Hermen Jan Hupkes

Stability of Pulse Solutions for the Discrete FitzHugh{Nagumo System

We show that the fast travelling pulses of the discrete FitzHugh{Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be veried involve linear operators that

Discontinuous initial value problems for functional differential-algebraic equations of mixed type

We study the well-posedness of initial value problems for nonlinear functional differential-algebraic equations of mixed type. We are interested in solutions to such problems that admit a single jump discontinuity at time zero. We focus specially on the question whether unstable equilibria can be stabilized by appropriately choosing the size of the jump discontinuity. We illustrate our techniques by analytically studying an economic model for the interplay between inflation and interest rates. In particular, we investigate under which circumstances the central bank can prevent runaway inflation by appropriately hiking the interest rate.

Traveling Pulse Solutions for the Discrete FitzHugh–Nagumo System

The existence of fast traveling pulses of the discrete FitzHugh–Nagumo equation is obtained in the weak-recovery regime. This result extends to the spatially discrete setting the well-known theorem that states that the FitzHugh–Nagumo PDE exhibits a branch of fast waves that bifurcates from a singular pulse solution. The key technical result that allows for the extension to the discrete case is the Exchange Lemma that we establish here for functional differential equations of mixed type.

Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes

The nonlinear stability of traveling Lax shocks in semidiscrete conservation laws involving general spatial forward-backward discretization schemes is considered. It is shown that spectrally stable semidiscrete Lax shocks are nonlinearly stable. In addition, it is proved that weak semidiscrete Lax profiles satisfy the spectral stability hypotheses made here and are therefore nonlinearly stable. The nonlinear stability results are proved by constructing the resolvent kernel using exponential dichotomies, which have recently been developed in this setting, and then using the contour integral representation for the associated Greens function to derive pointwise bounds that are sufficient for proving nonlinear stability. Previous stability analyses for semidiscrete shocks relied primarily on Evans functions, which exist only for one-sided upwind schemes.

Multi-dimensional stability of waves travelling through rectangular lattices in rational directions

We consider general reaction diusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Green’s functions estimates for functional dierential equations of mixed type (MFDEs), allowing our results

NEGATIVE DIFFUSION AND TRAVELING WAVES IN HIGH DIMENSIONAL LATTICE SYSTEMS

We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile ...