On Axisymmetric Traveling Waves and Radial Solutions of Semi‐linear Elliptic Equations

Abstract

ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementary approach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowly varying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisymmetric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensively discuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Many of the results contained in this article are known, and in presenting known results, it is intended that this article be expository in nature, providing elementary demonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radially symmetric solutions of semilinear elliptic problems, especially to those articles in which more advanced mathematical theory is developed.

Radial propagation in population dynamics with density-dependent diffusion

“…(3) as expected. Once again, we have seen that the front speed is altered by the curvature as found in other similar systems [7,22]. At sufficiently large distances that R γ /2, the front speed can be approximated as a constant c ≈ 1.…”

Quantifying the roles of cell motility and cell proliferation in a circular barrier assay

“…Our estimates of D and l allow us to predict the long-term front speed for the proliferative populations. Formally, equation (3.1) does not support travelling wave solutions [9,35]. However, the asymptotic result for the Fisher-Kolmogorov equation is approximately valid in an axisymmetric radial geometry for sufficiently large r [9].…”

Curvature dependences of wave propagation in reaction-diffusion models

“…It should be emphasised that this equation is well behaved as 𝑟 → 0. Indeed, by symmetry, lim 𝑟→0 𝑢 𝑟 = 0, (10) so that using Bernoulli-L'Hôpital's rule, we have lim…”

“…The time-stepping rule for 𝑖 = 0 is based on Eq. ( 11), and 𝑢 𝑘 −1 is set to 𝑢 𝑘 1 to satisfy the Neumann-like symmetry condition (10) Curvature and velocity. Numerical estimates of the mean curvature 𝜅 are calculated from:…”

“…In one spatial dimension, excitable media support the establishment of travelling waves with constant propagation speed 𝑐 in the long-time limit, under appropriate initial and boundary conditions [9]. In higher spatial dimensions, it is well known experimentally and theoretically, that the propagation speed of travelling fronts is modified by the front's curvature [1,3,5,7,10]. Singular perturbation theories of excitable reaction-diffusion systems show that the normal velocity 𝑣 of travelling fronts is given by…”