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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 Schrödinger / Gross-Pitaevskii Equation with a non-separable potential of finite contrast. We show that unlike in the case of separable… (More)

- Johannes Müller, Hannes Uecker
- Journal of mathematical biology
- 2013

Bacteria may change their behavior depending on the population density. Here we study a dynamical model in which cells of radius [R] within a diffusive medium communicate with each other via diffusion of a signalling substance produced by the cells. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior… (More)

We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx− ωt; k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx + φ±; k) as x → ±∞ with different phases φ− 6= φ+ at infinity for solutions that initially converge to… (More)

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… (More)

- Andreas Müller-Rettkowski, Hannes Uecker, Guido Schneider

4 The cubic case 19 4.1 Derivation and residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1Department of Physics and College of Optics & Photonics: CREOL & FPCE, University of Central Florida, Orlando FL 32816, USA 2new adress: Institut für Theoretische… (More)

- Hannes Uecker, Daniel Wetzel
- SIAM J. Applied Dynamical Systems
- 2014

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… (More)

- Hannes Uecker
- SIAM Journal of Applied Mathematics
- 2003

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 IBL… (More)

We consider dissipative systems on the real axis in situations when the evolution is dominated by a dynamics similar to the one of a linear diffusion equation. It is surprising that such a diffusive behavior occurs in relatively complicated systems. After a discussion of the linear and nonlinear diffusion equation, we give a brief introduction into the… (More)

The Integral Boundary Layer equation (IBLe) arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane. The trivial solution of the IBLe is linearly at best marginally stable, i.e., it has essential spectrum at least up to the imaginary axis. Here we show that in the stable case this trivial solution is in fact… (More)