Table of contents

We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.

In this paper we study the nonlinear stability of Ekman-Hartmann-type boundary layers in a rotating magnetohydrodynamics flow in a half-space and between two planes. We prove rigorously that if the Reynolds number defined on boundary-layer characteristics is smaller than a critical value, the boundary layer is nonlinearly stable. It is shown that the normal component of the magnetic field increases the critical Reynolds number for instability.

In this paper we present the second part of a study of spherically-symmetric solutions of the Schrödinger-Newton equations for a single particle (Penrose R 1998 Quantum computation, entanglement and state reduction Phil. Trans. R. Soc. 356 1-13). We show that there exists an infinite family of normalizable, finite energy solutions which are characterized by being smooth and bounded for all values of the radial coordinate. We therefore provide analytical support for our earlier numerical integrations (Moroz I M et al 1998 Spherically-symmetric solutions of the Schrödinger-Newton equations Classical and Quantum Gravity 1998 15 2733-42).

It will be shown that the smooth conjugacy class of an S-unimodal map which has neither a periodic attractor nor a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M Shub and D Sullivan (1985 Expanding endomorphism of the circle revisited Ergod. Theor. Dynam. Sys. 5 285-9) for smooth expanding maps of the circle.

Instantaneous gelation in the addition model with superlinear rate coefficients is investigated. The conjectured post-gelation solution is shown to arise naturally as the limit of solutions to some finite approximations as the number of equations grows to infinity. Non-existence of continuous solutions to the addition model is also established in that case.

The Skyrme-Faddeev system, a modified O(3) sigma model in three space dimensions, admits topological solitons with nonzero Hopf number. One may learn something about these solitons by considering the system on the 3-sphere of radius R. In particular, the Hopf map is a solution which is unstable if .

We study the recently proposed convection-diffusion model equation , with a bounded function . We consider both strictly monotone dissipation fluxes with , and nonmonotone ones such that . The novel feature of these equations is that large amplitude solutions develop spontaneous discontinuities, while small solutions remain smooth at all times. Indeed, small amplitude kink solutions are smooth, while large amplitude kinks have discontinuities ( subshocks). It is demonstrated numerically that both continuous and discontinuous travelling waves are strong attractors of a wide classes of initial data. We prove that solutions with a sufficiently large initial data blow up in finite time. It is also shown that if is monotone and unbounded, then u_x is uniformly bounded for all times. In addition, we present more accurate numerical experiments than previously presented, which demonstrate that solutions to a Cauchy problem with periodic initial data may also break down in a finite time if the initial amplitude is sufficiently large.

We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of such systems may be studied with the help of the SU(2) coherent states and the generalized Husimi distribution.

We establish the existence of a finite-energy solitary wave in a two-dimensional planar ferromagnet which moves rigidly at any constant velocity v that is smaller than the magnon velocity c. The shape of the calculated soliton depends crucially on the relative magnitude of v and c. For v<<c, the soliton describes a widely separated vortex-antivortex pair undergoing Kelvin motion at a relative distance . There exists a crossover velocity v₀ at which the vortex-antivortex character is lost and the energy-momentum dispersion develops a cusp. For v₀<v<c, the soliton becomes a lump with no apparent topological features and solves the modified KP equation in the limit . We also describe briefly a similar calculation of a vortex ring in a three-dimensional planar ferromagnet. These results together with the analytically known one-dimensional kink provide an interesting set of semitopological solitons whose physical significance is yet to be explored.

A 3D volume-preserving system is considered. The system differs by a small perturbation from an integrable one. In the phase space of the unperturbed system there are regions filled with closed phase trajectories, where the system has two independent first integrals. These regions are separated by a 2D separatrix passing through non-degenerate singular points. Far from the separatrix, the perturbed system has an adiabatic invariant. When a perturbed phase trajectory crosses the two-dimensional separatrix of the unperturbed system, this adiabatic invariant undergoes a quasi-random jump. The formula for this jump is obtained. If the geometry of the system allows for multiple separatrix crossings, the destruction of adiabatic invariance is possible, leading to chaotic behaviour in the system. An example of such a system is given.

Let M be a compact metric space and g: M --> M be an homeomorphism C⁰-close to an expansive map of M. In general, it is not true that g is also expansive, but it still has some properties resembling the expansivity. In fact, if we identify pairs of points whose g-orbits stay nearby, both for the future and the past, we obtain an equivalence relation~. The quotient space M / ~ is a compact, metric space and g induces an expansive homeomorphism on that quotient. If M is a surface, we show that for any the connected component of the local stable (unstable) set containing is nontrivial and arc-wise connected.

AMS classification scheme numbers: 58F30, 58F15, 54H20, 54F15

An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively.

Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper.

Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation.

Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows.

AMS classification scheme numbers: 35C05, 35Q51, 35Q53, 35Q58, 47D50, 47N20

The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincaré-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian.

AMS classification scheme number: 58F05

The classification of turning curves begun by Branner and Hubbard is completed with the critically recurrent case. Examples are constructed showing that it is possible for such a curve to be of finite length. The relation of persistent recurrence to the length of the turning curve is also addressed.

AMS classification scheme numbers: 58F23, 30D05

The von Neumann entropy production for a quantum mechanical kicked rotor coupled to a thermal environment is calculated. This rate of entropy increase is shown to be a good criterion to distinguish between quantum mechanical counterparts of chaotic and regular classical motion. We show that for high temperatures the entropy production rate increases linearly with the Kolmogorov-Sinai entropy of the classical system. However, for lower temperatures we also show that there are fluctuations in this linear behaviour due to dynamical localization.

PACS Numbers: 0545, 0365

In this paper, we made extensive use of a theorem by A Araújo and R Mañé (the theorem is stated in our article). When writing our paper, we had a detailed discussion with Ricardo Mañé about this (still unpublished) result and about the way we intended to use it. Unfortunately, Ricardo died in 1995, and the proof of this theorem was never published (although a manuscript exists that sketches part of the proof). Very recently, E Pujals and M Sambarino [1] gave a proof of a very similar result. This new result is weaker than the one announced by A Araújo and R Mañé, and cannot be used to prove the result of our paper.

Thus, for the time being, the Araújo-Mañé theorem, and consequently our theorem, remain conjectures.

The normal form of an axially symmetric perturbation of the isotropic harmonic oscillator is invariant under a 2-torus action and thus integrable in three degrees of freedom. The reduction of this symmetry is performed in detail, showing how the singularities of the reduced phase space determine the distribution of periodic orbits and invariant 2-tori in the original perturbation. To illustrate these results a particular quartic perturbation is analysed.

AMS classification scheme numbers: 34C20, 58F05, 58F30, 70H33, 70J05, 85A05