Existence of spiral waves in oscillatory media with nonlocal coupling

Abstract

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, $κ$, with expansion $κ\sim C e^{-a/\varepsilon}$, where $a$ is a positive constant, $\varepsilon$ is the small bifurcation parameter, and the positive constant $C$ 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.

Figures (4)

• Figure 1: Spiral chimeras.
• Figure 2: Simulation of FitzHugh-Nagumo system appearing in shima2004 on a square domain of length $L =100$, using a cosine spectral method and an implicit Euler time stepping scheme with $N = 1024$ nodes and a time step $h= 0.05$. Convolution kernel used has Fourier symbol $\hat{K}(\xi) = \frac{ -\eta | \xi|^2}{ 1+ \varepsilon^2 D |\xi|^2}$ with fixed $\varepsilon^2 D =0.1$ and varying $\eta$. From left to right we have $\eta =0.5, \eta= 1.0, \eta=1.5,$ and $\eta= 2.0$.
• Figure 3: Simulation of FitzHugh-Nagumo system appearing in shima2004 on a square domain of length $L =100$, using a cosine spectral method and an implicit Euler time stepping scheme with $N = 1024$ nodes and a time step $h= 0.05$. Convolution kernel used has Fourier symbol $\hat{K}(\xi) = \frac{ -\eta | \xi|^2}{ 1+ \varepsilon^2 D |\xi|^2}$ with fixed $\eta =1$ and varying $\tilde{D} =\varepsilon^2 D$. From left to right we have $\tilde{D} =0.5, \tilde{D}= 1.0, \tilde{D}=1.5,$ and $\tilde{D}= 2.0$. First row depicts full spiral, second row zooms in into core of spiral.
• Figure 4: Examples of algebraic decay/growth for weighted, $L^2_{\gamma}(\mathbb{R}^2)$, and doubly-weighted, $L^2_{\gamma, \sigma}(\mathbb{R}^2)$, Sobolev spaces. Parameter $\gamma$ encodes decay/growth properties of functions at infinity, while parameter $\sigma$ encodes decay/growth rates of functions near the origin.

Theorems & Definitions (68)

• Theorem 1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• proof