arXiv subjects

H. J. Hupkes

Publications and source records attributed to H. J. Hupkes.

6 recordsLinked to original sources

arXiv2020

Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small. By applying a stochastic phase-shift together with a time-transform, we obtain a semilinear sPDE that describes the fluctuations from the primary wave. We subsequently develop a semigroup approach to handle the nonlinear stability question in a fashion that is closely related to modern deterministic methods.

math.AP↗

arXiv2019

Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions

We establish the existence and nonlinear stability of travelling pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions close to the continuum limit. For the verification of the spectral properties, we need to study a functional differential equation of mixed type (MFDE) with unbounded shifts. We avoid the use of exponential dichotomies and phase spaces, by building on a technique developed by Bates, Chen and Chmaj for the discrete Nagumo equation. This allows us to transfer several crucial Fredholm properties from the PDE setting to our discrete setting.

math.DS↗

arXiv2018

Travelling waves for spatially discrete systems of FitzHugh-Nagumo type with periodic coefficients

We establish the existence and nonlinear stability of travelling wave solutions for a class of lattice differential equations (LDEs) that includes the discrete FitzHugh-Nagumo system with alternating scale-separated diffusion coefficients. In particular, we view such systems as singular perturbations of spatially homogeneous LDEs, for which stable travelling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multi-component functional differential equations of mixed type (MFDEs). In order to analyze these equations, we generalize the spectral convergence technique that was developed by Bates, Chen and Chmaj to analyze the scalar Nagumo LDE. This allows us to transfer several crucial Fredholm properties from the spatially homogeneous to the spatially periodic setting. Our results hence do not require the use of comparison principles or exponential dichotomies.

math.DS↗

arXiv2014

Multidimensional Stability of Waves Travelling Through Rectangular Lattices in Rational Directions

We consider general reaction diffusion 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 differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.

math.AP↗

arXiv2013

Entire Solutions for Bistable Lattice Differential Equations with Obstacles

We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in a paper of Berestycki, Hamel and Matano for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.

math.DS↗

arXiv2004

Unconditional Security of Practical Quantum Key Distribution

We present an extension of the first proof for the unconditional security of the BB84 quantum key distribution protocol which was given by Mayers. We remove the constraint that a perfect BB84 quantum source is required and the proof given here covers a range of practical quantum sources. Nothing is assumed about the detector except that the efficiency with which signals are detected is basis independent.

quant-ph↗