arXiv subjects

Christian Klein

Publications and source records attributed to Christian Klein.

29 records · Page 2Linked to original sources

arXiv2021

On a nonlinear Schr{ö}dinger equation for nucleons in one space dimension

We study a 1D nonlinear Schr{ö}dinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.

math.AP↗

arXiv2018

Numerical Approach to Painlevé Transcendents on Unbounded Domains

A multidomain spectral approach for Painlevé transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and approximates the solution on straight lines lying entirely within said sector without the need of evaluating truncations of the series at any finite point. The accuracy of the method is illustrated for the example of the tritronquée solution to the Painlevé I equation.

math.CA↗

arXiv2015

On critical behaviour in generalized Kadomtsev--Petviashvili equations

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.

math-ph↗

arXiv2013

Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations

The formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers' equation and for dispersive regularizations as the Korteweg-de Vries equation, the Fourier coefficients indicate as expected global regularity of the solutions. The Kadomtsev-Petviashvili (KP) equation can be seen as a dispersive regularization of the dKP equation. The behavior of KP solutions for small dispersion parameter $ε\ll 1$ near a break-up of corresponding dKP solutions is studied. It is found that the difference between KP and dKP solutions for the same initial data at the critical point scales roughly as $ε^{2/7}$ as for the Korteweg-de Vries equation.

math.AP↗

arXiv2012

Interplay between non equilibrium and equilibrium spin torque using synthetic ferrimagnets

We discuss the current induced magnetization dynamics of spin valves F0|N|SyF where the free layer is a synthetic ferrimagnet SyF made of two ferromagnetic layers F1 and F2 coupled by RKKY exchange coupling. In the interesting situation where the magnetic moment of the outer layer F2 dominates the magnetization of the ferrimagnet, we find that the sign of the effective spin torque exerted on the free middle layer F1 is controlled by the strength of the RKKY coupling: for weak coupling one recovers the usual situation where spin torque tends to, say, anti-align the magnetization of F1 with respect to the pinned layer F0. However for large coupling the situation is reversed and the spin torque tends to align F1 with respect to F0. Careful numerical simulations in the intermediate coupling regime reveal that the competition between these two incompatible limits leads generically to spin torque oscillator (STO) behavior. The STO is found in the absence of magnetic field, with very significant amplitude of oscillations and frequencies up to 50 GHz or higher.

cond-mat.mes-hall↗

arXiv2006

Unbiased Matrix Rounding

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics. We improve earlier solutions of different authors in two ways. For rounding matrices of size $m \times n$, we reduce the runtime from $O((m n)^2 Second, our roundings also have a rounding error of less than one in all initial intervals of rows and columns. Consequently, arbitrary intervals have an error of at most two. This is particularly useful in the statistics application of controlled rounding. The same result can be obtained via (dependent) randomized rounding. This has the additional advantage that the rounding is unbiased, that is, for all entries $y_{ij}$ of our rounding, we have $E(y_{ij}) = x_{ij}$, where $x_{ij}$ is the corresponding entry of the input matrix.

cs.DS↗

arXiv2005

Numerical solution of the small dispersion limit of Korteweg de Vries and Whitham equations

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $ε^2$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $ε$. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the KdV equation in the small dispersion limit and the corresponding approximate solution for values of $ε$ between $10^{-1}$ and $10^{-3}$. The numerical results are compatible with a difference of order $ε$ within the `interior' of the Whitham oscillatory zone, of order $ε^{1/3}$ at the left boundary outside the Whitham zone and of order $ε^{1/2}$ at the right boundary outside the Whitham zone.

math-ph↗

arXiv2002

Factual and Legal Risks regarding wireless Computer Networks

The IEEE 802.11b wireless ethernet standart has several serious security flaws. This paper describes this flaws, surveys wireless networks in the Cologne/Bonn area to get an assessment of the security configurations of fielded networks and analizes the legal protections provided to wireless ethernet operators by german law. We conclude that wireless ethernets without additional security measures are not usable for any transmissions which are not meant for a public audience.

cs.CY↗

arXiv2001

Exact relativistic treatment of stationary counter-rotating dust disks III. Physical Properties

This is the third in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the physical properties of a class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. It is parameterized by two physical parameters, the central redshift and the relative density of the two counter-rotating streams in the disk. We discuss the dependence of the metric on these parameters using a combination of analytical and numerical methods. Interesting limiting cases are the Maclaurin disk in the Newtonian limit, the static limit which gives a solution of the Morgan and Morgan class and the limit of a disk without counter-rotation. We study the mass and the angular momentum of the spacetime. At the disk we discuss the energy-momentum tensor, i.e. the angular velocities of the dust streams and the energy density of the disk. The solutions have ergospheres in strongly relativistic situations. The ultrarelativistic limit of the solution in which the central redshift diverges is discussed in detail: In the case of two counter-rotating dust components in the disk, the solutions describe a disk with diverging central density but finite mass. In the case of a disk made up of one component, the exterior of the disks can be interpreted as the extreme Kerr solution.

gr-qc↗

arXiv2001

Exact relativistic treatment of stationary counter-rotating dust disks II: Axis, Disk and Limiting Cases

This is the second in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. We discuss the metric functions at the axis of symmetry and the disk. Interesting limiting cases are the Newtonian limit, the static limit, and the ultra-relativistic limit of the solution in which the central redshift diverges.

gr-qc↗

arXiv2001

Exact relativistic treatment of stationary counter-rotating dust disks I: Boundary value problems and solutions

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.

gr-qc↗