Searcharxiv⌕ Search

arXiv subjects

C. Klein

Publications and source records attributed to C. Klein.

At least 55 records · Page 3Linked to original sources
arXiv2015

Computational approach to compact Riemann surfaces

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.

math-ph↗
arXiv2015

Spectral approach to D-bar problems

We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.

math.NA↗
arXiv2015

Numerical study of the stability of the Peregrine breather

The Peregrine breather is widely discussed as a model for rogue waves in deep water. We present here a detailed numerical study of perturbations of the Peregrine breather as a solution to the nonlinear Schrödinger (NLS) equations. We first address the modulational instability of the constant modulus solution to NLS. Then we study numerically localized and nonlocalized perturbations of the Peregrine breather in the linear and fully nonlinear setting. It is shown that the solution is unstable against all considered perturbations.

math.AP↗
arXiv2015

On the tritronquée solutions of P$_I^2$

For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,...,6$, called {\em tritronquée}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronquée solutions.

math-ph↗
arXiv2014

Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations

We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the $L_{2}$ critical case, the blow-up mechanism by Martel, Merle and Raphaël can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Raphaël is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time $t^{*}$ in dependence of the small dispersion parameter $ε$ and find an exponential dependence $t^{*}(ε)$ and that there is a minimal blow-up time $t^{*}_{0}$ greater than the critical time of the corresponding Hopf solution for $ε\to0$. To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg-de Vries equations. This allows to identify the type of the singularity.

math-ph↗
arXiv2014

Multidomain spectral method for Schrödinger equations

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

math.NA↗
arXiv2014

IST versus PDE, a comparative study

We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.

math.AP↗
arXiv2014

Computational approach to hyperelliptic Riemann surfaces

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the curve. A canonical basis of the homology is introduced algorithmically for this curve. The periods of the holomorphic differentials and the Abel map are computed with the Clenshaw-Curtis method in order to achieve spectral accuracy. The code can handle almost degenerate Riemann surfaces. This work generalizes previous work on real hyperelliptic surfaces with prescribed cuts to arbitrary hyperelliptic surfaces. As an example, solutions to the sine-Gordon equation in terms of multi-dimensional theta functions are studied, also in the solitonic limit of these solutions.

math.AG↗
arXiv2014

A numerical approach to Blow-up issues for Davey-Stewartson II type systems

We provide a numerical study of various issues pertaining to the dynamics of the Davey-Stewartson systems of the DS II type. In particular we investigate whether or not the properties (blow-up, radiation,...) displayed by the focusing and defocusing DS II integrable systems persist in the non integrable case.

math.AP↗
arXiv2014

Numerical Study of the semiclassical limit of the Davey-Stewartson II equations

We present the first detailed numerical study of the semiclassical limit of the Davey-Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing initial data with a single hump. The formal limit of these equations for vanishing semiclassical parameter $ε$, the semiclassical equations, are numerically integrated up to the formation of a shock. The use of parallelized algorithms allows to determine the critical time $t_{c}$ and the critical solution for these $2+1$-dimensional shocks. It is shown that the solutions generically break in isolated points similarly to the case of the $1+1$-dimensional cubic nonlinear Schrödinger equation, i.e., cubic singularities in the defocusing case and square root singularities in the focusing case. For small values of $ε$, the full Davey-Stewartson II equations are integrated for the same initial data up to the critical time $t_{c}$. The scaling in $ε$ of the difference between these solutions is found to be the same as in the $1+1$ dimensional case, proportional to $ε^{2/7}$ for the defocusing case and proportional to $ε^{2/5}$ in the focusing case. We document the Davey-Stewartson II solutions for small $ε$ for times much larger than the critical time $t_{c}$. It is shown that zones of rapid modulated oscillations are formed near the shocks of the solutions to the semiclassical equations. For smaller $ε$, the oscillatory zones become smaller and more sharply delimited to lens shaped regions. Rapid oscillations are also found in the focusing case for initial data where the singularities of the solution to the semiclassical equations do not coincide.

math-ph↗
arXiv2014

Numerical study of fractional Nonlinear Schrödinger equations

Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schrödinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be computed in one spatial dimension, only. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states, and the long time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions to the fractional nonlinear Schrödinger equation.

math.AP↗
arXiv2014

Numerical study of the long wavelength limit of the Toda lattice

We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the focusing nonlinear Schrödinger equation in $1+1$ dimensions. The full Toda system is then studied for finite small values of the dispersion parameter $ε$ in the vicinity of the shocks of the dispersionless Toda equations. We determine the scaling in $ε$ of the difference between the Toda solution for small $ε$ and the singular solution of the dispersionless Toda system. In the hyperbolic case, the same scaling proportional to $ε^{2/7}$ is found as in the small dispersion limit of the Korteweg-de Vries and the defocusing nonlinear Schrödinger equations. In the elliptic case, we obtain the same scaling proportional to $ε^{2/5}$ as in the semiclassical limit for the focusing nonlinear Schrödinger equation. We also study the formation of dispersive shocks for times much larger than the break-up time in the hyperbolic case. In the elliptic case, an $L_{\infty}$ blow-up is observed instead of a dispersive shock for finite times greater than the break-up time. The $ε$-dependence of the blow-up time is determined.

math-ph↗
arXiv2014

A numerical approach to Blow-up issues for dispersive perturbations of Burgers' equation

We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for "long" times, and the decomposition of the initial data into solitary waves plus radiation. We numerically construct solitons for fractionary Korteweg-de Vries equations.

math.AP↗
arXiv2013

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

math-ph↗
arXiv2013

Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations

We present a numerical study of solutions to the generalized Kadomtsev-Petviashvili equations with critical and supercritical nonlinearity for localized initial data with a single minimum and single maximum. In the cases with blow-up, we use a dynamic rescaling to identify the type of the singularity. We present a discussion of the observed blow-up scenarios.

math.AP↗
arXiv2013

On the integrable elliptic cylindrical Kadomtsev-Petviashvili equation

There exist two versions of the Kadomtsev-Petviashvili equation, related to the Cartesian and cylindrical geometries of the waves. In this paper we derive and study a new version, related to the elliptic cylindrical geometry. The derivation is given in the context of surface waves, but the derived equation is a universal integrable model applicable to generic weakly-nonlinear weakly-dispersive waves. We also show that there exist nontrivial transformations between all three versions of the KP equation associated with the physical problem formulation, and use them to obtain new classes of approximate solutions for water waves.

nlin.PS↗
arXiv2012

Computation of the topological type of a real Riemann surface

We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution $τ$, namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the $\mathcal{A}$-cycles are invariant under the anti-holomorphic involution $τ$.

math.AG↗