Table of contents

The authors consider the damped sine-Gordon equation perturbed by (thermal) space-time noise in the form it arises in the theory of the Josephson junction and charge density waves. They announce a rigorous proof that the coupling constant expansion of the solution of the initial value problem converges for small coupling. Taking the limit of the initial time t₀ to - infinity they obtain a stationary solution. It is shown that the stationary solution is localized both in time and space even if the solution at finite time is not.

A toy random field model in one dimension is solved exactly by a mapping onto a stochastic differential equation.

To derive an approximate solution of the damped Burgers equation, the author uses the tanh method as a perturbation technique. As a result, a damped shock-wave structure appears which moves with a decreasing velocity. In particular a bump type of behaviour appears, after a certain time, in the tails of the solution.

The authors obtain the exact Bethe ansatz solution of an O(n) model on the honeycomb lattice with open boundaries across a finite strip. Several quantities relevant to the surface critical behaviour of the O(n) model are derived from this solution. The exact surface critical exponents are in agreement with those obtained by Cardy (1987) and by Duplantier and Saleur (1986) based on conformal invariance arguments.

Low-temperature interfacial properties of several driven, non-equilibrium lattice gases in steady states are studied with computer simulations. The drives are directed along the tangent to the interface. By focusing on the small-momentum behaviour of the height-height correlation, G(q), the authors determine that the longitudinal spatial isotropy symmetry is relevant to the long-distance behaviour. For the case where the sign of the drive is random in time, the analytically predicted G(q) approximately q^-1 is observed for the first time. For uniform drive, they determine numerically G(q) approximately q^-2+n, with eta approximately= 1.33. This suggests that the effective stiffness of the interfaces is enhanced on large scales, consistent with roughness being severely suppressed.

The two-particle annihilation reaction, A+A to inert, for immobile reactants on the Bethe lattice is solved exactly for the initially random distribution. The process reaches an absorbing state in which no nearest-neighbour reactants are left. The approach of the concentration to the limiting value is exponential. The solution reproduces the known one-dimensional result which is further extended to the reaction A+B to inert.

We consider critical asymptotically hierarchical quantum models and show that the probability distribution of the appropriately scaled square of the total spin converges, as the number of spins tends to infinity, to the same function in the corresponding classical systems.

The authors describe the finite-dimensional functional manifold which contains all meromorphic solutions to the multidimensional generalization of the Lame equation.

The authors discuss the role of the microscopic stochastic dynamics in the macroscopic properties of a simple spin-flip non-equilibrium mean field model where ferromagnetic and antiferromagnetic interactions are competing. They have found that different analytical forms for the dynamic mechanism give rise to different stationary states at low temperature, in contrast to what happens at equilibrium. They also find an effective free-energy functional which allows them to classify the stable and metastable stationary solutions.

Takayasu and Tretyakov (1992) studied branching annihilating random walks with n=1-5 offspring. These models exhibit a continuous phase transition to an absorbing state. For odd n the models belong to the universality class of directed percolation. For even n the particle number is conserved modulo two and the critical behaviour is not compatible with directed percolation. Branching annihilating random walks with n=4 additional process (spontaneous annihilation) which breaks the conservation law are studied. The inclusion of spontaneous annihilation, even at very small rates, leads to directed percolation critical behaviour.

General relations and constraints which must be satisfied by the topological correlations in 2D space-filling random cellular structures are discussed and a topological short-range order coefficient is defined. Topological models of 2D structures are associated with planar tessellations with topologically unstable sites which belong to z)3 polygons. The stable configurations, called states, are obtained by replacing every vertex by z-3 added sides. The topological properties of the latter models are calculated exactly for a distribution of independent and equiprobable states on the various sites and for any value of z. The case of the structures associated with tilings by triangles is thoroughly considered. The calculated correlations are compared with the correlations in alumina cuts and in random Voronoi froths. The variability of the topological properties of 2D random cellular structures is discussed.

The percolation model studied in finite strips (L << M) at criticality is characterized by the preferential growth of percolating clusters in the L-direction. The authors present a Monte Carlo and finite-size scaling study of the density profiles of the percolating clusters. The behaviour of the pair-connectedness function is also analysed at criticality within both the algebraic and the exponential regimes.

The scaling behaviour of the percolation hull rotationally constrained site percolation on the square and triangular lattices is studied. The single cluster growth method has been used to determine the values of some critical exponents. Evidence is obtained for a scaling form of the hull-size distribution function.

The authors consider hysteresis in isotropic N-vector models in d dimensions in an external spatially uniform field varying sinusoidally in time. They use renormalization group arguments to show that for d ) 2 and N >or= 2, for small frequencies omega , and small amplitudes H₀ of the field, the area of the hysteresis loop scales as (H₀ omega )¹2/. with logarithmic corrections. For N =1 and d ) 1, using nucleation theory they show that the area for omega << H₀ scales as mod Tln(H₀ omega ) mod ^-1(d-1/). The power-law dependence of the area of hysteresis loops in continuous spin systems is a manifestation of their self-organized criticality.

The generalization ability of the Hopfield model of neural networks trained with examples is studied in mean-field theory in the presence of synaptic noise. Although the latter improves the generalization ability for a finite number of concepts, it does not for a macroscopic number of them. Nevertheless, the network performance is still robust against synaptic noise. Numerical simulations are performed to verify the mean-field theory predictions.

The authors examine the possibility of improving the performance of discrete-synapse neural networks, functioning as content-addressable memories, by the inclusion of noise in their training procedure, and study the effects on the training itself. Pattern stability field distributions for optimized networks are illustrated for various levels of training noise, including the noiseless, maximally stable, regime. They show that the clipped Hebb rule is optimal in the high training noise limit, but that simulated annealing cannot be relied upon to identify a well defined optimal network for an arbitrary, finite, training-noise, in contrast to the case for continuous-synapse systems. Training by use of a continuous-synapse network, whose synapses are subsequently clipped, is also addressed.

A q-deformed osp(1,2) superalgebra is defined by the use of a pair of q-boson annihilation and creation operators. A kind of two-component coherent state representation of the q-deformed superalgebra is found and a q-differential realization of the q-deformed osp(1,2) superalgebra is obtained.

The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.

The characteristic algebra of generalized Ermakov systems is sl(2,R). The structure of these systems in three dimensions is obtained. A subset in the form of an equation of motion with the additional requirement of so(3) symmetry is studied. It includes the classical equation of the magnetic monopole. The existence of three vectors of Poincare type is established. Consideration is given to weak generalized Ermakov systems in which the symmetry breaking occurs in the radial equation.

The star operations and reality conditions for the complex quantum algebra U_q(sl(4;C)) providing real quantum algebras U_q(o(6-k,k)) k = 0, 1, 2, 3 and U_q(su(3,1)) are classified. Standard and non-standard star operations are considered. It appears that only four choices of real forms (one with mod q mod =1, three with q real) provide real Hopf algebra U_q(su(2,2)) approximately=U_q(o(4,2)) describing D = 4 conformal quantum algebras. The authors show that only the antipode-extended Cartan-Weyl basis of U_q(sl(4;C)) permits one to define real q-deformed D = 4 conformal algebra generators. In order to obtain the real D = 4 Weyl algebra as the Hopf subalgebra of U_q (su(2,2)) only the non-standard real forms can be employed.

Power series for the parabolic cylinder function, D_v(z), are derived from known addition theorems, thus enabling a more compact and efficient expression of the function. One of the expansions is applied to find an asymptotic approximation for the function as mod v mod to infinity with mod arg(-v) mod ( pi . Next, a large-argument (z) asymptotic addition theorem is derived from an integral representation of D_v(z). Before extending this type of summation theorem to the two general confluent hypergeometric functions, motivating mathematical applications to three integration problems are given as examples of the practical usefulness of the formulae.

The theory of value distribution is applied, through a study of boundary value distribution for the Weyl-Titchmarsh m-function, to the spectral analysis of differential operators on the half-line. Particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra, and new identities for the spectral measure in terms of solutions of the associated differential equation.

Electromagnetic fields produced by the spike pulse of hard radiation are discussed. The authors propose a model from which the electromagnetic field of a running pulse generated by a line current can be derived. The solution of the electrodynamic problem is found by the Smimov method of incomplete separation of variables by means of the Riemann method.

For the fifth-order Korteweg-de Vries equation with a small parameter multiplying the highest-derivative term it is known that solitary waves are non-local, and are accompanied by co-propagating oscillatory waves of small amplitude and short wavelength. Here the authors report on some preliminary results for mutual interactions of these waves. First they impose a quantization condition upon a chain of solitary waves to obtain a periodic solution. Then they compute the interaction force between two neighbouring waves and hence estimate the conditions for a bound state to occur.

The authors categorize classes of coupled nonlinear Schrodinger equations which allow generalized similarity solutions, using the approach of Clarkson and Kruskal (1989). In all cases, the resulting pair of ordinary differential equations belongs to a single class. Certain cases allowing solution in terms of familiar functions are identified. An alternative approach shows that only two conditions need be placed on the four real and two complex coefficients in the governing equations in order that solutions generated by an arbitrary solution of the (integrable) NLS equation exist. Applications to some standard coupled systems arising from fibre optics are given.

Dynamical properties of classical Hamiltonian systems and their Moyal quantizations are related.

For continuous Schrodinger equations the discretization process is defined by preserving the Heisenberg equation of motion rather than the Schrodinger equation itself. For instance, strong changes are obtained for one particle in DC electric fields. In the old discretization process, all eigenstates are factorially localized and the spectrum becomes discrete. On the other hand, in the model, the author conjectured that the spectrum becomes continuous. The author remarks that discrete systems play an important role in physics because they are, in many cases, a first approach to real systems. The goal is to study the equivalence between continuous and discrete Schrodinger equations by preserving the Heisenberg equations of motion.

The integrability of the cubic-quintic nonlinear Schrodinger equation is investigated numerically. By analytically studying the linear stability of the homogeneous state and numerically solving such a continuum Hamiltonian dynamic system, the authors show that the quintic nonlinear term leads to a spatiotemporal complexity of wave fields, and illustrate that this-behaviour is associated with the stochastic partition of energy in Fourier modes. In addition, they show the presence of the stochastic motion is due to the homoclinic orbit crossings.

The problem of the exact solutions of the Dirac equation in the presence of the scalar and pseudoscalar fields is investigated by means of an algebraic method. Such an approach allows the author to find all the configurations of the scalar and pseudoscalar fields where separation of variables is possible. Some general recommendations for the search of exact solutions of the Dirac equation in Cartesian coordinates are proposed and some exact solutions in curvilinear coordinates are discussed.

Using the operator formalism, Ward identities are derived for super W₃ conformal field theories on a supertorus. Differential equations of correlation functions on the supertorus are also derived by using the Ward identities and null vectors.