Table of contents

A system of two equations governing the irrotational flow of a capillary fluid is shown, for a particular class of free energy functions, to reduce to a nonlinear Schrödinger equation.

We examine the Fourier - Gauss transformation properties of a two-parameter family of the continuous big q-Hermite polynomials.

We discuss the non-uniqueness of the generating partitions of two-dimensional maps, and we give new explanations for ambiguities that are to be expected in the construction of generating partitions if a parameter is varied. Both points are discussed with the help of the connecting lines of homoclinic tangencies in the entire phase space, i.e. also outside the attractor, that can be calculated very easily with a new algorithm introduced recently.

An analogue electronic experiment is used to demonstrate that large fluctuations in a thermal equilibrium system exhibit time-reversal symmetry and, furthermore, that they arise through the action of an optimal force that can be predicted on the basis of Hamiltonian fluctuation theory.

We present a large-scale computer simulation of the prototypical three-dimensional continuum percolation model consisting of a distribution of overlapping (spatially uncorrelated) spheres. By using simulations of up to particles and studying the finite-size scaling of various effective percolation thresholds, we obtain a value of . This value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems. Employing this value of and systems of size L = 160 (relative to a sphere of unit radius), we also obtain estimates of the critical exponents , and for the continuum system and show that the values are different than those obtained using previous values of .

The correlations of the free-energy landscape of mean-field spin glasses at different temperatures are investigated, concentrating on models with a first-order freezing transition. Using a `potential function' we follow the metastable states of the model in temperature, and discuss the possibility of level crossing (which we do not find) and multifurcation (which we find). The dynamics at a given temperature starting from an equilibrium configuration at a different temperature is also discussed. In the presence of multifurcation, we find that the equilibrium is never achieved, leading to an aging behaviour at slower energy levels than usual aging. The relevance of the observed mechanisms for real structural glasses is discussed, and some numerical simulations of a soft sphere model of glass are presented.

The statistical properties of the length of the cycles and of the weights of the attraction basins in fully asymmetric neural networks (i.e. with completely uncorrelated synapses) are computed in the framework of the annealed approximation which we previously introduced for the study of Kauffman networks. Our results show that this model behaves essentially as a random map possessing a reversal symmetry. Comparison with numerical results suggests that the approximation could become exact in the large-size limit.

A consistent theory of the ground-state energy and its splitting due to the process of tunnelling for the Lipkin - Meshkov - Glick (LMG) model is presented. We calculate accurately the trivial and the instanton saddle-point contributions to the functional integral for the partition function of the model, in terms of the spin coherent states. We show that such a calculation has to be performed very accurately taking into account the discrete nature of the functional integral. This accurate consideration leads to the replacement of the magnitude of the spin s by s + 1/2, in the formula for the ground-state splitting obtained by a naive continuous method. We compare the numerical calculation of the ground-state energy and the splitting due to tunnelling with the results obtained by the quasiclassical method and obtain excellent agreement.

A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann - Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some , and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals 1). The surface susceptibility has been evaluated exactly. For we find that is finite at the bulk critical temperature , in contrast to the recently derived value of in the case of just one global spherical constraint. The result is only recovered if , where is the dimensionless critical coupling. When , diverges exponentially as . An effective Hamiltonian is also proposed which leads to an exactly solvable model with , the value for the limit of the corresponding model.

We consider steady states of the one-mode quantized field interacting with two independent baths, each characterized by the one- and two-photon absorption and emission processes. In the absense of two-photon emission, using an exact analytical solution to the master equation for the diagonal elements of the density matrix in the Fock basis in terms of the confluent hypergeometric function, we obtain simple explicit expressions for the photon distribution function and for the factorial moments in the limiting cases of weak and strong two-photon absorption. If the two-photon absorption is strong enough, the steady state exhibits a sub-Poissonian photon statistics characterizing nonclassical behaviour, but Mandel's -parameter cannot be less . However, the distribution depends essentially on the temperature of the `one-photon bath'. For weak two-photon absorption, the stationary distribution is Gaussian, provided that the temperature of the `one-photon' bath is high enough. For an inversely populated `one-photon' bath, the -parameter is close to . In a generic case of nonzero two-photon emission probability, approximate asymptotic expressions for the factorial moments are found.

A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low-density inhomogeneous phase to a high-density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly. The phase transition is shown to persist in both cases with the critical densities being higher than that for random sequential dynamics. The steady-state velocities are related to the fugacity of a Bose system suggesting a principle of minimization of velocity. A generalization of the dynamics, to the case where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.

Based on the bosonization of vertex operators for the face model by Asai, Jimbo, Miwa and Pugai, using vertex-face correspondence we obtain vertex operators for the -symmetric Belavin model, which are constructed by deformed boson oscillators.

On - off intermittency is an ubiliquitous phenomenon observed beyond the instability of a particular chaotic motion. The system below the instability reveals a typical transient chaos. It is found that near the instability point the inverse characteristic first passage time satisfies the scaling law with the transverse Lyapunov exponent (TLE) , and being respectively an initial condition-dependent characteristic TLE and the fluctuation intensity of local transverse expansion rate. Numerical simulation for several systems suggests that the scaling function g(z) is a universal function.

We calculate connected correlators in Gaussian orthogonal, unitary and symplectic random matrix ensembles by the replica method in the 1/N-expansion. We obtain averaged one-point Green's functions up to the next-to-leading order , wide two-level correlators up to the first non-trivial order and wide three-level correlators up to the first non-trivial order by carefully treating fluctuations in saddle-point evaluation.

A new method to analyse Glauber dynamics of the Sherrington - Kirkpatrick (SK) spin glass model is presented. The method is based on ideas used in the classical kinetic theory of fluids. It is applied to study spin correlations in the high-temperature phase of the SK model at zero external field. The zeroth-order theory is equivalent to a disorder-dependent local equilibrium approximation. Its predictions agree well with computer simulation results. The first-order theory involves coupled evolution equations for the spin correlations and the dynamic (excess) parts of the local field distributions. It qualitatively accounts for the error made in the zeroth approximation.

As models for polymer diffusion we consider the motion in two dimensions and three dimensions of four random walkers restricted by different holonomic constraints. The random walkers perform uncorrelated steps, which obey algebraic waiting-time distributions. We provide numerical results for the centre-of-mass motion and analytical approximations for the short- and long-time diffusion constants. Distinct from the two- and three-walkers problem we encounter here - depending on the constraints - partial nontrivial decoupling of the motion.

Fermi edge absorption singularities (FES) are studied using a combination of conformal field theory (CFT), an exact sum rule and numerical work on a tight-binding model which is shown to exhibit remarkable simplifying features. The relationship between FES and the Anderson orthogonality exponent is established in great generality, using CFT, including the case where the core hole potential produces a bound state. Universal results on the absorption intensity in a finite-sized sample are obtained. Various predictions are checked numerically and the evolution of the absorption intensity with electron density is studied.

A momentum representation for p-adic quantum mechanics is constructed. It uses a Fourier transform in the -space with respect to the p-adic Lebesgue distribution and a study of its properties. The momentum representation is used to investigate the spectral properties of the p-adic momentum operator and to study the (p-adic valued) Schrödinger equation for the p-adic harmonic oscillator.

We show how the widely used concept of spontaneous symmetry breaking can be explained in causal perturbation theory by introducing a perturbative version of quantum gauge invariance. Perturbative gauge invariance, formulated exclusively by means of asymptotic fields, is discussed for the simple example of Abelian U(1) gauge theory (Abelian Higgs model). Our findings are relevant for the electroweak theory, as pointed out elsewhere.

The projection of optical images on warped screens is a canonical transformation of phase space between flat and warped evolution-parameter surfaces. In mechanics, the evolution parameter is time; in geometric optics it is the optical axis of coordinate space. We consider the specific problem of bringing to focus an axis-symmetric aberrating optical system by warping the output screen. The solution for the surface curvature coefficients is given in terms of the Lie aberration coefficients of the system; a linear optimization strategy applies.

The `n-component Eckhaus equation',

is C-integrable. Here the dependent variable is an n-vector, and and are two constant -matrices restricted by the condition that both and be Hermitian. Some special cases, and some explicit solutions, are displayed.

The Drinfeld - Sokolov reduction method has been used to associate extensions of the matrix r-KdV system with the Lie algebra . We present reductions of these systems to the fixed point sets of involutive Poisson maps, implementing the reduction of to classical Lie algebras of type B, C, D. Modifications corresponding, in the first place to factorization of the Lax operator, and then to Wakimoto realizations of the current algebra components of the factorization, are also described.

We present a method for calculating the energies of bound and quasibound states of quantum-mechanical problems expressed in terms of coupled-channel equations. The approach is based on a rational approximation to the logarithmic derivative of the eigenfunction and leads to a simple quantization condition.

A geometrical description of mechanical systems subjected to impulsive non-holonomic constraints is given. Their motions are determined by means of suitable projection operators which allow us to evaluate the instantaneous jumps of momenta due to the action of impulsive reaction forces. Several paradigmatic examples are investigated from this viewpoint: a disk falling into a plane, a rolling ball hitting a wall and the collision of two balls.

In the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic. In the governing equations we keep linear terms that account for both second- and third-order dispersion and nonlinear terms describing both nonlinear dispersion and self-steepening of the pulse edge. We then perform the Painlevé singularity structure analysis of the most general system of coupled partial differential equations we derived. In a specific case, when third-order dispersion is negligible, by using a Hirota-like method, we found zero- and one-parameter families of bright (fundamental frequency) and dark (second harmonic) solitary waves which travel at a locked group velocity.

In this paper we first reformulate a non-integrability criterion obtained by Yoshida for Hamiltonian systems with two degrees of freedom in order to make it easier to handle those problems whose natural formulation is given in polar coordinates, as occurs with those that have harmonic potential. Among other applications, we prove the non-integrability of the satellite problem under McCullagh's approximation of the potential, i.e. truncated at the term that, in most cases, is the main problem of the satellite of a triaxial primary body, hence its importance.

Recently Ismail, Masson and Suslov established a continuous orthogonality relation and some other properties of a -Bessel function on a q-quadratic grid. Askey suggested that the `Bessel-type orthogonality' found in the above paper at the -level really has a general character and can be extended up to the -level. Very well poised -fuctions are known as a nonterminating version of the classical Askey - Wilson polynomials. In this paper we prove Askey's conjecture and discuss some properties of the orthogonal -functions. Another type of the orthogonality relation for a very well poised -function was recently found by Askey, Rahman and Suslov.