Table of contents

For the two-parametric quantized group gl_q,s(2), the authors calculate the explicit form of the universal R-matrix.

A Miura transformation is presented relating the discrete Painleve II equation to what plays the role of the modified d-P_II (equation (34) in the Painleve-Gambier classification). The study of this transformation makes possible the derivation of the auto-Backlund transform of the discrete P_II and allows one to obtain particular solutions to the latter. Moreover the use of this transformation makes it possible to construct a new mapping, in the parameter space of P_II. The latter is a novel method for the construction of integrable discrete systems.

The author shows that for plane parallel capacitors with a large ratio of (area/perimeter²) and large plate separation, the excess capacitance due to fringing fields decreases as the electrode spacing decreases. This is contrary to intuition, but is supported by numerical work on disc-shaped capacitors.

A supersymmetric multiple three-wave interaction model is proposed, and its quantum integrability is demonstrated in a straightforward way.

The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.

The authors present surface properties of a nonlinear random deposition model taking into account anisotropy in diffusion, which appears in some experimental situations. They report the results of simulations made in a bidimensional substrate. They find that the surface width w(t, L) scales as t^beta with beta =0.35 for small t and the saturation value does not depend on L. A crossover phenomenon associated with large length scales is studied.

The effects of self-amplification of a vorticity field in 3D turbulence and a vorticity gradient in 2D turbulence are considered. The conditionally averaged tensor of strain rates (with fixed vorticity) is obtained for 3D turbulence. The corresponding tensor (with fixed vorticity gradient) is obtained for 2D turbulence. These results are discussed in the context of observations and direct numerical simulations of turbulent flows. In particular, the appearance of 'vortex strings' in 3D turbulent flows is in accord with the obtained formula. The presented method is quite general and can be applied to a variety of physical systems with strong interaction.

A recently proposed mesoscopic description of fluid dynamics leads to a new approach to turbulence. In contrast to the classical statistical theory of turbulence the new approach introduces a probabilistic time evolution of the random velocity governed by a master equation. The mesoscopic approach is explained by means of the (1+1)-dimensional Burger's model of turbulence. By a continuous time stochastic simulation, realizations of turbulent velocity fields are generated. Correlation functions and energy spectra are evaluated from appropriate ensemble averages.

The authors show how the method of Lie symmetries can be used to obtain first integrals and for the identification of completely or partially integrable cases of the reduced three-wave interaction problem.

Hydrodynamic reductions are found for the lattice KP hierarchy and its zero-dispersion limit. This is similar to the continuous case where the reductions result in dispersive water waves and Benney hierarchies respectively.

Exploiting the supersymmetric connection between the statistics of branched polymers and the problem of the Yang-Lee edge singularity, the authors study universal properties of the adsorption transition of diluted branched polymers in a three-dimensional solution at a hard wall. They calculate various scaling functions for the crossover between the non-adsorbed and adsorbed state exactly. In particular, the crossover exponent is found to be phi =1/2.

The author shows that all equilibrium statistical properties of an interface confined in a strip geometry, with arbitrary aspect ratio, exactly at a second-order, fluctuation dominated, interfacial unbinding (wetting) transition are determined by a single number q for interfacial binding potentials that are conformally mapped from the semi-infinite plane. The parameter q distinguishes the fluctuation regimes describing the wetting transition and can be directly related to wetting critical exponents. In the strong-fluctuation regime one finds q=0 whilst in the weak-fluctuation regime q=1. The values of q at all intermediate-fluctuation scaling regimes are also determined. He shows that the eigenstates of the transfer differential operator for conformally mapped marginal long-ranged potentials are the simplest possible generalization of the eigenstates corresponding to systems with short-ranged forces. He speculates that the universal parametrization of the finite-size effects at fluctuation dominated wetting transitions is a consequence of local scale invariance.

The time-dependent properties of an inclined interface separating up and down spin regions in a two-dimensional nearest-neighbour Ising model evolving under Glauber dynamics in a non-zero field are studied. In the limit of large exchange coupling, the model reduces to the single-step model for ballistic growth and thence to the asymmetric exclusion process which describes a driven diffusive system of hard core particles on a one-dimensional lattice. The drift velocity of the interface is found as a function of field, temperature and inclination, and interface correlation functions are related to sliding tag correlation functions in the particle system. The existence of a critical value of the sliding-tag velocity implies that there is an inclination-dependent easy direction along which temporal interface fluctuations grow subdiffusively. This direction is found, as is the asymptotic behaviour of the correlation function in all other directions.

The authors present a model for the study of the synchronization between coupled limit-cycle oscillators. This model is motivated by the recent observations that revealed the existence of synchronization in the firing patterns of neural cells. The study is based on a particular coupling architecture where all the oscillators are coupled to a comparator unit that feeds back to each oscillator the mean value of the total phase. The various regimes of this model are analysed in the presence of non-uniform external driving.

The equation of motion method for the Green's functions is applied to study the behaviour of pairs of interacting two-level tunnelling units, coupled with a phonon field to calculate, within the framework of the small-polaron theory, the dynamic susceptibilities and relaxation rates of the system. Explicit expressions for these quantities are obtained in the Debye approximation.

The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O⁽ⁿ⁾ model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.

Using constrained path integrals and perturbation theory, the authors study the algebraic area, A, enclosed by closed Brownian curves on various domains such as rectangles or strips. In the limit of infinitely long curves, the probability distribution P(A) is shown to be Gaussian. The standard deviation (A²)^1/2 is simply expressed in terms of the length scales of the problem.

The relaxation of random walks and the autocorrelation function on the landscape of the graph-bipartitioning problem are calculated.

The Q-states gauge Potts model is exactly solved on a special infinite-dimensional lattice which is a Bethe lattice of plaquettes. The critical properties are studied. The model exhibits a first-order transition for any number of states Q and coordination number gamma +1. The critical values of coupled constant beta _c is shown to be in a good agreement with known results.

The surface roughness of a growing crystal is studied by Monte Carlo simulation in a kinetic six-vertex model. Although for equilibrium the results are in good agreement with the exact solution, even a small disequilibrium breaks down the roughening transition. For temperatures below the equilibrium roughening temperature T_R and arbitrary large disequilibrium, as well as for temperatures above T_R but small disequilibrium the surface is logarithmically rough, whereas, for temperatures above T_R and sufficiently large disequilibrium, the surface roughness increases as a power-law of size. A crossover from logarithmic to power-law roughness occurs when the disequilibrium is increased at a fixed temperature above T_R, or the temperature is increased for disequilibrium fixed and sufficiently large.

The authors study the manifold of fixed points of the generalized weak-graph transformation on lattices of even coordination number for the most general vertex model respecting spin-flip symmetry. They conjecture these fixed points to be the loci of phase transitions. As an example, they turn to coordination number six and find phase transitions in certain regions of the manifold of fixed points: first by investigating a gauge-invariant Ising model on a three-dimensional SC lattice with the help of Monte-Carlo simulations; and second for the ice-type zero-field ferroelectric model, in which the transition between frozen ordered and disordered phase is of first order.

The authors study a ferromagnet in a random magnetic field using the replica method. They find that the mean field equations have localized solutions (i.e. instantons), which are not invariant under rotations in replica space. The relevance and the properties of these solutions are studied.

The author studies the effects of topology on the free energy of uniform lattice animals interacting with a surface. Topology is specified by fixing an abstract graph, tau , and the lattice animals considered are embeddings of tau in the square and simple cubic lattice. He proves that such embeddings in the simple cubic lattice and interacting with a plane have the same free energy as self-avoiding walks independent of the choice of tau and independent of whether or not the embeddings are restricted to be uniform. For embeddings in the square lattice and interacting with a line, he proves that the free energy may differ from that for walks depending on whether tau has a cut edge. Further restricting the embeddings to be uniform forces the free energy to be different from that for walks for all tau (except the tau corresponding to walks) and we obtain bounds on the free energy which depend on the number of branches, cycles and vertices of degree 3 and 4 in the graph.

The anomalous diffusion in a unidirectional random velocity field with long-range correlations is analysed in directions transverse to the direction of the field or along the field direction. Critical values of the exponents, which characterize the power-like falloff of the correlations in the transverse and longitudinal directions, and the critical dimension of space are determined. The anomalous dimension of the longitudinal diffusion coefficient is also calculated in the first order of the in -expansion for several cases of a long-range correlated random velocity field.

Recently, very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed. To test such methods, detailed information about exactly solvable models can be very helpful. To this purpose the authors present results for the stationary, 1D linear BGK equation, obtained by the singular eigenfunction method, that extend results already known in the literature, both in scope and in precision. Special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary; this singularity cannot be reproduced exactly by the numerical methods available. Explicit expressions are presented for the Milne problem and for the albedo problem with input particles of a single velocity. They compare the results with those of a recently developed variant of the two-stream moment method. They find excellent agreement, except close to the singularity; for many quantities of interest, the accuracy obtainable by the novel two-stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution.

Two-species diffusion-limited annihilation in less than four dimensions is well known to have a concentration decaying as t^-d4/, as opposed to the rate equation result of t^-1. This result had not, however, been demonstrated numerically in three dimensions, due to very strong transient effects. A variant of the model is proposed here, and is shown to reach the exactly known asymptotic behaviour in numerically accessible times. This also allows one to investigate open questions in this system with some certainty that the results are indeed asymptotic. In particular, it is shown that the distance between two particles of the same species scales identically to the distance between particles of different species. This is in contradistinction to recent (numerical and scaling) results in one and two dimensions. Some evidence is also presented that, contrary to previous suggestions, the domains formed in three-dimensional two-species annihilation are indeed regular objects with smooth interfaces.

The authors extend their previous derivation (1991) of similarity laws in directional solidification of eutectics to general non-axisymmetric growth patterns. This involves mathematical subtleties which are not encountered in the symmetric case. The result explains the observation that numerical solutions describing tilted eutectic growth share the basic similarity property of axisymmetric solutions. They find additional scaling relations of the form lambda approximately V^-1/2g(G/V), e.g. for the wavelength lambda of a tilted pattern at fixed tilt angle phi (V is the pulling velocity, G the temperature gradient). Discussing the question of universality of the scaling function g for different distinguished wavelengths, the authors are led to the prediction that the transition to a parity-broken state takes place at roughly twice the selected wavelength of the symmetric pattern for sufficiently large velocities and that the ratio of these wavelengths increases with decreasing velocity.

The authors study a one-dimensional abstract model for classical and quantum irregular scattering in which the interacting dynamics is defined by the standard map. This model allows for a direct comparison of classical and quantum transport properties. Whereas the classical model is characterized by chaotic diffusion, in the quantum case the interplay of diffusion and localization determines a transition from a ballistic regime to a localized one, with an intermediate ohmic regime in the crossover region. The scattering matrix is numerically computed by solving a Lippman-Schwinger equation. In the ballistic regime the S-matrix fluctuations are found to share some typical features with the Ericson fluctuations, with correlation lengths close to the classical rates of exponential decay. Qualitative modifications occurring in the diffusive regime, including universal transmission fluctuations, are discussed.

The authors extend the semiclassical theory of spectral fluctuations to include composite systems. These systems are characterized by some number of weakly communicating, highly chaotic subsystems. They contain new and longer time scales, depending on the connecting fluxes, in addition to those associated with the rapid internal mixings. The new time scales produce discernible modifications in the nature of the level fluctuations of the corresponding quantum systems. The theory is applied to two coupled quartic oscillators.

The SU_q(2) symmetry of the q-rotator model, which for rotational bands predicts squeezing of the energy levels equivalent to the variable moment of inertia (VMI) model, predicts an increase with angular momentum of the B(E2) transition probabilities among these levels, while the rigid rotor model predicts saturation and the interacting boson model predicts a decrease. Some evidence supporting the SU_q(2) prediction is presented. The possible usefulness of the quantum algebraic method in extending the VMI concept to B(E2) transition probabilities is pointed out.

A formula for the energy corresponding to a potential phi obeying (- Del ²+ kappa ²) phi =0 is derived as an expansion in kappa ^-1 for manifolds with arbitrary smooth boundaries and either Neumann or Dirichlet boundary conditions. The first three terms in the expansion are explicitly determined and they agree with results of Duplantier (1990) in three dimensions for flat space. Careful attention in the Dirichlet case is given to ensuring a well defined regularized expression for the kernel on the boundary surface derived from the Green function so as to ensure an unambiguous finite expression for the energy.

Considers the inverse scattering problem for a scalar wavefield incident at grazing angles on a one-dimensional rough surface. The problem is formulated first as a pair of coupled integral equations in two unknown functions, knowledge of which immediately yields the surface. A method is described for the direct approximate solution of this system. Preliminary results are presented in groups of complicated rough surfaces which are closely recaptured in all details except for scale.

The heat kernel K(s)=Tr(e^{-sH( lambda )}-e^-sH(0)) is determined exactly for the operator H( lambda )=- delta _x²- lambda ( lambda +1) sech² x. The behaviour of K(s) is extracted for both large and small s. It is shown that for all s( infinity , K(s) is a continuous function of lambda although new bound states are formed when lambda passes through the positive integers. This implies that the scattering states provide a discontinuous contribution to the heat kernel such that the sum of bound and scattering contributions is continuous. For small s, a general expression is derived for the coefficients K_m in the small s expansion of the heat kernel: K(s)= Sigma K_ms^m-12/. With E_n( lambda ) denoting the eigenvalues of H( lambda ), the spectral function N(E)= Sigma ( Theta (E-E_n( lambda ))- Theta (E-E_n(0))) is found. It is proved for general potentials that the coefficients of the large E expansion of N(E), N_m, given by N(E)= Sigma N_mE^-(m-12)/ are related to those of the small s expansion of K(s) by N_m=K_m/ Gamma (3/2-m) and this is demonstrated explicitly for H( lambda ) given above. A discussion is given on the use of the small s expansion to reproduce K(s).

A numerical technique of solving the Schrodinger equation with two-Coulomb centers plus the oscillator Hamiltonian has been developed. A scheme of evaluating energy levels and wavefunctions of such a Hamiltonian is described in some detail. The proposed algorithm also allows the study of analytical properties of energy levels in a complex plane of the internuclear distance, i.e. to find positions of branch points in the complex plane. The prolate ellipsoidal coordinates enable separation of variables in the equation studied which greatly facilitates the solution. A numerical method for finding branching point positions for an arbitrary analytical function is outlined.

For autonomous two-dimensional conservative dynamical systems the author derives four necessary and sufficient conditions which the potential function U(x, y) has to satisfy in order that it is integrable with the second constant of motion quartic in the velocity components. The author also develops the method by means of which he finds the quartic invariant for a given potential satisfying these conditions. Certain degenerate cases leading to pseudo-quartic integrals are discussed. Two examples and a counter-example are presented.

A geometric classification of degenerate time-dependent Lagrangian systems is given and the reduction of evolution space is analysed. General properties of semiregular Lagrangians (type II) are discussed and particular attention is paid to the reduction of completely degenerate Lagrangians (type III) which are considered in detail.

An energy-conserved soliton cellular automaton is proposed. It is a generalization of the Park, Steiglitz and Thurston model (1986), and is shown to contain richer solitonic phenomena. A systematic comparison of their collision statistics is also given.

The Schrodinger equation for an electron and a single-mode photon field with interactions is solved by a direct method. A unique feature of these solutions is the inclusion of retardation effects in the photon field. Some interesting physical questions arising from the solutions are discussed. The Keldysh-Faisal-Reiss formula for the transition rate of multiphoton ionization modified by the inclusion of retardation effects is simplified by averaging the degenerate initial states. The result shows that the retardation effects can be calculated in terms of the radial part of the momentum wavefunction of the initial state. The physical significance of the inclusion is analysed in the near-threshold case of multiphoton ionization. The result shows that in the near-threshold case, retardation effects depend exponentially on the orbital angular momentum of the initial state. The effect vanishes for s-states, but is significant for states with high orbital angular momentum.

The highly degenerate eigenspaces of the two-dimensional isotropic harmonic oscillator are proven to contain eigenstates that are optimally localized on the closed trajectories of the classical dynamics. As h(cross) to 0, their phase space probability density converges to the unique probability density on the corresponding trajectory which is invariant under the classical flow.

It is shown here that the non-stationary Schrodinger equation can be solved exactly for two quantum models subject to Dirichlet boundary conditions. One of them is a modified problem of a quantum bouncer, i.e. the problem of a particle falling down in the gravitational field on a moving (oscillating) platform such as a loudspeaker. The second model is a 'cuff-off oscillator' with a moving infinite potential wall and a time-dependent frequency. In both cases exact solutions are given in closed forms, easy to use. Their possible applications are also indicated. In each of the models extra coordinate- and time-dependent phase factors are generated by moving boundaries in the former case giving rise to a non-local effect in quantum mechanics.

Eigenvalues of the ground state of the radial Schrodinger equation for a spiked harmonic-oscillator potential have been evaluated employing two methods: numerically, via the Lanczos/grid technique, and by means of standard Pade approximants constructed from an expansion of large coupling parameter series for the energy. Numerical results are compared for several values of the parameters characterizing the spiked singular potential.

Feynman's diagrammatic approach to perturbative quantum field theory is not easily applied unless the interactions have simple power series expansions. In particular, when the interaction involves non-integer powers of the field, as happens when carrying out the so-called delta -expansion, the diagrammatic approach must be supplemented with some prescription for the analytic continuation of the exponents. Here the authors propose a new approach to perturbative field theory when bypasses Wick's theorem, and uses instead the fact that the joint probability distribution function for the fields at a finite set of points can be determined exactly from their expectation values, variances and mutual covariances. One can then calculate expectation values for products of operators at these points, or at least express them as finite-dimensional definite integrals. This technique is illustrated by calculating expectation values for non-polynomial O(n)-invariant operators.

The properties of Weyl ordered products of operators are investigated and the technique of integration within Weyl ordered product (IWWP) is introduced. The overcompleteness relation of the coherent state is then recast into Weyl ordered form. In so doing, a new approach for Weyl ordering quantum mechanical operators is presented.