Wolf-Patrick Düll

Validity of Whitham's Equations for the Modulation of Periodic Traveling Waves in the NLS Equation

We prove that slow modulations in time and space of periodic wave trains of the NLS equation can be approximated via solutions of Whitham’s equations associated with the wave train. The error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like existence and uniqueness theorem, and energy estimates.

Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear KleinGordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems.

Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation

Abstract We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order O ( e ) in Sobolev norms exist for a time span of order O ( e − 2 ) for sufficiently small e. Secondly, we derive the Nonlinear Schrodinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation. The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances.

A WAITING TIME PHENOMENON IN PATTERN FORMING SYSTEMS

Waiting time phenomena are well known to occur in degenerate parabolic equations like the porous medium equation. In this paper we prove that such a phenomenon also occurs approximately in the Ginzburg–Landau equation for modulations of the pattern with a basic wavenumber close to the boundaries of the Eckhaus stable region.

The Validity of Whitham's Approximation for a Klein--Gordon--Boussinesq Model

In this paper we prove the validity of a long wave Whitham approximation for a system consisting of a Boussinesq equation coupled with a Klein--Gordon equation. The proof is based on an infinite series of normal form transformations and an energy estimate. We expect that the concepts of this paper will be a part of a general approximation theory for Whithams equations which are especially used in the description of slow modulations in time and space of periodic wave trains in general dispersive wave systems.

Analysis of the embedded cell method in 1D for the numerical homogenization of metal-ceramic composite materials

Abstract In this paper, we analyze the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials. We show the convergence of the iteration scheme of this algorithm and the coincidence of the material properties predicted by the limit with the effective material properties provided by the analytical homogenization theory in two situations, namely for a one-dimensional linear elasticity model and a simple one-dimensional plasticity model.

On the Mathematical Description of Time-Dependent Surface Water Waves

This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and global well–posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models.

The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification

We consider a spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification. We are interested in solutions bifurcating from the spatially homogeneous ground state in the case when pairs of imaginary eigenvalues simultaneously cross the imaginary axis. For this system we prove the bifurcation of a family of invariant tori which may contain quasiperiodic solutions. There is a serious difficulty in obtaining this result, because the linearization at the ground state possesses an essential spectrum up to the imaginary axis for all values of the bifurcation parameter. To construct the invariant tori, we use their invariance under the flow which manifests in a condition in PDE form. The nonlinear terms of this resulting PDE exhibit a loss of regularity. Since the linear part of this PDE is not smoothing, an adaption of the hard implicit function theorem (or Nash-Moser scheme) and energy estimates will be used to prove our result.