Table of contents

It is shown that the geometrical optics limit of the Maxwell equations for certain nonlinear media with slow variation along one axis and particular dependence of the dielectric constant on frequency and fields gives rise to the dispersionless Veselov–Novikov equation for the refractive index. It is demonstrated that the dispersionless Veselov–Novikov hierarchy is amenable to the quasiclassical -dressing method. A connection is noted between the geometrical optics phenomena under consideration and the quasiconformal mappings on the plane.

A scaling invariance in the Lorenz model allows one to consider the usually discarded case σ = 0. We integrate it with the third Painlevé function.

We present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half-filled ring of size L by using the Bethe ansatz. We show that, in the large-L limit, the Bethe equations reduce to a simple transcendental equation involving the polylogarithm, a classical special function. By solving that equation, the gap and the dynamical exponent are readily obtained. Our method can be extended to a system with an arbitrary density of particles.

A two-species particle model on an open chain with dynamics which is non-conserving in the bulk is introduced. The dynamical rules which define the model obey a symmetry between the two species. The model exhibits a rich behaviour which includes spontaneous symmetry breaking and localized shocks. The phase diagram in several regions of parameter space is calculated within the mean-field approximation, and compared with Monte Carlo simulations. In the limit where fluctuations in the number of particles in the system are taken to be zero, an exact solution is obtained. We present and analyse a physical picture which serves to explain the different phases of the model.

In this paper we have calculated the escape rate from a meta stable state for coloured and correlated noise driven open systems based on the Fokker–Planck description of the stochastic process. We consider the effect of two correlation times due to the additive coloured noise and the correlation between additive coloured and multiplicative white noises. The effect of the noise correlation strength on the rate has also been investigated.

The one-dimensional saddle point equation of the Ginzburg–Landau Hamiltonian with random temperature is studied with a numerical method. The random temperature is correlated with a finite range l. The distribution width of the random temperature is Δ. The ground state of the saddle point equation is solved. The average, fluctuation and auto-correlation of the order parameter are obtained. It is found that the auto-correlation function behaves like ∼exp(−x²/ξ²_ϕ). For Δ ≫ 1/l², where l is dimensionless, the correlation length is given by ξ_ϕ ∝ l. For Δ < 1/l², as t = 0, the correlation length ξ_ϕ ∝ lΔ^−α, where α = 0.65. All the saddle point solutions for Δ < 1/l² can be mapped to that for Δ = 1/l² by using a coarse-grained approximation.

The influence of time-dependent control parameters on time-delayed feedback schemes for control of chaos is investigated by analytical means. For the logistic map the linear stability of the period-two orbit subjected to a modulated time-delayed feedback loop is calculated. We find enhanced control performance due to phase lags between the periodic orbit and the controller.

When calculating molecular electronic energies, the contributions involving the Coulomb operator for bielectronic terms are required rapidly and to high chemically significant accuracy. The atomic orbital basis functions chosen in the present work are Slater-type functions (STFs). These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier-transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of two- or three-dimensional integral representations. In this work, we present a generalized algorithm for a precise and fast numerical evaluation of molecular integrals over STFs. Numerical results obtained with C₂H₂, C₂H₄ and CH₄ molecules show the efficiency of the approach presented in this work. Comparisons with the existing codes are also listed.

We consider the Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann 'windows' of the same length, the centres of which are 2l apart, and study the asymptotic behaviour of the discrete spectrum as l → ∞. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit l → ∞. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue.

Introducing the associated hypergeometric functions in terms of two non-negative integers, we factorize their corresponding differential equation into a product of first-order differential operators by four different ways as shape invariance equations. These shape invariances are realized by four different types of raising and lowering operators. This procedure gives four different pairs of recursion relations on the associated hypergeometric functions.

The distribution of atoms in quasi-crystals lacks periodicity and displays point symmetry associated with non-crystallographic modules. Often it can be described by quasi-periodic tilings on built from a finite number of prototiles. The modules and the canonical tilings of five-fold and icosahedral point symmetry admit inflation symmetry. In the simplest case of stone inflation, any prototile when scaled by the golden section number τ can be packed from unscaled prototiles. Observables supported on for quasi-crystals require symmetry-adapted function spaces. We construct wavelet bases on for the icosahedral Danzer tiling. The stone inflation of the four Danzer prototiles is given explicitly in terms of Euclidean group operations acting on . By acting with the unitary representations inverse to these operations on the characteristic functions of the prototiles, we recursively provide a full orthogonal wavelet basis of . It incorporates the icosahedral and inflation symmetry.

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation.

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a†] = 1) monomials of the form exp[λ(a†)^ra], r = 1, 2, ..., under the composition of their exponential generating functions. They turn out to be of Sheffer type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a) the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobiński-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobiński-type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.

A method is proposed for obtaining certain solutions (TE-polarized electromagnetic waves) of the Helmholtz equation, describing the reflection and transmission of a plane monochromatic wave at a (linear or nonlinear) dielectric film situated between two linear semi-infinite media. All three media are assumed to be lossless, nonmagnetic and isotropic. The permittivity of the film is modelled by (i) a continuously differentiable real-valued function of the transverse coordinate, and by (ii) a Kerr-nonlinearity. It is shown that the solution of the Helmholtz equation exists in the form of a uniformly convergent series (in case (i)) and in the form of a uniformly convergent sequence (in case (ii)) of iterations of the equivalent Volterra integral equation. Numerical results of the approach are presented.

It is proposed to use the Lie group theory of symmetries of differential equations to investigate the system of equations describing a static star in a radiative and convective equilibrium. It is shown that the action of an admissible group induces a certain algebraic structure in the set of all solutions, which can be used to find a family of new solutions. We have demonstrated that, in the most general case, the equations admit an infinite parameter group of quasi-homologous transformations. We have found invariants of the symmetry groups which correspond to the fundamental relations describing a physical characteristic of the stars such as the Hertzsprung–Russell diagram or the mass–luminosity relation. In this way we can suggest that group invariants have not only purely mathematical sense, but their forms are closely associated with the basic empirical relations.

We present a new method for the solution of the Schrödinger equation applicable to problems of a non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: an asymptotic scale, which depends uniquely on the form of the potential at large distances; an intermediate scale, still characterized by an exponential decay of the wavefunction; and, finally, a short distance scale, in which the wavefunction is sizable. The notion of optimized perturbation is then used in the last two regimes. We apply the method to the quantum anharmonic oscillator and find it suitable to treat both energy eigenvalues and wavefunctions, even for strong couplings.

Here, we show how the molecular structure appears and becomes stable for supercritical physical conditions. In particular we consider, for ammonia-type molecules, a simplified model based on a standard non-linear double-well Schrödinger equation with a dissipative term and a perturbative term representing weak collisions.

The challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states (density operators). A family of tripartite states satisfying equality is derived.

We investigate the vacuum expectation values of the energy–momentum tensor and the fermionic condensate associated with a massive spinor field obeying the MIT bag boundary condition on a spherical shell in the global monopole spacetime. In order to do that, we use the generalized Abel–Plana summation formula. As we shall see, this procedure allows us to extract from the vacuum expectation values the contribution coming from the unbounded spacetime and to explicitly present the boundary induced parts. As regards the boundary induced contribution, two distinct situations are examined: the vacuum average effects inside and outside the spherical shell. The asymptotic behaviour of the vacuum densities is investigated near the sphere centre and near the surface, and at large distances from the sphere. In the limit of strong gravitational field corresponding to small values of the parameter describing the solid angle deficit in the global monopole geometry, the sphere induced expectation values are exponentially suppressed. We discuss, as a special case, the fermionic vacuum densities for the spherical shell on the background of the Minkowski spacetime. Previous approaches to this problem within the framework of the QCD bag models have been global and our calculation is a local extension of these contributions.