Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scalingGoh, Ryan; Kaper, Tasso J.; Scheel, Arnd
doi: 10.1111/sapm.12702pmid: N/A
We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed c∼ε1/3$c\sim \varepsilon ^{1/3}$, where ε$\varepsilon$ is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of c$c$, that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings–McLeod solution of the Painlevé‐II equation to Painlevé‐II equations with a drift term.
Orthogonal polynomials on domains of revolutionXu, Yuan
doi: 10.1111/sapm.12703pmid: N/A
We study orthogonal polynomials (OPs) for a weight function defined over a domain of revolution, where the domain is formed from rotating a two‐dimensional region and goes beyond the quadratic domains. Explicit constructions of orthogonal bases are provided for weight functions on a number of domains. Particular attention is paid to the setting when an orthogonal basis can be constructed explicitly in terms of known polynomials of either one or two variables. Several new families of OPs are derived, including a few families that are eigenfunctions of a spectral operator and their reproducing kernels satisfy an addition formula.
From primary HPV infection to carcinoma in situ: A mathematical approach to cervical intraepithelial neoplasiaBitsouni, Vasiliki; Gialelis, Nikolaos; Stratis, Ioannis G.; Tsilidis, Vasilis
doi: 10.1111/sapm.12697pmid: N/A
Cervical intraepithelial neoplasia (CIN) is the development of abnormal cells on the surface of the cervix, caused by a human papillomavirus (HPV) infection. Although in most of the cases it is resolved by the immune system, a small percentage of people might develop a more serious CIN which, if left untreated, can develop into cervical cancer. Cervical cancer is the fourth most common cancer in women globally, for which the World Health Organization (WHO) recently adopted the Global Strategy for cervical cancer elimination by 2030. With this research topic being more imperative than ever, in this paper, we develop a nonlinear mathematical model describing the CIN progression. The model consists of partial differential equations describing the dynamics of epithelial, dysplastic, and immune cells, as well as the dynamics of viral particles. We use our model to explore numerically three important factors of dysplasia progression, namely, the geometry of the cervix, the strength of the immune response, and the frequency of viral exposure.
Revivals, or the Talbot effect, for the Airy equationPelloni, B.; Smith, D. A.
doi: 10.1111/sapm.12699pmid: N/A
We study Dirichlet‐type problems for the simplest third‐order linear dispersive partial differential equations (PDE), often referred to as the Airy equation. Such problems have not been extensively studied, perhaps due to the complexity of the spectral structure of the spatial operator. Our specific interest is to determine whether the peculiar phenomenon of revivals, also known as Talbot effect, is supported by these boundary conditions, which for third‐order problems are not reducible to periodic ones. We prove that this is the case only for a very special choice of the boundary conditions, for which a new type of weak cusp revival phenomenon has been recently discovered. We also give some new results on the functional class of the solution for other cases.
Well‐posedness of the Green–Naghdi model for an uneven bottom in presence of the Coriolis effect and surface tensionBerjawi, Marwa; El Arwadi, Toufic; Israwi, Samer; Talhouk, Raafat
doi: 10.1111/sapm.12725pmid: N/A
The objective of this work is to derive and analyze a Green–Naghdi model with Coriolis effect and surface tension in nonflat bottom geometry. Gui et al. derive a Green–Naghdi‐type model in flat bottom geometry under the gravity and Coriolis effect. Chen et al. proved the existence and uniqueness of solution in Sobolev space under a condition depending on the initial velocity and the Coriolis effect. In this paper, we provide a rigorous derivation of Green–Naghdi model under the influence of the two mentioned effects, with nonflat bottom. After that, the existence and construction of solutions for the derived model will be proved under two alternative conditions: the first one is the same condition as in Chen et al. and Berjawi et al. and the second one concerns only the Coriolis coefficient Ω$\Omega$ that supposed to be only of order O(μ)$O({\sqrt {\mu }})$. This existence and uniqueness result ameliorate the result of Chen et al. and Berjawi et al. in the sense that no condition on the velocity is needed. We also prove the continuity of the associated flow map.
Stabilized time‐series expansions for high‐order finite element solutions of partial differential equationsDeeb, Ahmad; Dutykh, Denys
doi: 10.1111/sapm.12708pmid: N/A
Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time‐Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high‐order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time‐series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher‐order FEs. Our approach involves the incorporation of artificial diffusion terms on the left‐hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
An analytical and numerical study of the Diaz–Solovchuk–Sheu acoustic model: How does it compare with Blackstock's in approximating the Euler system?Jordan, Pedro M.
doi: 10.1111/sapm.12721pmid: N/A
Employing primarily numerical methods, and working in 1D, we seek to determine which of two competing finite‐amplitude acoustic models, specifically, those of Blackstock and Diaz et al, best approximates the acoustic special case of the Euler system. Working in the context of the classical signaling problem with sinusoidal input, we perform our assessment using not only velocity profile plots, but also a number of metrics. Our findings show, without equivocation, that the simpler Diaz et al model outperforms Blackstock's vis‐à‐vis all comparisons performed and metrics considered.