Hannes Uecker

pde2path - A Matlab Package for Continuation and Bifurcation in 2D Elliptic Systems

pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the Matlab pdetoolbox, and is explained by a number of examples, including Bratu’s problem, the Schnakenberg model, Rayleigh-Benard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via Matlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parameter arclength-continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary time-integration for the associated parabolic problem are also supported. The continuation, branch-switching, plotting etc are performed via Matlab command-line function calls guided by the AUTO style. The software can be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path , where also an online documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems.

An integral boundary layer equation for film flow over inclined wavy bottoms

We study the flow of an incompressible liquid film down a wavy incline. Applying a Galerkin method with only one ansatz function to the Navier–Stokes equations, we derive a second-order weighted residual integral boundary layer equation, which, in particular, may be used to describe eddies in the troughs of the wavy bottom. We present numerical results which show that our model is qualitatively and quantitatively accurate in wide ranges of parameters, and we use the model to study some new phenomena, for instance, the occurrence of a short wave instability (at least in a phenomenological sense) for laminar flows which does not exist over a flat bottom.

Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential

Abstract Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schrodinger/Gross–Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable potentials [T. Dohnal, D. Pelinovsky, G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonlinear Sci. 19 (2009) 95–131] the CME derivation has to be carried out in Bloch rather than physical coordinates. Using the Lyapunov–Schmidt reduction we then give a rigorous justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and provide H s estimates for this approximation. The results are confirmed by numerical examples, including some new families of CMEs and gap solitons absent for separable potentials.

Diffusive stability of rolls in the two–dimensional real and complex swift–hohenberg equation

We show the nonlinear stability of small bifurcating stationary rolls u , , for the Swift–Hohenberg–equation on the domain R2. In Bloch wave representation the linearization around a marginal stable roll u∊,x, has continuous spectrum up to 0 with a locally parabolic shape at the critical Bloch vector 0. Using an abstract renormalization theorem we show that small spatially localized integrable perturbations decay diffusively to zero. Moreover we estimate the size of the domain of attraction of a roll u∊,x, in terms of its modulus and Fourier wavenumber. To explain the method we also treat the nonlinear stability of stationary rolls for the complex Swift–Hohenberg equation on R2

Numerical Results for Snaking of Patterns over Patterns in Some 2D Selkov--Schnakenberg Reaction-Diffusion Systems

For a Selkov--Schnakenberg model as a prototype reaction-diffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the Ginzburg-Landau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg-Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non-snaking behaviour appears to be related to too narrow bistable ranges with rather small Ginzburg-Landau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized atterns over patterns, for instance hexagon patches embedded in radial stripes, and fully localized hexagon patches over straight stripes.

Perturbation theory for plasmonic eigenvalues

We develop a perturbative approach for calculating, within the quasistatic approximation, the shift of surface resonances in response to a deformation of a dielectric volume. Our strategy is based on the conversion of the homogeneous system for the potential which determines the plasmonic eigenvalues into an inhomogeneous system for the potential’s derivative with respect to the deformation strength and on the exploitation of the corresponding compatibility condition. The resulting general expression for the first-order shift is verified for two explicitly solvable cases and for a realistic example of a deformed nanosphere. It can be used for scanning the huge parameter space of possible shape fluctuations with only quite small computational effort.

APPROXIMATION OF THE INTEGRAL BOUNDARY LAYER EQUATION BY THE KURAMOTO-SIVASHINSKY EQUATION ∗

In suitable parameter regimes the integral boundary layer equation (IBLe) can be formally derived as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane. For very long waves with small amplitude, the IBLe can be further reduced to the Kuramoto--Sivashinsky equation (KSe). Here we justify this reduction of the IBLe to the KSe. Using energy estimates, we show that solutions of the KSe approximate solutions of the IBLe over sufficiently long time scales. This is a step towards understanding the approximation properties of the KSe for the full Navier--Stokes system describing the inclined film flow.

Desertification by front propagation

Understanding how desertification takes place in different ecosystems is an important step in attempting to forecast and prevent such transitions. Dryland ecosystems often exhibit patchy vegetation, which has been shown to be an important factor on the possible regime shifts that occur in arid regions in several model studies. In particular, both gradual shifts that occur by front propagation, and abrupt shifts where patches of vegetation vanish at once, are a possibility in dryland ecosystems due to their emergent spatial heterogeneity. However, recent theoretical work has suggested that the final step of desertification - the transition from spotted vegetation to bare soil - occurs only as an abrupt shift, but the generality of this result, and its underlying origin, remain unclear. We investigate two models that detail the dynamics of dryland vegetation using a markedly different functional structure, and find that in both models the final step of desertification can only be abrupt. Using a careful numerical analysis, we show that this behavior is associated with the disappearance of confined spot-pattern domains as stationary states, and identify the mathematical origin of this behavior. Our findings show that a gradual desertification to bare soil due to a front propagation process can not occur in these and similar models, and opens the question of whether these dynamics can take place in nature.

A Hopf-Bifurcation Theorem for the Vorticity Formulation of the Navier–Stokes Equations in ℝ3

We prove a Hopf-bifurcation theorem for the vorticity formulation of the Navier–Stokes equations in ℝ3 in case of spatially localized external forcing. The difficulties are due to essential spectrum up to the imaginary axis for all values of the bifurcation parameter which a priori no longer allows to reduce the problem to a finite dimensional one.

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross–Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.