6 research outputs found
Dynamics of spiral waves in the complex Ginzburg-Landau equation in bounded domains
Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau
equation in bounded domains are considered. We investigate the effect of the
boundaries on spiral motion under homogeneous Neumann boundary conditions, for
small values of the twist parameter . We derive explicit laws of motion for
rectangular domains and we show that the motion of spirals becomes
exponentially slow when the twist parameter exceeds a critical value depending
on the size of the domain. The oscillation frequency of multiple-spiral
patterns is also analytically obtained
Existence of spiral waves in oscillatory media with nonlocal coupling
We prove existence of spiral waves in oscillatory media with nonlocal
coupling. Our starting point is a nonlocal complex Ginzburg-Landau (cGL)
equation, rigorously derived as an amplitude equation for integro-differential
equations undergoing a Hopf bifurcation. Because this reduced equation includes
higher order terms that are usually ignored in a formal derivation of the cGL,
the solutions we find also correspond to solutions of the original nonlocal
system. To prove existence of these patterns we use perturbation methods
together with the implicit function theorem. Within appropriate parameter
regions, we find that spiral wave patterns have wavenumbers, , with
expansion , where is a positive constant,
is the small bifurcation parameter, and the positive constant
depends on the strength and spread of the nonlocal coupling. The main
difficulty we face comes from the linear operators appearing in our system of
equations. Due to the symmetries present in the system, and because the
equations are posed on the plane, these maps have a zero eigenvalue embedded in
their essential spectrum. Therefore, they are not invertible when defined
between standard Sobolev spaces and a straightforward application of the
implicit function theorem is not possible. We surpass this difficulty by
redefining the domain of these operators using doubly weighted Sobolev spaces.
These spaces encode algebraic decay/growth properties of functions, near the
origin and in the far field, and allow us to recover Fredholm properties for
these maps.Comment: 57 pages, 4 figure