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C. Klein

Publications and source records attributed to C. Klein.

At least 73 records · Page 4Linked to original sources
arXiv2012

WKB analysis of Bohmian dynamics

We consider a semi-classically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime which illustrate the above results.

math-ph↗
arXiv2012

Asteroid rotation periods from the Palomar Transient Factory survey

The Palomar Transient Factory (PTF) is a synoptic survey designed to explore the transient and variable sky in a wide variety of cadences. We use PTF observations of fields that were observed multiple times (>=10) per night, for several nights, to find asteroids, construct their lightcurves and measure their rotation periods. Here we describe the pipeline we use to achieve these goals and present the results from the first four (overlapping) PTF fields analyzed as part of this program. These fields, which cover an area of 21 deg^2, were observed on four nights with a cadence of ~20 min. Our pipeline was able to detect 624 asteroids, of which 145 (~20%) were previously unknown. We present high quality rotation periods for 88 main-belt asteroids and possible period or lower limit on the period for an additional 85 asteroids. For the remaining 451 asteroids, we present lower limits on their photometric amplitudes. Three of the asteroids have lightcurves that are characteristic of binary asteroids. We estimate that implementing our search for all existing high-cadence PTF data will provide rotation periods for about 10,000 asteroids mainly in the magnitude range ~14 to ~20.

astro-ph.EP↗
arXiv2011

Numerical Study of Blowup in the Davey-Stewartson System

Nonlinear dispersive partial differential equations such as the nonlinear Schrödinger equations can have solutions that blow-up. We numerically study the long time behavior and potential blowup of solutions to the focusing Davey-Stewartson II equation by analyzing perturbations of the lump and the Ozawa exact solutions. It is shown in this way that the lump is unstable to both blowup and dispersion, and that blowup in the Ozawa solution is generic.

math.AP↗
arXiv2011

Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations

Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schrödinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial to resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kadomtsev-Petviashvili and the Davey-Stewartson equations, two integrable equations in 2+1 dimensions: exponential time-differencing, integrating factors, time-splitting, implicit Runge-Kutta and Driscoll's composite Runge-Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.

math.NA↗
arXiv2011

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schrödinger equations.

math-ph↗
arXiv2011

New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation

An independent derivation of solutions to the Camassa-Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay's identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.

math-ph↗
arXiv2011

Transverse stability of periodic traveling waves in Kadomtsev-Petviashvili equations: A numerical study

We numerically investigate transverse stability and instability of so-called cnoidal waves, i.e., periodic traveling wave solutions of the Korteweg-de Vries equation, under the time-evolution of the Kadomtsev-Petviashvili equation. In particular, we find that in KP-I small amplitude cnoidal waves are stable (at least for spatially localized perturbations) and only become unstable above a certain threshold. In contrast to that, KP-II is found to be stable for all amplitudes, or, equivalently, wave speeds. This is in accordance with recent analytical results for solitary waves given in \cite{RT1, RT2}.

math.AP↗
arXiv2011

Efficient computation of the branching structure of an algebraic curve

An efficient algorithm for computing the branching structure of a compact Riemann surface defined via an algebraic curve is presented. Generators of the fundamental group of the base of the ramified covering punctured at the discriminant points of the curve are constructed via a minimal spanning tree of the discriminant points. This leads to paths of minimal length between the points, which is important for a later stage where these paths are used as integration contours to compute periods of the surface. The branching structure of the surface is obtained by analytically continuing the roots of the equation defining the algebraic curve along the constructed generators of the fundamental group.

cs.CG↗
arXiv2011

Multiwavelength campaign on Mrk 509. I. Variability and spectral energy distribution

(Abridged) Active galactic nuclei show a wealth of interesting physical processes, some of which are poorly understood. We want to address a number of open questions, including the location and physics of the outflow from AGN, the nature of the continuum emission, the geometry and physical state of the X-ray broad emission line region, the Fe-K line complex, the metal abundances of the nucleus and finally the interstellar medium of our own Galaxy. We study one of the best targets for these aims, the Seyfert 1 galaxy Mrk 509 with a multiwavelength campaign using five satellites (XMM-Newton, INTEGRAL, Chandra, HST and Swift) and two ground-based facilities (WHT and PAIRITEL). Our observations cover more than five decades in frequency, from 2 um to 200 keV. The combination of high-resolution spectroscopy and time variability allows us to disentangle and study the different components. Our campaign covers 100 days from September to December 2009, and is centred on a simultaneous set of deep XMM-Newton and INTEGRAL observations with regular time intervals, spanning seven weeks. We obtain a continuous light curve in the X-ray and UV band, showing a strong, up to 60% flux increase in the soft X-ray band during the three weeks in the middle of our deepest monitoring campaign, and which is correlated with an enhancement of the UV flux. This allows us to study the time evolution of the continuum and the outflow. By stacking the observations, we have also obtained one of the best X-ray and UV spectra of a Seyfert galaxy ever obtained. In this paper we also study the effects of the spectral energy distribution (SED) that we obtained on the photo-ionisation equilibrium. Thanks to our broad-band coverage, uncertainties on the SED do not strongly affect the determination of this equilibrium.

astro-ph.CO↗
arXiv2010

Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations

This article is concerned with a conjecture by one of the authors on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasilinear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behaviour of their solution near the point of gradient catastrophe for the transport equation is described locally by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blow up can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $ε^2$ where $ε^2$ is the small dispersion parameter. Corrections up to order $ε^4 $ are obtained and tested numerically.

math-ph↗
arXiv2010

On linear degeneracy of integrable quasilinear systems in higher dimensions

We investigate $(d+1)$-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case $d\geq 3$ we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi-dimensional integrable systems in question. In particular, our results imply the non-existence of 2-component integrable systems of hydrodynamic type for $d\geq 6$. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2+1 dimensions.

nlin.SI↗
arXiv2009

Measuring the magnetization of three monolayer thick Co islands and films by X-ray dichroism

Co islands and films are characterized by x-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM). The spatial resolution capabilties of the technique together with atomic growth control permit obtaining perfectly flat triangular islands with a given thickness (3 ML), very close to an abrupt spin-reorientation transition. The magnetic domain configurations are found to depend on island size: while small islands can be magnetized in a single-domain state, larger islands show more complex patterns. Furthermore, the magnetization pattern of the larger islands presents a common chirality. By means of dichroic spectro-microscopy at the Co L absorption edges, an experimental estimate of the ratio of the spin- and orbital magnetic moment for three monolayer thick films is obtained.

cond-mat.mes-hall↗
arXiv2009

Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions

The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture. We present a quantitative numerical comparison between the CH and the asymptotic solution. The dependence on the small dispersion parameter $ε$ is studied in the interior and at the boundaries of the Whitham zone. In the interior of the zone, the difference between CH and asymptotic solution is of the order $ε$, at the trailing edge of the order $\sqrtε$ and at the leading edge of the order $ε^{1/3}$. For the latter we present a multiscale expansion which describes the amplitude of the oscillations in terms of the Hastings-McLeod solution of the Painlevé II equation. We show numerically that this multiscale solution provides an enhanced asymptotic description near the leading edge.

math-ph↗
arXiv2008

Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlevé-II equation

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. 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. Whereas the difference between the KdV and the asymptotic solution decreases as $ε$ in the interior of the Whitham oscillatory zone, it is known to be only of order $ε^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $ε^{2/3}$.

math-ph↗
arXiv2007

On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\it tritronquée} solution to the Painlevé-I equation

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $ iεψ_t +\frac{ε^2}2ψ_{xx}+ |ψ|^2 ψ=0$ with analytic initial data of the form $ψ(x,0;ε) =A(x) e^{\frac{i}ε S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.

math.AP↗
arXiv2007

Numerical study of a multiscale expansion of KdV and Camassa-Holm equation

We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlevé I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equation

math-ph↗
arXiv2006

Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods we identify four non-degenerate critical points of this function and compute the signature of the Hessian at these points. The curve with the maximal number of automorphisms (the Burnside curve) turns out to be the point of the absolute maximum. Our results agree with the mass formula for orbifold Euler characteristics of the moduli space. A similar analysis is performed for the Bolza's strata of symmetric Riemann surfaces of genus two.

math.SP↗