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- On the convergence to critical scaling profiles in submonolayer deposition modelsPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelIn this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $n\geq 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $x=\tau$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $\xi:= (x − \tau )/ \tau$ , corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $\Phi_{2,n}(\xi)$ when $x, \tau \to +\infty$ with $\xi$ fixed, as well as the rate at which the limit is approached.
- Steady state solutions in a model of a cholesteric liquid crystal samplePublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Grinfeld, Michael; Mottram, Nigel; Xayxanadasy, KedtysackMotivated by recent mathematical studies of Fréedericksz transitions in twist cells and helix unwinding in cholesteric liquid crystal cells [(da Costa et al. in Eur J Appl Math 20:269–287, 2009), (da Costa et al. in Eur J Appl Math 28:243–260, 2017), (McKay in J Eng Math 87:19–28, 2014), (Millar and McKay in Mol Cryst Liq Cryst 435:277/[937]–286/[946], 2005)], we consider a model for the director configuration obtained within the framework of the Frank-Oseen theory and consisting of a nonlinear ordinary differential equation in a bounded interval with non-homogeneous mixed boundary conditions (Dirichlet at one end of the interval, Neumann at the other). We study the structure of the solution set using the depth of the sample as a bifurcation parameter. Employing phase space analysis techniques, time maps, and asymptotic methods to estimate integrals, together with appropriate numerical evidence, we obtain the corresponding novel bifurcation diagram and discuss its implications for liquid crystal display technology. Numerical simulations of the corresponding dynamic problem also provide suggestive evidence about stability of some solution branches, pointing to a promising avenue of further analytical, numerical, and experimental studies.
- Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial conditionPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe establish rates of convergence of solutions to scaling (or similarity) profiles in a coagulation type system modeling submonolayer deposition. We prove that, although all memory of the initial condition is lost in the similarity limit, information about the large cluster tail of the initial condition is preserved in the rate of approach to the similarity profile. The proof relies on a change of variables that allows for the decoupling of the original infinite system of ordinary differential equations into a closed two-dimensional nonlinear system for the monomer--bulk dynamics and a lower triangular infinite dimensional linear one for the cluster dynamics. The detailed knowledge of the long time monomer concentration, which was obtained earlier by Costin et al. in [Commun. Inf. Syst., 13 (2013), pp. 183--200] using asymptotic methods and is rederived here by center manifold arguments, is then used for the asymptotic evaluation of an integral representation formula for the concentration of j-clusters. The use of higher order expressions, both for the Stirling expansion and for the monomer evolution at large times, allow us to obtain not only the similarity limit, but also the rate at which it is approached.