Invariant manifolds and applications for functional differential equations of mixed type

Differential equations posed on discrete lattices have by now become a popular modelling tool used in a wide variety of scientific disciplines. Such equations allow the inclusion of non-local interactions into models and lead to interesting dynamical and pattern-forming behaviour. Although many numerical results have already been obtained for such lattice differential equations (LDEs), we are still far removed from a rigorous mathematical theory that is able to confirm and predict many of the interesting phenomena that these studies have uncovered. It is a basic and well-established mathematical practice to start the investigation of LDEs by looking for travelling waves. These are patterns that have a fixed shape and travel through the lattice at a fixed speed. Even this simple scenario however poses a significant challenge, due to the fact that a type of equation is encountered that defies treatment using classical techniques. This thesis describes several contributions towards the development of a new mathematical framework that will help to meet this challenge. The application range of the results is illustrated by discussing several problems encountered in various fields of research.Thomas Stieltjes Institute for MathematicsUBL - phd migration 201

Micropterons, nanopterons and solitary wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou problem

Analysis and Stochastic

Travelling Waves for spatially discrete Systems of FitzHugh-Nagumo Type with periodic Coefficients

We establish the existence and nonlinear stability of traveling 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 traveling wave solutions are known to exist in various settings. The two-periodic waves considered in this paper are described by singularly perturbed multicomponent 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.Analysis and Stochastic

Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory

Analysis and Stochastic

Travelling waves for adaptive grid discretizations of reaction diffusion systems: I: well-posedness

Analysis and Stochastic

Scaling relations for auxin waves

We analyze an 'up-the-gradient' model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi-Pasta-Ulam-Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.Analysis and Stochastic

Carrier dynamics in conducting polymers: Case of PF6 doped Polypyrrole

Quantum Matter and Optic

Curvature-driven front propagation through planar lattices in oblique directions

Analysis and Stochastic

Travelling waves for reaction-diffusion equations forced by translation invariant noise

Inspired by applications, we consider reaction-diffusion equations on R that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in Hamster and Hupkes (2018,2019) for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions. (C) 2019 Elsevier B.V. All rights reserved.Analysis and Stochastic

Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting