eta is determined as a function of coupling constant and compared with theoretical expectations.

Real-space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the Heisenberg ferromagnet and antiferromagnet on the infinite Vicsek snowflake fractal. The authors have calculated the associated dynamic exponents z_F and z_A by linearising about the respective fixed points of the ferromagnetic/antiferromagnetic recursion relations and find them to satisfy the relation z_A=z_F/2. In addition, using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho _F( omega ) and rho _A( omega ) have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho _alpha ( omega ) approximately b^{z alpha -df} rho ( lambda _alpha omega ) near their respective fixed points ( alpha =A,F), df denoting the fractal dimension. Finally, the amplitudes g_alpha ( omega ) appearing in the solution to the scaling equation rho _alpha ( omega ) approximately g_alpha ( omega ) omega ^dfz alpha -1/, for the density of states rho _alpha ( omega ), were found to be periodic functions of ln( omega ) with period ln( lambda _alpha ) where lambda _alpha is the eigenvalue associated with ferromagnetic ( alpha =F) or antiferromagnetic ( alpha =A) fixed point.

The modified direct correlation functions C_cond and C_sym are studied for a two-dimensional SOS system (M* infinity ) in a zero external field G identical to 0. The asymptotic limit W to infinity of the interface width W=(M+1)/ pi is considered in particular, also in connection with the Yvon-Triezenberg-Zwanzig (YTZ) formula for the interfacial tension Gamma and with its modification obtained by Ciach et al (1987). The successive contributions to the interfacial tension Gamma , resulting from various terms of the derived relations, are computed and discussed. In the asymptotic limit W to infinity the interfacial tensions obtained from the YTZ formula and from the Ciach formula agree with each other and with Gamma calculated earlier by Evans (1979) and extrapolated to G=0 by Stecki and Dudowicz (1985).

The role of solvent velocity fluctuations on the transport properties of polymer solutions is studied using renormalisation group expansions to order in ² of the coupled Langevin equation model within a Fokker-Planck equation formulation. Introduction of the timescale separation approximation between polymer and solvent characteristic relaxation times leads to an additional expansion in powers of a small dimensionless parameter which may heuristically be interpreted as the ratio of characteristic polymer (rouse-Zimm) to solvent relaxation rates. Retaining zeroth-order terms in the latter expansion reduces the order in ² solution of the Fokker-Planck equation identically to the corresponding expanded solution of the Kirkwood diffusion equation representation of the Rouse-Zimm model. Explicit expressions are derived for corrections to the Kirkwood diffusion equation due to solvent velocity fluctuations. The bare leading corrections for the intrinsic viscosity of preaveraged Gaussian chains are analysed and are shown to have a magnitude consistent with heuristic timescale separation arguments and to be negligibly small for typical polymer-solvent systems. It is therefore conjectured that the renormalised parameter is also sufficiently small to validate its neglect for these polymer systems.

Taking into account spatial fluctuations in the discrete model for certain types of immune response, the model of Weisbuch and Atlan (1988) is studied on a square lattice. In general, the authors end up with a healthy carrier state '29' where at all sites all cell types except activated killer cells are present. They study the time it takes to reach this fixed point, as well as effects of disorder and noise.

The authors study the retrieval properties of Hopfield's model (1982) when there is a finite correlation between two of the memorised patterns.

It is demonstrated numerically that the percolation transition of the unstable sites and the onset of chaos in the Kauffman model (1969) happens for the same value of the bias on the randomly chosen rules in the two-dimensional triangular lattice and in the three- and four-dimensional hypercubic lattices. The percolation thresholds for the stable sites are different for the three- and presumably also four-dimensional lattices. On the triangular lattice the differences between the thresholds are too small to be resolved. The fractal dimension of the damage is calculated and also the spreading time on the four-dimensional hypercubic lattice.