Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations

Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to polynomial systems of ordinary and partial differential equations. The second and third algorithms allow one to explicitly compute polynomial conserved densities and higher-order symmetries of nonlinear evolution and lattice equations. The first algorithm is implemented in the symbolic syntax of both Macsyma and Mathematica. The second and third algorithms are available in Mathematica. The codes can be used for computer-aided integrability testing of nonlinear differential and lattice equations as they occur in various branches of the sciences and engineering. Applied to systems with parameters, the codes can determine the conditions on the parameters so that the systems pass the Painlev\'e test, or admit a sequence of conserved densities or higher-order symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style files elsart.sty and elsart12.st

Exact Solutions of the One-Dimensional Quintic Complex Ginzburg-Landau Equation

Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlev\'e test for integrability. These solutions are expressed in terms of hyperbolic functions, and include the pulses and fronts found by van Saarloos and Hohenberg. We also find previously unknown sources and sinks. The emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc Ans\"atze.Comment: 24 pages, regular LaTeX, no figure

Painleve singularity analysis applied to charged particle dynamics during reconnection

For a plasma in the collisionless regime, test-particle modelling can lend some insight into the macroscopic behavior of the plasma, e.g conductivity and heating. A common example for which this technique is used is a system with electric and magnetic fields given by B = {dollar}\delta y{dollar}cx x + xcx y + {dollar}\gamma{dollar}cx z and E = {dollar}\epsilon{dollar}cx z, where {dollar}\delta{dollar}, {dollar}\gamma{dollar}, and {dollar}\epsilon{dollar} are constant parameters. This model can be used to model plasma behavior near neutral lines, ({dollar}\gamma{dollar} = 0), as well as current sheets ({dollar}\gamma{dollar} = 0, {dollar}\delta{dollar} = 0). The integrability properties of the particle motion in such fields might affect the plasma\u27s macroscopic behavior, and we have asked the question For what values of {dollar}\delta{dollar}, {dollar}\gamma{dollar}, and {dollar}\epsilon{dollar} is the system integrable? to answer this question, we have employed Painleve singularity analysis, which is an examination of the singularity properties of a test particle\u27s equations of motion in the complex time plane. This analysis has identified two field geometries for which the system\u27s particle dynamics are integrable in terms of the second Painleve transcendent: the circular O-line case and the case of the neutral sheet configuration. These geometries yield particle dynamics that are integrable in the Liouville sense (i.e. there exist the proper number of integrals in involution) in an extended phase space which includes the time as a canonical coordinate, and this property is also true for nonzero {dollar}\gamma{dollar}. The singularity property tests also identified a large, dense set of X-line and O-line field geometries that yield dynamics that may possess the weak Painleve property. In the case of the X-line geometries, this result shows little relevance to the physical nature of the system, but the existence of a dense set of elliptical O-line geometries with this property may be related to the fact that for {dollar}\epsilon{dollar} positive, one can construct asymptotic solutions in the limit {dollar}t \to \infty{dollar}

Understanding complex dynamics by means of an associated Riemann surface

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.Comment: 25 pages, 14 figures, typed in AMS-LaTe

Bose-Einstein condensates with long-range dipolar interactions

Bose-Einstein condensation is a phase transition which atoms undergo when cooled near absolute zero temperature Since the theoretical prediction in 1924, and the spectacular experimental confirmation of Bose-Einstein condensation in 1995, a rich new field in physics has emerged studying ultracold degenerate quantum gases. Although these ultracold gases are very dilute, their properties are nevertheless strongly influenced by interatomic interactions. Usually, these interactions are dominated by short range, isotropic contact interactions. In contrast, the recently realised Bose-Einstein Condensate (BEC) of Chromium atoms contains long-range, anisotropic dipolar interactions leading to interesting new physics. In this graduation project, stationary states of such dipolar BECs in harmonic traps are investigated for various experimentally relevant parameters. Furthermore, the elementary excitations of the BEC are calculated, as well as its response to a rotating perturbation. Finally, some more advanced topics such as vortex interactions and condensate response to impurities are investigated. Bose-Einstein condensation is a phase transition which atoms undergo when cooled near absolute zero temperature Since the theoretical prediction in 1924, and the spectacular experimental confirmation of Bose-Einstein condensation in 1995, a rich new field in physics has emerged studying ultracold degenerate quantum gases. Although these ultracold gases are very dilute, their properties are nevertheless strongly influenced by interatomic interactions. Usually, these interactions are dominated by short range, isotropic contact interactions. In contrast, the recently realised Bose-Einstein Condensate (BEC) of Chromium atoms contains long-range, anisotropic dipolar interactions leading to interesting new physics. In this graduation project, stationary states of such dipolar BECs in harmonic traps are investigated for various experimentally relevant parameters. Furthermore, the elementary excitations of the BEC are calculated, as well as its response to a rotating perturbation. Finally, some more advanced topics such as vortex interactions and condensate response to impurities are investigated

Komplexe Analysis

The aim of this workshop was to discuss recent developments in several complex variables and complex geometry. Special emphasis was put on the interaction between model theory and the classification theory of complex manifolds. Other topics included Kähler geometry, foliations, complex symplectic manifolds and moduli theory