Table of contents

A class of dynamical systems is presented which includes, as special cases, both the (autonomous) Ermakov system and central force problems of Kepler type with angular dependence of the force. It is shown that all members of this class are linearizable up to a pair of quadratures.

The authors examine the dynamics of interface growth under the combined influence of a stabilizing and a destabilizing surface current in 1+1 dimensions. It has been shown that the surface is weakly stabilized if a newly deposited particle can move locally to maximize the total number of nearest-neighbour contacts with the surface (curvature-driven current). The introduction of a weak, destabilizing surface tension term results in an almost periodic, grooved surface. The width of the surface grows linearly in time and does not reach a steady state. It appears that such a formation cannot be described by a continuous Langevin equation.

Motivated by recent simulation results of the suppression of interfacial roughening, the authors study analytically the interfaces in non-equilibrium steady states for systems with bulk conservation. The general method of deriving interfacial properties from the bulk is developed, and the necessity of employing the full dynamics, even for static properties, is exhibited. Applying this to the randomly driven diffusion system, they obtained novel results. Specifically, height-height correlation diverges weakly as 1/q for small wavevector q, thus displaying the first analytical evidence of roughening suppression for driven systems.

The Green function of a tight-binding model with mixed impurities is calculated exactly. The spectral properties of the one-dimensional lattice, including both extended and localized states, are also analysed exactly. The authors find at most two localized states which are always non-degenerate. The density of states per site in the continuum is calculated. Using a constructive procedure they obtain the localized eigenfunctions for the infinite and semi-infinite chain. Some results for three-dimensional lattices are presented.

The thermal processes in the hydrodynamic stage are investigated within the formalism of nonequilibrium thermofield dynamics (NETFD) where the concepts in nonequilibrium thermodynamics are implanted. The treatment gives a good indication for an understanding of the new concept, the spontaneous creation of dissipation, in conjunction with the realization of the representation space within NETFD.

Different contractions are applied to the quantum superalgebra osp_q(1 mod 2). In the first of them the graded analogue s-H_q(1) of the one-dimensional Heisenberg quantum algebra is obtained and its R-matrix explicitly calculated. In the second contraction a Z₂-graded version of the q-oscillator is proposed and finally the supersymmetric two-dimensional Euclidean quantum algebra s-E_q(2) is found.

Explicit realizations of the quantum groups GL_p,q(2) and GL_p,q(1 mod 1) corresponding to unimodular values of the deformation parameters p and q are given in terms of the canonically conjugate (X, P) operators, using the Heisenberg-Weyl commutation relations. Matrix representations are also discussed. Some observations are made on similar realizations of the non-commutative coordinate spaces on which the quantum groups act as endomorphisms.

A new method for finding generators for the normalizer of an n-dimensional crystallographic (arithmetic) point group is described. First a set of generators for the centralizer is determined, whereafter the completeness of the found set is checked. After evaluating all inner automorphisms, representatives of the outer automorphisms, if existent, are determined. A complete generating set for the normalizer of some point groups for n=5,6 is determined with use of an algorithm, based upon this method.

The author exploits local and non-local symmetries of the two equations of the title and of some related equations. Some new exact solutions are derived and some specific boundary value problems of physical significance are considered.

Using two different methods of calculation of the imaginary part of some Feynman integral, the author obtains an explicit formula for a definite integral (over z from 0 to a) with modified Bessel function I₀(z) and weight function w(z)=z cosh (AZ) mod Z, where Z= square root (a²-z²), and A and a are real parameters.

Lie operator techniques and the Magnus expansion are developed in the framework of classical mechanics. This leads to an exponential perturbation theory that preserves the canonical character at each order of approximation. The treatment is kept as close as possible to the quantum mechanical case in order take full advantage of the properties of the expansion. The explicit relationship with secular perturbation theory is established and a recursive procedure for obtaining higher-order approximants is provided. Finally, the formalism is applied to two problems of physical interest.

A system of dissipative modes in an incompressible flow of space dimensions d>or=2 is considered. The self-induced phase chaos is shown to arise in the motions of very small scales, which are fed by the large-scale ones through repeated nonlinear interactions. This property is used to derive the equations for the Fourier amplitudes. Solutions similar to those derived previously for turbulent fluctuations in the dissipation range are obtained. Properties of the short-scale intermittency are analysed. The authors show that no coherence and intermittency can be built up at asymptotically high wavenumbers.

The Cartan structure of the dynamical superalgebra of the t-J model of strongly correlated electrons is found. This is used to define and explicitly construct the coherent states for this algebra and calculate the invariant measure for the completeness relation for the coherent states.