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1. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

   https://doi.org/10.1007/s10884-022-10143-4  

   Hupkes, H. J.; Van Vleck, E. S. (April 2022, Journal of Dynamics and Differential Equations)

   Abstract 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. 

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2. Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems II: Linear Theory

   https://doi.org/10.1007/s10884-021-09942-y  

   Hupkes, H. J.; Van Vleck, E. S. (September 2022, Journal of Dynamics and Differential Equations)

   Abstract In this paper we consider an adaptive spatial discretization scheme for the Nagumo PDE. The scheme is a commonly used spatial mesh adaptation method based on equidistributing 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 Equ 28:955, 2016). For small spatial grid-sizes, we establish some useful Fredholm properties for the operator that arises after linearizing our system around the travelling wave solutions to the original Nagumo PDE. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is a spatially stretched and twisted version of the standard second order differential operator that is associated to the PDE waves. 

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3. Travelling colourful patterns in self-organized cellulose-based liquid crystalline structures

   https://doi.org/10.1038/s43246-021-00182-7  

   Silva, Pedro E.; Chagas, Ricardo; Fernandes, Susete N.; Pieranski, Pawel; Selinger, Robin L.; Godinho, Maria Helena (December 2021, Communications Materials)

   null (Ed.)

   Abstract Cellulose-based systems are useful for many applications. However, the issue of self-organization under non-equilibrium conditions, which is ubiquitous in living matter, has scarcely been addressed in cellulose-based materials. Here, we show that quasi-2D preparations of a lyotropic cellulose-based cholesteric mesophase display travelling colourful patterns, which are generated by a chemical reaction-diffusion mechanism being simultaneous with the evaporation of solvents at the boundaries. These patterns involve spatial and temporal variation in the amplitude and sign of the helix´s pitch. We propose a simple model, based on a reaction-diffusion mechanism, which simulates the observed spatiotemporal colour behaviour. 

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4. Spatial dynamics of higher order rock-paper-scissors and generalisations

   https://doi.org/10.1088/1751-8121/ad3bf6  

   Griffin, Christopher; Feng, Li; Wu, Rongling (April 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract We introduce and study the spatial replicator equation with higher order interactions and both infinite (spatially homogeneous) populations and finite (spatially inhomogeneous) populations. We show that in the special case of three strategies (rock–paper–scissors) higher order interaction terms allow travelling waves to emerge in non-declining finite populations. We show that these travelling waves arise from diffusion stabilisation of an unstable interior equilibrium point that is present in the aspatial dynamics. Based on these observations and prior results, we offer two conjectures whose proofs would fully generalise our results to all odd cyclic games, both with and without higher order interactions, assuming a spatial replicator dynamic. Intriguingly, these generalisations for $N\text{⩾}5$strategies seem to require declining populations, as we show in our discussion. 

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5. The leading edge of a free boundary interacting with a line of fast diffusion

   L. A. Caffarelli, J.M. Roquejoffre (January 2020, Algebra i analiz)

   null (Ed.)

   The goal of this work is to explain an unexpected feature of the expanding level sets of the solutions of a system where a half-plane in which reaction-diffusion phenomena take place exchanges mass with a line having a large diffusion of its own. The system was proposed by H. Berestycki, L. Rossi and the second author as a model of enhancement of biological invasions by a line of fast diffusion. It was observed numerically by A.-C. Coulon that the leading edge of the front, rather than being located on the line, was in the lower half-plane. We explain this behavior for a closely related free boundary problem. We construct travelling waves for this problem, and the analysis of their free boundary near the line confirms the predictions of the numerical simulations. 

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