66 research outputs found
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Traveling pulse solutions in a three-component FitzHugh-Nagumo Model
We use geometric singular perturbation techniques combined with an action
functional approach to study traveling pulse solutions in a three-component
FitzHugh--Nagumo model. First, we derive the profile of traveling -pulse
solutions with undetermined width and propagating speed. Next, we compute the
associated action functional for this profile from which we derive the
conditions for existence and a saddle-node bifurcation as the zeros of the
action functional and its derivatives. We obtain the same conditions by using a
different analytical approach that exploits the singular limit of the problem.
We also apply this methodology of the action functional to the problem for
traveling -pulse solutions and derive the explicit conditions for existence
and a saddle-node bifurcation. From these we deduce a necessary condition for
the existence of traveling -pulse solutions. We end this article with a
discussion related to Hopf bifurcations near the saddle-node bifurcation
Linear stability of radially symmetric equilibrium solutions to the singular limit problem of three-component activator-inhibitor model
We show linear stability or instability for radially symmet-ric equilibrium solutions to the system of interface equation and two parabolic equations arising in the singular limit of three-component activator-inhibitor models
Large amplitude radially symmetric spots and gaps in a dryland ecosystem model
We construct far-from-onset radially symmetric spot and gap solutions in a
two-component dryland ecosystem model of vegetation pattern formation on flat
terrain, using spatial dynamics and geometric singular perturbation theory. We
draw connections between the geometry of the spot and gap solutions with that
of traveling and stationary front solutions in the same model. In particular,
we demonstrate the instability of spots of large radius by deriving an
asymptotic relationship between a critical eigenvalue associated with the spot
and a coefficient which encodes the sideband instability of a nearby stationary
front. Furthermore, we demonstrate that spots are unstable to a range of
perturbations of intermediate wavelength in the angular direction, provided the
spot radius is not too small. Our results are accompanied by numerical
simulations and spectral computations
Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system
This manuscript extends the analysis of a much studied singularly perturbed
three-component reaction-diffusion system for front dynamics in the regime
where the essential spectrum is close to the origin. We confirm a conjecture
from a preceding paper by proving that the triple multiplicity of the zero
eigenvalue gives a Jordan chain of length three. Moreover, we simplify the
center manifold reduction and computation of the normal form coefficients by
using the Evans function for the eigenvalues. Finally, we prove the unfolding
of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to
stable periodic front motion, including stable traveling breathers, and these
results are illustrated by numerical computations.Comment: 39 pages, 7 figure
Constructive proofs for localized radial solutions of semilinear elliptic systems on
Ground state solutions of elliptic problems have been analyzed extensively in
the theory of partial differential equations, as they represent fundamental
spatial patterns in many model equations. While the results for scalar
equations, as well as certain specific classes of elliptic systems, are
comprehensive, much less is known about these localized solutions in generic
systems of nonlinear elliptic equations. In this paper we present a general
method to prove constructively the existence of localized radially symmetric
solutions of elliptic systems on . Such solutions are essentially
described by systems of non-autonomous ordinary differential equations. We
study these systems using dynamical systems theory and computer-assisted proof
techniques, combining a suitably chosen Lyapunov-Perron operator with a
Newton-Kantorovich type theorem. We demonstrate the power of this methodology
by proving specific localized radial solutions of the cubic Klein-Gordon
equation on , the Swift-Hohenberg equation on , and
a three-component FitzHugh-Nagumo system on . These results
illustrate that ground state solutions in a wide range of elliptic systems are
tractable through constructive proofs
Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
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