275 research outputs found
Changing shapes: adiabatic dynamics of composite solitary waves
We discuss the solitary wave solutions of a particular two-component scalar
field model in two-dimensional Minkowski space. These solitary waves involve
one, two or four lumps of energy. The adiabatic motion of these composite
non-linear non-dispersive waves points to variations in shape.Comment: 21 pages, 15 figures. To appear in Physica D: Nonlinear Phenomen
Spacetime approach to force-free magnetospheres
Force-Free Electrodynamics (FFE) describes magnetically dominated
relativistic plasma via non-linear equations for the electromagnetic field
alone. Such plasma is thought to play a key role in the physics of pulsars and
active black holes. Despite its simple covariant formulation, FFE has primarily
been studied in 3+1 frameworks, where spacetime is split into space and time.
In this article we systematically develop the theory of force-free
magnetospheres taking a spacetime perspective. Using a suite of spacetime tools
and techniques (notably exterior calculus) we cover 1) the basics of the
theory, 2) exact solutions that demonstrate the extraction and transport of the
rotational energy of a compact object (in the case of a black hole, the
Blandford-Znajek mechanism), 3) the behavior of current sheets, 4) the general
theory of stationary, axisymmetric magnetospheres and 5) general properties of
pulsar and black hole magnetospheres. We thereby synthesize, clarify and
generalize known aspects of the physics of force-free magnetospheres, while
also introducing several new results.Comment: v2: numerous improvements; v3: further improvements, matches
published versio
Mini-Workshop: Algebraic and Analytic Techniques for Polynomial Vector Fields
Polynomial vector fields are in the focus of research in various areas of mathematics and its applications. As a consequence, researchers from rather different disciplines work with polynomial vector fields. The
main goal of this mini workshop was to create new and consolidate existing interdisciplinary exchange on the subject
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis
(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived.
(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.
(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for -series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations.
(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.
(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed
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