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Christian Klein

Publications and source records attributed to Christian Klein.

At least 19 recordsLinked to original sources
arXiv2026

The 3D critical Zakharov--Kuznetsov equation: blow-up and soliton dynamics

We study the full three-dimensional dynamics of the $L^2$-critical Zakharov-Kuznetsov equation with the fractional nonlinearity $|u|^{4/3}u$, equivalently $u^{7/3}$ for real-valued functions. This equation is a higher-dimensional extension of the generalized Korteweg-de Vries equation. In the critical setting solutions to this 3D ZK equation may blow up in finite time or exhibit global time dynamics. The novelties of this work is to treat a non-integer power and to study the dynamics of solutions in a higher dimension. We first review the finite time blow-up in 2D critical ZK, then do a formal analysis of the slightly mass-supercritical blow-up dynamics, deriving the corrections to the blow-up rate and profile for the critical ZK equation in any dimension. We then perform a computational study of solutions, utilizing full 3D numerical simulations. In particular, we use a Fourier pseudospectral discretization and an integrating factor fourth-order Runge-Kutta method on a full three-dimensional grid. A multi-GPU implementation makes it possible to follow blow-up solutions in a full 3D setting. We examine perturbations of the ground state, Gaussian data, and nonsymmetric two-bump configurations. The computations show dispersive and concentrating regimes, in both cases with radiation emitted in a conic-type region opposite to the direction of propagation and convergence of the concentrating core toward a rescaled ground-state profile. The two-bump experiments also demonstrate that total mass alone does not determine the blow-up dynamics. We discuss the numerical evidence for the predicted blow-up rate and identify the pre-asymptotic and resolution limitations that remain near the blow-up time.

math.AP↗
arXiv2026

Numerical study of the 2D Kaup-Broer-Kuperschmidt Boussinesq system

In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.

math.AP↗
arXiv2026

Multi-domain spectral approach for Zakharov-Kuznetsov equations in 3D with cylindrical symmetry

We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves). Using the three-dimensional generalized Zakharov-Kuznetsov (gZK) equation as a model, we convert it into cylindrical coordinates and implement a domain decomposition strategy. By partitioning the computational domain into distinct regions based on expected solution behavior, we significantly reduce computational complexity while maintaining the high resolution necessary for capturing small-scale dynamics. Another key innovation of our method is the ability to efficiently handle fractional nonlinearities, specifically, the critical power $p = 7/3$ in 3D, which typically introduces significant computational overhead and numerical instabilities that compromise simulation accuracy. Using this framework, we are able to investigate the dynamics of solutions (with cylindrical symmetry) close to the ground state soliton and show that for the 3D critical ZK equation, the ground state serves as the sharp threshold for global vs. finite time existence of solutions. Our method successfully tracks the profiles of these singular solutions, providing new insights into the dynamics of wave collapse in three-dimensional magnetized media.

math.NA↗
arXiv2026

The double Schwarzschild solution in bispherical coordinates

The double Schwarzschild solution in the equal mass case is studied in bispherical coordinates. An explicit conformal transformation from cylindrical Weyl coordinates to bispherical coordinates is given in terms of elliptic functions. A multi-domain spectral method for spacetimes in bispherical coordinates is presented to numerically reconstruct this solution.

gr-qc↗
arXiv2026

Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase $\omega_{k}$, $k=1,\ldots, N$ are far from the singularity $\zeta$ of the integrand, a polarization approach is proposed for the case that $|\zeta-\omega_{k}|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in $\mathbb{C}^{2}$ ($\widetilde{\omega}:=\overline{\omega}$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotic expressions in terms of special functions are given.

math.AP↗
arXiv2025

On visible effects in the double Schwarzschild solution

Physical aspects of a static solution to the Einstein equations with two black holes are studied via ray tracing. The exact solution for this double Schwarzschild solution is known in explicit form. The black holes are separated by a singularity called \emph{Weyl strut}. The effect of this strut on null geodesics is shown to be defocusing in contrast to the focusing effect of the black holes. It is shown that black holes with a large separation essentially lead to similar behavior of the null geodesics as a single black hole, whereas nearby holes display a widely changed behavior due to the Weyl strut.

gr-qc↗
arXiv2025

Nonlinear Schr\"odinger equation on a unit ball in one and two dimensions

We consider the nonlinear Schr\"odinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are stable in subcritical and critical cases, and in the supercritical case the ground state solutions split into a stable and an unstable branch. Perturbations of a ground state on the stable branch keep solutions near a corresponding ground state with very small oscillation around it, while perturbations of the unstable branch make solutions either blow up in finite time, if perturbations have an amplitude large than the height of the ground state, or oscillate between two states, if perturbations have an amplitude smaller than the original ground state. We also observe that this equation does not have any scattering or radiation, and thus, the soliton resolution holds for all data, splitting solutions into coherent structures such as ground state solutions even for very small initial data.

math.AP↗
arXiv2025

Dynamics of solutions in the 1d bi-harmonic nonlinear Schr\"odinger equation

We consider the one dimensional 4th order, or bi-harmonic, nonlinear Schr\"odinger (NLS) equation, namely, $i u_t - \Delta^2 u - 2a \Delta u + |u|^{\alpha} u = 0, ~ x,a \in \R$, $\alpha>0$, and investigate the dynamics of its solutions for various powers of $\alpha$, including the ground state solutions and their perturbations, leading to scattering or blow-up dichotomy when $a \leq 0$, or to a trichotomy when $a>0$. Ground state solutions are numerically constructed, and their stability is studied, finding that the ground state solutions may form two branches, stable and unstable, which dictates the long-term behavior of solutions. Perturbations of the ground states on the unstable branch either lead to dispersion or the jump to a stable ground state. In the critical and supercritical cases, blow-up in finite time is also investigated, and it is conjectured that the blow-up happens with a scale-invariant profile (when $a=0$) regardless of the value of $a$ of the lower dispersion. The blow-up rate is also explored.

math.AP↗
arXiv2025

Numerical study of transverse (in-)stability of solitary waves in the cubic-quintic nonlinear Schr\"odinger equation

We study the nonlinear Schr\"odinger equation with a competing cubic-quintic power law nonlinearity on the waveguide domain $\mathbb R_x \times \mathbb T_{L_y}$. This model is globally well-posed and admits line solitary wave solutions, whose transverse (in-)stability is numerically investigated. We consider both spatially localized perturbations and periodic deformations of the line solitary wave and numerically confirm that there exists a critical torus length $L_y>0$ above which instability appears.

math.AP↗
arXiv2025

A numerical study of stability for solitary waves of a quasi-linear Schr{\"o}dinger equation

We discuss the (in)stability of solitary waves for a quasi-linear Schr{\"o}dinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the nonlinearity, we determine analytically the asymptotic behavior of the $L^2$-mass of the solution as a function of the frequency close to the critical frequencies, which leads to natural conjectures concerning their stability. Depending on the exponent and the dimension, we expect all solitary waves to be stable, or the emergence of both a stable and an unstable branch of solutions. We investigate our conjectures numerically, and find compatible results both for the mass-energy relation and the dynamics. We observe that perturbations of solitary waves on the unstable branch may converge dynamically to the stable solution of a similar mass, or disperse. More general initial conditions show a similar behavior.

math.AP↗
arXiv2024

Optimally truncated WKB approximation for the 1D stationary Schr\"odinger equation in the highly oscillatory regime

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schr\"odinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter $\varepsilon$. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of $\varepsilon$ and the truncation order $N$. For any fixed $\varepsilon$, this allows to determine the optimal truncation order $N_{opt}$ which turns out to be proportional to $\varepsilon^{-1}$. When chosen this way, the resulting error of the optimally truncated WKB series behaves like $\mathcal{O}(\exp(-r/\varepsilon))$, with some parameter $r>0$. The theoretical results established in this paper are confirmed by several numerical examples.

math.NA↗
arXiv2023

Optimally truncated WKB approximation for the highly oscillatory stationary 1D Schr\"odinger equation

We discuss the numerical solution of initial value problems for $\varepsilon^2\,\varphi''+a(x)\,\varphi=0$ in the highly oscillatory regime, i.e., with $a(x)>0$ and $0<\varepsilon\ll 1$. We analyze and implement an approximate solution based on the well-known WKB-ansatz. The resulting approximation error is of magnitude $\mathcal{O}(\varepsilon^{N})$ where $N$ refers to the truncation order of the underlying asymptotic series. When the optimal truncation order $N_{opt}$ is chosen, the error behaves like $\mathcal{O}(\varepsilon^{-2}\exp(-c\varepsilon^{-1}))$ with some $c>0$.

math.NA↗
arXiv2023

Numerical study of fractional Camassa-Holm equations

A numerical study of fractional Camassa-Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localised initial data from the Schwartz class of rapidly decreasing functions. The appearence of dispersive shock waves is explored.

math.AP↗
arXiv2022

Numerical study of the transverse stability of line solitons of the Zakharov-Kuznetsov equations

We present a detailed numerical study of the stability under periodic perturbations of line solitons of two-dimensional, generalized Zakharov-Kuznetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical speed, below which the line soliton is stable. For higher velocities, the numerical results indicate an instability against the formation of lumps, solitons localized in both spatial directions. In the $L^2$-critical and supercritical cases but subcritical for the 1D generalized Korteweg-de Vries equation), the line solitons are shown to be numerically stable for small velocities, and strongly unstable for large velocities, with a blow-up observed in finite time.

math.AP↗
arXiv2022

Large $|k|$ behavior for the reflection coefficient for Davey-Stewartson II equations

The study of complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations for large values of the spectral parameter $k$ in \cite{KlSjSt20} is extended to the reflection coefficient. For the case of potentials $q$ with compact support on some domain $\Omega$ with smooth strictly convex boundary, improved asymptotic relations are provided.

math.AP↗
arXiv2020

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated. We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.

math.AP↗
arXiv2020

On a nonlinear Schr{\"o}dinger equation for nucleons in one space dimension

We study a 1D nonlinear Schr{\"o}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↗