Table of contents

Using Painleve property analysis the authors obtained the Lax pair, auto-Backlund transformation and recursion operator of a perturbed Korteweg-de Vries equation, u_t+6uu_x+u_xxx+f(t)u=0. The exact form of f(t) for which the system is completely integrable is reported.

It is proved that cantori for area-preserving twist maps have dimension zero, only finitely many orbits of gaps, and gap widths go to zero exponentially in each orbit, if they are hyperbolic.

A method for the estimation of the topological dimension of a manifold from time series data is presented. It is based on the approximation of the manifold near a point chi by its tangent space at chi . The dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition. The method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state.

Theoretical and computer simulation results are presented for the effect of fluctuations on the dynamics of layer growth during complete wetting. For example, in two dimensions the authors predict a crossover in the dynamics of the layer thickness W, from Lipowsky's result, W approximately t^1/4, for all non-zero temperatures in a lattice-gas system, to W approximately (ln t)^1/2 at zero temperature (the roughening transition temperature) for intermediate times. Simulations of wetting in the three-state chiral Potts model give W approximately t^psi , where Psi is 0.25+or-0.03 for temperatures T>or=0.6 T_c, where T_c is the critical temperature, while the effective exponent Psi monotonically decreases with T below this temperature.

A stochastic centre manifold treatment is applied to study a subcritical inverted bifurcation scenario. The author demonstrates that the intrinsic fluctuations are subject to scaling relations involving the small parameters characteristic of the particular unfolding. A stochastic counterpart of the Lorenz system reduced to Poincare normal form is analysed.

The author studies the time evolution of two three-dimensional Ising configurations by Monte Carlo simulation, which differ only by the central spin. By monitoring the spreading of the damaged spins, a phase transition is observed at a value of the temperature in agreement with the Ising-correlated site percolation.

Scaling properties of the cluster distribution function and mean cluster size in the ensemble of clusters with fixed perimeter are analysed. The relevant scaling exponents are determined at all three regions of interest: lattice animals (p<p_c) percolation (p=p_c) and the compact clusters (p>p_c). Also, a form of the lattice animal distribution function in the statistical ensemble of clusters with fixed perimeter is presented. In particular, the authors compare the exponents alpha and beta , defined through (t)_s approximately s^alpha and (s)_t approximately t^beta , where (t)_s is the mean perimeter t of s-site clusters (the s-ensemble) and (s)_t is the corresponding quantity in the t-ensemble.

The author uses transfer matrices to calculate the free energy and various thermodynamic quantities of the two- and three-state Potts models with random bond interactions at the critical temperature. He verifies the Harris criterion for the three-state Potts model and estimates the new random critical exponents. For the Ising case, the results are consistent with randomness being irrelevant for the critical behaviour.

The author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi-infinite lattices when the exponentiated coupling is extended into the complex plane. He shows however, that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy. This unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit. The author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature.

The critical relaxation of the four-state Potts model on L*L lattices with L<or=64 is studied by the Monte Carlo method. The decay of the magnetisation from the nonequilibrium initial state exhibits a crossover from a power-law relaxation for t<<t^c(L) to an exponential one for t>>t^c(L), where t^c(L) is some crossover time. This crossover is discussed within the context of dynamic finite-size scaling theory. As a result, the dynamic critical exponent is estimated to be Z=2.94+or-0.29.

Sinai has considered a novel one-dimensional walk with a random bias field E on each site. He has shown that when the field is taken with equal probability to be +E₀ or -E₀ the RMS displacement R identical to (( chi ²))^1/2 increases with time t by the anomalously slow law R approximately (log t)². Here the authors introduce long-range correlation between the random fields on each site by choosing a 'string' of k sites to a have the same value of E, where k is chosen from the power law distribution P(k)=k^{- beta} . They find that the Sinai law is generalised to the form R approximately (log t)^y, where y sticks at the Sinai value y=2 for beta >2. However, for 1< beta <2, y varies continuously with beta as t= beta /( beta -1). The authors interpret this result physically in terms of the novel crossover between the physics underlying the Sinai effect and the physics of biased diffusion.

It has been observed that the viscosity of water is dramatically enhanced in various hydrophilic confinements such as lipid bilayers and micelles of microemulsions. Now computer simulations of the hydrogen bond connectivity based on a percolation model demonstrate that this effect exists up to characteristic dimensions of some 10 nm. This theoretical result is in fact in agreement with the experimental data and predicts similar influences on water confined in ultranarrow capillaries.

Using a vector coherent state theory, it is shown that the construction of Gel'fand bases for the unitary group is particularly simple. The very specific rules for construction of the states greatly facilitate the subsequent computation of matrix elements of the generators of the u(n) Lie algebra.

Using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous Lie groups of mathematical physics: (i) the conformal group; (ii) the semi-direct product of the Weyl group and the one-parameter group of Psi translation; (iii) the direct product of the Poincare group and the one parameter group of Psi scaling; (iv) the Weyl group; (v) the 14-parameter maximal symmetry group of the Schrodinger equation for the Coulombic system of atoms and molecules; and (vi) SU(3) of elementary particle physics. the first four groups are the only groups which, together with the Poincare group, are the maximal symmetry groups of all forms of the non-linear Klein-Gordon equation.

For pt.I see ibid vol.20, p.769-84, 1987. The new solution to the SU(n) external state labelling problem, proposed in the first paper of this series, is analysed in detail in the SU(3) case in terms of the two complementary chains U(3)*U(3) contains/implies U(3) and U(2, 2) contains/implies U(2)*U(2). The classification of SU(3) coupled states is completed by the labels of an intermediate U(2) irreducible representation h^s=(h^s₁h^s₂), directly related to King's branching rule for U( 3)*U(3)U(3). For pedagogical purposes, the SU(2) coupled state construction, based upon the complementary chains U(2)*U(2) contains/implies U(2) and U(1, 1) contains/implies U(1)*U(1), is considered first. For both SU(2) and SU(3), it is proved that, in addition to the standard recursion relations, the corresponding Wigner coefficients satisfy recursion relations of a different type, arising from the action of the U(1, 1) or U(2, 2) complementary group generators. A detailed example shows that such complementary recursion relations are quite useful for numerical purposes.

The generalised Madelung problem, that of finding the spatial electric potential within a crystal lattice, is often approached via analytic function theory. But new results can be achieved through careful application of standard theorems from classical electrostatics. In this way the authors show that the celebrated Madelung constant for the NaCl crystal can be rigorously bounded through symmetry arguments devoid of summations. They derive new expansions of Madelung constants of general crystals: an absolutely convergent 'sine' series which has the advantage of non-alternating summands, and an 'exponential' series which agrees in the simple cubic cases with the 'cosech' expansions of previous authors. Finally, the authors show how the NaCl Coulomb singularity may be removed to yield a regular power series expansion for the potential well at the origin.

The introduction of local optical potential wells of a generalized Wood-Saxon (WS) type in optical-model studies, and also in some developments of the energy density formalism, is necessary to deal with the moments of a Wood-Saxon distribution raised to a (in general non-integral) real power (WS)^beta . The analytic structure of these moments D_{n, beta} (a, R) is analysed. In fact, an expansion is derived for them that for intermediate and large values of chi =R/a reduces to a polynomial of order n in chi . At the same time, rapidly converging series expansions are obtained in order to represent the coefficients appearing in this polynomial and, in addition, a recursion relation that makes their numerical computation easier is established. Finally, by assuming a (WS)^beta parametrisation, this polynomial is applied to the study of the A dependence of several quantities of relevance in the geometrical characterisation of density distributions and optical potential wells corresponding to medium-mass and heavy nuclei.

A method of solving the initial value problem delta _t phi =( delta ²_Omega -V( Omega )) phi in an arbitrary interval is presented, given auxiliary boundary conditions which induce a unique solution. Although the Green propagator for the problem may be impossible to find directly, progress is made by finding the propagator for the same evolution equation in a different interval and with simpler boundary conditions. This primitive propagator propagates the Green propagator for the actual problem forward from its initial value. The initial value is chosen outside the interval so as to satisfy the actual boundary conditions. Examples are presented.

Studies by Handy and Bessis (1985) on the effectiveness of a Hankel-Hadamard determinant quantisation analysis are generalised to include wavefunction reconstruction. For a given bosonic ground-state wavefunction, Psi ( chi )=e^s(x)excellent pointwise convergence to dS/d chi can be readily achieved with very few moments. This approach also yields very good lower and upper bounds to dS/d chi .

The authors give rigorous bound inequalities for the universal scaling factors with their numerical values (n=3, 4, 5), of the power spectrum for a renormalisation group equation of arbitrary prime order eta by using a generalisation of the method of Collet-Eckmann-Thomas.

The concept of Painleve chains is extended to chains of ordinary differential equations obtained by successive and simultaneous differentiation of both sides of equations of the general type f( chi , u, u_x, u_2x,. . .,u_nx)= g( chi , u, u_x, u_2x,. . ., u_mx) where m<or=n-1. The three Painleve chains are thus generated by such successive differentiation of u_2x=ku² (KdV chain), u_2x=ku³ (modified KdV chain) and u_x=ku² (Burgers chain). Hybrid Painleve chains can be analogously obtained by successive differentiation of hybrid differential equations; such chains obtained from u_2x=-uu_x+u³ and u_2x=-3uu_x-u³ are described in detail. Passive differential equations in which the balancing exponent must depend upon the coefficients can also lead analogously to Painleve chains as illustrated by those obtained from u_2x=ku²_x/u where k=2 or 3/2. The Schwarzian derivative, (u_3x/u_x)-3/2(u_2x/u_x)² generates a Painleve chain in which the members have consecutive integral resonances starting with the ubiquitous -1. Chains of higher-order differential equations exhibiting some, but not all, of the features of Painleve chains can be obtained from the second-order eigenvalue problem. The dominant truncations of most evolution equations as well as the Painleve canonical equations including the irreducible Painleve transcendents appear in Painleve chains generated these methods.

The Smoluchowski coagulation equation with the kernel K_ij=A(i+j)+B is solved exactly for arbitrary initial conditions. The author obtains a compact form of the size distribution for monodisperse initial conditions. For polydisperse initial conditions, a simple form of the size distribution, including a parameter N_kl determined by a recursive relation, is obtained. In the special case with K/sub /i_j=i+j, the author obtains the compact form of the size distribution for arbitrary initial conditions.

For the tight-binding model of electron motion in a perfect crystal the authors propose a method for obtaining an ordinary differential equation satisfied by the local Green function. Such an equation probably exists for every case.

The author considers the propagation of surface Love waves in a random layered elastic medium. The material parameters in the model are stochastic functions of the vertical coordinates chi ₃: this includes the general case when the wave propagates through a statistically inhomogeneous layer. The author applies the averaging method for stochastic equations. The method essentially involves approximating the expectation of the solution with averages formed by integrating various moments of the random coefficients over 'long' times (long relative to the correlation time). This method provides higher-order corrections to the purely asymptotic approximations which are often used to solve equations of this type. It is shown that the mean wave amplitude always attenuates exponentially on the depth scale in ² chi ₃ where in is the fluctuation parameter. The author analyses the dispersion relation for a statistically homogeneous layer and finds a criterion which determines whether the mean phase velocity is decreased by the fluctuations.

The problem of finding the maximum entropy solution which satisfies some poorly defined constraints is central to data processing. Hitherto all attempts to solve it have been conceptually somewhat incomplete. The authors consider the traditional Boltzmann-Gibbs approach to entropy. When data are available the various outcomes can be assigned a priori probabilities (or weights) according to how much each of them agrees with the measured distribution when errors are taken into account. The reasoning yields a statistic S- chi ², which is a precise expression for the entropy. The maximum S- chi ² solution defines a distribution uniquely. Even in circumstances when there is no information apart from those provided by the data, the inferred distribution differs from the given by the data mean values and bears a closer resemblance to the underlying population than do the raw data. Where additional constraints are present, this maximum entropy procedure is more effective in incorporating them than are traditional maximum likelihood techniques. S- chi ² maximisation may also be carried out iteratively up to a certain limit. It offers the most advantage in situations when one has a thorough understanding of the inherent noise behaviour.

An algorithm is described for obtaining successive approximations to geometric properties K_j of a closed boundary B (such as its length L or the area A within it), given the lowest N eigenvalues (E_n) of some wave operator defined on the domain bounded by B. The technique is based on the asymptotic expansion of the partition function for small t: Phi (t)= Sigma _n=1^infinity exp(-E_nt) approximately t^-1 Sigma _j=0^infinity K_jt^j/2. Four different billiards are employed to illustrate the method. The first is the rectangular membrane, which is classically integrable; for the other three, B is an Africa shape, which is classically chaotic: Africa membrane, Africa Aharonov-Bohm billiard and Africa neutrino (massless Dirac) billiard. A typical result is that A and L can be reconstructed from 125 eigenvalues to a few parts in 10⁴ (for the rectangle the accuracy is even higher). The efficiency of the reconstruction algorithm appears to be independent of the classical chaology of B.

The equation of motion of an electron undergoing cyclotron emission in a uniform magnetic field is considered. A solution for the orbit is obtained in a fairly simple form for relativistic velocities. The conflict of basic conservation laws, as noted previously, is no longer evident in the present treatment. Difficulties in earlier works have arisen because the deviation of the electron orbit from circularity was not estimated properly. As particular consequences of the more complete theory, it is shown that (i) when the fractional energy loss per cycle approaches unity there are significant changes in the particle orbit, and (ii) when the fractional energy loss per pulse (or beam) crossing approaches unity, the radiation characteristics also exhibit modifications.

In the semiclassical limit, the sum S(E, Delta E)= Sigma _nm mod A_nm mod ² delta (E-1/2(E_n+E_m)) delta ( Delta E-(E_n-E_m)) of matrix elements of an arbitrary operator -A can be related to the classical correlation function of the Weyl symbol A(q, p) of -A: C_A(E, t)= integral d alpha delta (E-H( alpha ))A( alpha )A( alpha _t). S(E, Delta E) is proportional to the Fourier transform of C_A(E, t) over t, plus a set of correction terms associated with periodic trajectories in phase space. If the system has a chaotic classical limit, the matrix elements are independently Gaussian distributed with mean value zero, and S(E, Delta E) gives the variance of this distribution.

Taking into account Takeno and Homma's model Hamiltonian the authors have studied the influences of the longitudinal vibration on the properties of solitons in B-DNA with infinite length. It is shown that the longitudinal vibration will influence the amplitude and the width of the soliton. However, the influences due to the longitudinal vibration may be neglected ordinarily because the corrections have the magnitude of O( delta ²), where delta is a small coupling constant.

Exactly solvable generalisations of the Jaynes-Cummings model are considered which are non-linear both in bosonic and spin variables. Exact wavefunctions and energy levels are found for the corresponding systems.

An outline of a quantum approach to multi-kink profiles as coherent states in a ferromagnetic chain is given. Relying on the continuum and harmonic approximation the influence of the out-of-plane fluctuations on the stability and dynamics of multi-kink structures in a finite chain is studied and compared with the stability and dynamics of single-kink profiles in an infinite chain. Even in the harmonic approximation, very few modes are diagonal. The multi-kink structure does not exhibit a 'zero-frequency' Goldstone mode. This is a consequence of the out-of-plane fluctuations.

For pt.I see ibid vol.19, p.2297, 1986. It has been shown previously that a potential V₀( chi ) in one dimension which supports no bound states may be used as a reference potential from which, by successive applications of the concept of a supersymmetric partner to a given Hamiltonian, it is possible to find a potential V_n( chi ) which supports any specified number n of bound states at any chosen energies E_j, j=1,. . .,n. The reflection coefficient of V_n is related to the reflection coefficient of V₀. Various alternative representations of the potentials constructed by this procedure are presented. An illustrative example in which V_n is constructed by using a sech² chi barrier as the reference potential is discussed.

Using the sextic anharmonic oscillator a chi ²+b chi ⁴+c chi ⁶ as a testing ground it is shown why the Hill determinant method has a limited range of applicability. A possible improvement is suggested and discussed.

The exact dynamics of a system of N hard rods of different masses and the Boltzmann equation for an infinitely extended mixture of two kinds of hard rods are used to derive formulae for the equilibrium velocity distributions the collision frequency and the velocity autocorrelation function to first order in time. These formulae and a symmetry relation for the self-diffusion coefficients of a binary mixture are used to discuss some of the results of the molecular dynamics simulations performed by Marro and Masoliver (1985).

This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.

The authors give a rigorous proof that infinite AB percolation does not occur for any parameter value in a certain class of bipartite planar graphs. In particular, infinite AB percolation cannot occur on the square lattice.

Infinite AB percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values. It was not previously known that infinite AB percolation occurred with positive probability on any planar lattice.

The shapes of percolation clusters and lattice animals are investigated. The universal quantities Delta _d and S_d, which were introduced to measure the average asymmetry and degree of prolate- or oblateness, respectively, of long-chain polymers, are here computed in an in expansion for percolation clusters and for lattice animals. Delta _d is computed to O( in ), while S_d is computed to O(1). The clusters are shown to be on average anisotropic and prolate, but less so than polymers. Percolation clusters and lattice animals have identical shapes above eight dimensions. Below d=8 animals are slightly more anisotropic than percolation clusters.

The partition function of a finite Z-invariant six-vertex lattice model (with a prescribed arrow configuration at the boundary) is given. The expression is of the same type that occurs in the Bethe ansatz, but there are no 'wavenumber' equations to solve.

The authors simulated self-avoiding walks on the square lattice with fixed endpoints by means of a dynamic Monte Carlo algorithm. From these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v. They can therefore test the hyperscaling relation 2- alpha =dv with a careful estimate of systematic and statistical errors.

Corrections to finite-size scaling are determined numerically for several levels of the three-state Potts quantum chain with various boundary conditions. It is found that the leading correction term behaves like N^-0.8. In the case of periodic and twisted boundary conditions the coefficients of the N^-0.8 term are determined by the three-point correlation functions of the conformal theory.

The surface critical behaviour of the semi-infinite ferromagnetic n-vector Kac-Baker models is obtained. Near the ordinary and special transition points the self-consistency equations (which couple all the unknowns) are shown to be a small perturbation of a two-step recursion relation. The critical behaviour is thus shown to be controlled by the behaviour of a dynamical system near the bifurcation of a new hyperbolic fixed point. This provides both the local critical behaviour and the scaled magnetisation profile. Thereby, the range of validity of the continuum approximation is established.

The Z₂ lattice Abelian Higgs model in D=2+1 is studied. New cluster expansion techniques which involve consideration of two perturbative terms are developed. These are used to obtain strong coupling expansions, for the vacuum energy density and the mass gaps of both 'meson' and 'plaquette' excitations. The expansions are used, via Pade approximant techniques and duality considerations, to explore the phase diagram of the theory. The location of the second-order phase boundaries are in good agreement with those obtained by other methods. The behaviour of the line of first-order transitions is examined and is estimated to terminate at lambda >or approximately=4.5.

The application of quasispin in nuclear, atomic and ligand field theory is extended here to systems of electron and hole states in a metallic systems. Conversely, results known in connection with particle-hole conjugation in metals are generalised to shell theoretic applications of quasispin. The analogy between the relationship of particle hole conjugation with quasispin and the relationship of time reversal with ordinary spin is extended to the development of selection rules parallelling those associated through Kramers with time reversal. Hamiltonians such as the BCS Hamiltonian which are composed of quasispin operators with real or imaginary coefficients have definite conjugation signature and so have a special interest. Several old and new results for matrix elements within a half-filled shell are derived from this viewpoint.

The authors have justified a point-by-point optimisation procedure for parametrised perturbative approximations to wavefunctions by appealing to a principle of minimal sensitivity. Here it is shown that, if the diagonal matrix elements of the perturbation vanish (which can always be arranged via a simple redefinition of the unperturbed Hamiltonian), then the optimisation procedure is also justified by an underlying stationary principle, with the nth-order perturbative approximations being stationary to that order.

It is pointed out here that solenoidal solutions of the equation Del *a=ka exist in magnetoelectric media, in which case a must be identified as a linear combination of the electric and the magnetic fields.

It is shown that the Hamiltonian limit of a spin-¹/₂ Ising model on a D-dimensional lattice with three-spin interactions in one direction and two-spin interactions in the other directions is equivalent to the same limit of a nearest-neighbour four-state spin model.

The recently found breakdown of dynamic scaling of the kinetic Ising model (at the percolation threshold) is here shown to be exhibited, also, by the single-spin flip, dynamical q-state Potts model. The authors find A(q) mod A(2)=2(q-1) mod q, in good agreement with very recent Monte Carlo simulations.