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Spatially localized states arise in many physical systems, examples of
which include solitary waves (solitons) in fluids, bounded gas
discharge spots, cavity solitons in optics, and oscillons in vibrating
granular media. In addition, spatially localized states play a
paramount role in a variety of biological systems. While the most
famous example is related to insights that emerged out of the seminal
work on excitable pulses by Hodgkin and Huxley, which is central today
to neuroscience and cardiac arrhythmia, there are many other significant
examples. These include resonant responses in the cochlea,
aggregations by molecular motors inside cellular protrusions, and
vegetation patches in semi-arid climates.
Localized states arise in both conservative and dissipative systems
and impose fundamental questions and challenges in the field of
nonlinear partial differential equations. For example, extensive
studies of dissipative solitons, i.e., localized states that obey a
reflective spatial symmetry, have led to identification of a novel
bifurcation structure that is known today as homoclinic snaking. While
the basic mathematical understanding of homoclinic snaking exploits
the gradient Swift-Hohenberg model through a Maxwell construction, it
was shown that reaction-diffusion (RD) models are capable of capturing this
phenomenon as well. In fact, both cases show that homoclinic snaking
arises through the subcritical finite wave-number instability, which
is also referred to as a Turing instability in the RD context. Further details on homoclinic snaking and dissipative
solitons can be found in a recent review by Knobloch in
Annu. Rev. Condens. Matter Phys. (2015).
Localized patterns can also occur for RD systems in the singularly
perturbed limit corresponding to a large diffusivity ratio between two
solution components in the system. In this context the physical
dimensions introduce a further complexity: the singularly perturbed RD
system (e.g., Brusselator model), posed on a surface of a sphere,
allows for the existence of quasi steady-state localized spot
patterns that can exhibit self-replication and other instabilities in
various parameter regimes. Prior analytical studies
have focused on using weakly
nonlinear and equivariant bifurcation theory to derive normal form
equations characterizing the development of small amplitude spatial
patterns that bifurcate from a spatially uniform
steady-state. However, due to the typical high degree of degeneracy of
the eigenspace associated with spherical harmonics of large mode
number, these amplitude equations typically consist of a large coupled
set of nonlinear ODEs. The latter are known to have an intricate
subcritical bifurcation structure. As a result, the preferred spatial
pattern that emerges from an interaction of these weakly nonlinear
modes is difficult to predict theoretically. Alternatively, asymptotic
analysis can be used to derive a differential algebraic system (DAE)
of ODEs characterizing the time-dependent spatial locations of a
collection of localized spots. The DAE system is vaguely similar to
that for the motion of Eulerian point vortices on the sphere. In
certain cases, the equilibria of this DAE system for spot dynamics
have a close connection to the classical problem of determining a set
of elliptic Fekete points, corresponding to globally minimizing the
discrete logarithmic energy for N points on the sphere. Similar DAE
systems for spot dynamics can be derived for other RD models.
A common unifying theme of the talks in this Featured Minisymposium is
that, due to the non-gradient nature of RD models, localized states
can exhibit more complex behavior than energy driven systems, since
they are not restricted by the gradient dynamics of an associated
variational structure. Consequently, the objective of this Featured
Minisymposium is two-fold: (i) to report on novel mathematical
phenomena and methods that are related to localized states in RD
models and (ii) to link between the mathematics and applications that
cross a range of problems in chemistry, biology, and approximation
theory (Fekete point problem). The minisymposium constitutes four
talks on distinct aspects of dynamical systems and bifurcation theory
that are related to recent advances on the subject. The organizer,
Arik Yochelis (Ben-Gurion University), will reveal the conditions at
which homoclinic snaking may arise in excitable media using extended
FitzHugh-Nagumo model and show why these conditions are connected to
the generation of finite pulse trains out of a single localized
perturbation. Justin Tzou (Dalhousie University) will show the
organization of time-oscillatory patterns in a snaking region near a
codimension-two Turing-Hopf bifurcation in the Brusselator
model. Michael Ward (University of British Columbia) will present a
singular perturbation theory characterizing the slow dynamics of
localized spots on the surface of the sphere for the Brusselator
model, and the connection between this theory and the elliptic Fekete
Point problem of distributing point charges on the surface of the
sphere. Finally, Nicolas Verschueren Van Rees (University of Bristol)
will demonstrate the connection between the snaking scenario and front
dynamics in a model for cell polarization. Notably, talks in this
Featured Minisymposium are carefully designed to provide both
accessibility to a general mathematical audience and to elaborate on
the rapidly evolving field of spatially localized states.
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Top left: Slanted homoclinic snaking structure in a model for excitable medium. (Arik Yochelis, from Phys. Rev. E 91, 032924 (2015)).
Top right: Localized Turing state embedded in uniform Hopf oscillations. (Justin Tzou, from Phys. Rev. E 87, 022908 (2013)).
Bottom left: Self-replicating spot patterns on a slowly growing sphere. (Michael Ward, from SIADS 13, 564 (2014)).
Bottom right: Several snaking regions in the prototype model for cell polarization. (Nicolas Verschueren Van Rees, unpublished)
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