Math @ Duke
Publications [#9634] of Thomas P. Witelski
- Thomas P. Witelski, K. Ono, T. J. Kaper, On Axi-symmetric Traveling waves and Radial solutions of semi-linear elliptic equations,
Natural Resource Modeling 13, 3, 2000, pp. 339-387
(last updated on 2000/07/19)
Combining analytical techniques
from perturbation methods
and dynamical systems theory,
we present an elementary approach
to the detailed construction
of axi-symmetric diffusive interfaces
in semi-linear elliptic equations.
Solutions of the resulting non-autonomous radial
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
These axi-symmetric problems are fundamental examples of
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.
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