12 research outputs found
Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system
This manuscript extends the analysis of a much studied singularly perturbed
three-component reaction-diffusion system for front dynamics in the regime
where the essential spectrum is close to the origin. We confirm a conjecture
from a preceding paper by proving that the triple multiplicity of the zero
eigenvalue gives a Jordan chain of length three. Moreover, we simplify the
center manifold reduction and computation of the normal form coefficients by
using the Evans function for the eigenvalues. Finally, we prove the unfolding
of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to
stable periodic front motion, including stable traveling breathers, and these
results are illustrated by numerical computations.Comment: 39 pages, 7 figure
Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves
AbstractWe give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets
Butterfly catastrophe for fronts in a three-component reaction-diffusion system
Abstract We study the dynamics of front solutions in a three-component reaction-diffusion system via a combination of geometric singular perturbation theory, Evans function analysis and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically
Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients
Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the breakdown of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multipulse solutions
RIGOROUS DESCRIPTION OF MACROSCOPIC WAVE PACKETS IN INFINITE PERIODIC CHAINS OF COUPLED OSCILLATORS BY MODULATION EQUATIONS
Abstract. It is the purpose of this paper to prove error estimates for the approximate description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations, like the Korteweg–de Vries (KdV) or the Nonlinear Schrödinger (NLS) equation. The proofs are based on a discrete Bloch wave transform of the underlying infinite-dimensional system of coupled ODEs. After this transform the existing proof for the associated approximation theorem for the NLS approximation used for the approximate description of oscillating wave packets in dispersive PDE systems transfers almost line for line. In contrast, the proof of the approximation theorem for the KdV approximation of long waves is less obvious. In a special situation we prove a first approximation result. 1. Introduction. Macroscopi
Butterfly catastrophe for fronts in a three-component reaction-diffusion system
We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically