Table of contents

The authors consider the isospectral flows of ( delta ²+ Sigma ₁^N upsilon _i lambda ⁱ) Psi = alpha Psi . Using an unusual form of the 'Lax approach' they derive in a particularly simple manner: (a) the 'second half' of the Lax pair; (b) the associated Hamiltonian structures; (c) an infinite hierarchy of Poisson commuting Hamiltonians. In this way they show that these equations possess (N+1) compatible, purely differential Hamiltonian structures. The case N=1 is just the bi-Hamiltonian Harry Dym hierarchy. They thus extend the recent results on multi-Hamiltonian coupled KdV equations.

The structure of generalised Lie-Backlund symmetries for the coupled equations of Langmuir solitons are analysed in detail. The form of these symmetries, when compared with those of the conservation laws, yields the first symplectic form. The spectra gradient method is then seen to generate the recursion operator for these symmetries which, on factorisation, leads immediately to the second Hamiltonian structure. The same recursion operator, when used along with the (x,t)-dependent symmetries, yields a new class of generalised symmetries for the equations under consideration. Lastly it is observed that these symmetries are in involution with respect to a Jacobi bracket.

The supersymmetric extension of the system consisting of a relativistic scalar particle interacting with a gauge field is discussed. The construction is based on the ground-state wavefunction representation for supersymmetric quantum mechanics. The authors exploit the fact that any system invariant under spacetime reparametrisations has a vanishing Hamiltonian.

The properties of metastable states at zero temperature are examined numerically for the infinite-ranged Ising spin-glass model. It is shown that the energy levels of the metastable states behave like a random energy model. Furthermore, it is found that the barrier energy between them is an increasing function of the Hamming distance. The result is consistent with the prediction that the metastable states have an ultrametric organisation in the phase space.

Considers a far-from-equilibrium dissipative system sustaining a centre manifold. The decomposition in correlated subsystems inducing organisation by means of a nucleation process is analysed. The average lifetime of the organised subsystems determines the width of the strip of probability about the centre manifold in macrostate space.

The transition from random fractal structures to dendritic patterns in crack propagation in an elastic medium is shown to be a consequence of averaging out fluctuations, as already found for the growth of structures described by scalar fields (such as for diffusion-limited aggregation or dielectric breakdown).

The authors study the moments of the first passage time t of random walks on hierarchical structures using the exact enumeration method. They find that the different moments scale with the size of the system with an infinite hierarchy of exponents characterising a multifractal behaviour.

An Ising model on a regular Cayley tree, with competing ferro- and antiferromagnetic nearest-neighbour interactions, can be formulated as a discrete two-dimensional mapping. The authors use this mapping to obtain sequences of modulated phases, at low temperatures, associated with complete devil's staircases. The fractal dimensionality of the staircases increase with temperature. At high temperatures the incommensurate phases may occupy regions of finite measure of the phase diagram.

A new algorithm recently proposed by Pomeau (1984) and Herrmann (1986) for the microcanonical simulation of the Ising model is very efficiently implemented on a DAP computer. Its dynamical properties are studied and compared with those of the standard Metropolis algorithm.

Invariant and parameter-invariant solutions of the classical Euclidean SU(2) Yang-Mills equation are studied. It is shown that many interesting solutions of the Yang-Mills equation are parameter invariant and that thereby the solutions with finite action value (instantons) correspond to the parameter-invariant solutions with maximal symmetry. Some new classes of exact solutions are also obtained.

The authors study the stability problem for bound states of (1+1)-dimensional scalar fields considering different definitions of stability. Sharp stability and instability conditions for Liapunov stability are established by improving the Shatah-Strauss formalism (1985) and a recent technique of Grillakis, Shatah and Strauss (1987) based on an analysis of the linearised operators. The linear dynamical stability is investigated for the first time and a sharp stability condition is obtained. Explicit results for the regimes of stability and instability are given for power-like self-interactions and for the phi ⁴- phi ⁶ model.

As a specific example of billiards in a moving boundary, the authors discuss the classical motion on a circular table rotating uniformly about a point on its edge. Computer evidence for periodic and chaotic motion is displayed and they expose the origin of the chaotic motion as due to the effects of curvature of trajectories in the rotating frame.

The authors construct, by using the KdV-Virasoro connection, Z₃-symmetric conformal algebra from the KdV-type equation; the Boussinesq equation. Also they find that the Miura transformation of the Boussinesq equation is equivalent to the definition of the energy-momentum tensor, T, and the spin-3 field, W.

The statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles. The author explains this correspondence, including the fact that long-ranged spectral statistics show deviations from the results of random matrix theory. The method depends on the ambiguity of quantisation of a given classical system; although the energy levels depend on the particular quantisation used, the spectral statistics are assumed to depend only on the classical motion. The ambiguity of quantisation can be represented by a small perturbation acting on the Hamiltonian. This perturbation executes a random walk, which causes its matrix elements to undergo a random walk. If the matrix elements are completely uncorrelated, the spectral statistics are those of random matrix theory. In the case of classically chaotic systems, the matrix elements are constrained by sum rules related to classical periodic orbits. This leads to deviations of the long-ranged spectral statistics from the predictions of random matrix theory.

The problem of mixing of degenerated sublevels of a hydrogen-like system by a charged particle is reconsidered from the point of view of the theory of integrable systems on Lie algebras. The Lax representation is constructed both in classical and quantum mechanics. The corresponding dynamical system is a special limit case of a four-sites xxx magnet considered by Gaudin (1976).

Proves that for dimensions d=1, 2 the polaron (with general dispersion relation and couplings) has a unique ground state for any value of the total momentum (particle+Bose field). For d>or=3 and for sufficiently small total momentum the author shows the existence of a unique ground state, extending however the domain of uniqueness known previously.

A fourth-order nonlinear Thomas-Fermi equation describing an assembly of fermions moving under the action of two-body exponential forces is obtained. Its physical boundary conditions and numerical solutions are discussed.

The authors present a detailed study of the emergence and the long-time behaviour of coherent vortices in two-dimensional decaying turbulence. By high-resolution numerical experiments they find that the coherent structures are self-similar, i.e. their energy, enstrophy and size satisfy scaling laws. Moreover, the knowledge of the statistical distribution of the size of these vortices is sufficient to compute the energy spectrum of the 2D turbulent flow and to explain the significant deviations from the Kraichnan-Batchelor theory. At long times the motion of the fluid is dominated by the vortex dynamics, which is strikingly similar to the Hamiltonian motion of few point vortices, as is confirmed by a comparison between the numerical simulations of the two systems.

A detailed investigation has been carried out of the relaxation of fluctuations in the steady state of the Stratonovich model, also known as the random growing rate model. The autocorrelation time of this system, driven by parametric white noise in the linear term, has been measured for an electronic circuit model, computed by digital simulation and calculated by use of a continued fraction expansion method. The results are all consistent with each other, and with those obtained from the matrix continued fraction method of Jung and Risken (1985), provided that explicit account is taken of the ways in which any macroscopic real physical system is bound to differ from the idealisation represented by the original form of the Stratonovich model. In particular, it is necessary to recognise the existence of weak additive noise and a small additive constant. The physical origins of some earlier, seemingly discrepant, calculations and experimental data are discussed.

The scattering of a soliton from a mass impurity in a Morse or Toda one-dimensional lattice with periodic boundary conditions is studied numerically. The energy of the soliton as a function of time exhibits either a fast decay (chaotic behaviour) or a recurrence, or an intermediate 'metastable' behaviour, which consists of a time period when recurrence takes place, followed by a fast decay. A model is developed to explain the recurrence and predict the recurrence time. A semiquantitative argument is also presented to explain the 'metastable' case. Finally the generality of this type of behaviour is discussed.

To investigate the role of fluctuations in diffusion-limited growth the authors apply the noise reduction algorithm to the problem of two-dimensional deposition on a substrate consisting of a straight line. The authors consider the scaling of the height and the width of trees growing on the square lattice. For the sizes studied they find two length scales characterised by different exponents depending on the noise reduction parameter m. Using these exponents the effective fractal dimension D_eff of trees is calculated and for m>>1 they obtain D_eff approximately=1.57. The density profile in the direction of growth and the scaling properties of the lateral correlation function are also determined.

A stochastic cellular automata rule obtained by adding two rule components with probabilities 1-c1 and 1-c4 to Conway's 'Life' is studied both by computer simulation and theoretically. A line of phase transitions from an ordered domain-structured phase to a disordered one is observed and explained by introducing an inhomogeneous mean-field approximation which incorporates important correlations.

The authors show the equivalence between the two canonical formulations in any coordinate system of magnetic field line equations of Janaki and Ghosh (1987), and of Cary and Littlejohn (1983), through the expression of the magnetic field in terms of the vector potential.

The MFRG method is used to study the critical behaviour of the q-state Potts antiferromagnet in three dimensions. The q dependence of the critical temperature T_c as well as the exponent v are derived, and the results are compared with Monte Carlo data and predictions of mean-field theory.

The expression bs^x=constant is proposed as a suitable and possibly correct description of site-bond percolation phase boundaries. Bond thresholds, b, as a function of site thresholds, s, for this model agree with published numerical results to better than +or-2%. The exponent X and the constant are determined solely by the site-only and bond-only thresholds, s_c and b_c, so that the model may be applied to any percolation problem for which these are known.