Rigidly rotating spiral waves organize the spatio-temporal dynamics of excitable and oscillatory media modeled by reaction–diffusion systems. The memoir of Sandstede and Scheel established a comprehensive linear theory — existence, robustness, far-field asymptotics, and spectral characterization of spiral waves and of their truncations to large disks — but the nonlinear stability theory on the full plane remains open: standard semigroup and center-manifold arguments on finite disks give basins of attraction that shrink exponentially with the disk radius, because the resolvent grows exponentially with domain size and the essential spectrum generated by the emitted wave trains sits on the imaginary axis. The memoir poses, as the leading item of its open-problems section, a nonlinear stability theory on the unbounded plane that incorporates the outward transport encoded in the essential and absolute spectra, with sharp decay rates and basins that do not deteriorate with domain size, in the spirit of one-dimensional source-defect results. The nonlinear stability question is presently open; the known adjacent results — one-dimensional source defects, reduced curvature-flow and Ginzburg–Landau settings, linear and numerical spectral computation, and heuristic multi-spiral interaction laws — do not settle it.

Background

Setting. Reaction–diffusion systems

ut=D Δu+f(u),u∈RN, x∈R2,u_t = D\,\Delta u + f(u), \qquad u \in \mathbb{R}^N,\ x \in \mathbb{R}^2,

with a positive diagonal diffusion matrix DD and smooth nonlinearity ff, model spatially extended excitable and oscillatory media such as the Belousov–Zhabotinsky reaction, cardiac tissue, and neural tissue. A rigidly rotating spiral wave is a solution u(x,t)=u∗(r,φ−ω∗t)u(x,t) = u_*(r,\varphi - \omega_* t) in polar coordinates (r,φ)(r,\varphi) with rotation frequency ω∗>0\omega_* > 0: the pattern rotates rigidly about a localized core while radiating waves outward. In the far field r→∞r \to \infty, such a spiral is asymptotic to an emitted wave train, i.e. a periodic traveling wave u∞(k x−ω∗t)u_\infty(k\,x - \omega_* t); the spiral acts as a wave source when the wave train's group velocity (the derivative of the temporal frequency with respect to wavenumber along the dispersion relation) points away from the core.

The linear theory. Sandstede and Scheel's memoir develops a spectral theory of spiral waves in general reaction–diffusion systems using radial spatial dynamics — the radius rr is treated as an evolution variable in an exponentially weighted function space, and the resulting operators are analyzed with exponential dichotomies, which provide stable/unstable splittings along this radial evolution, the tool classically used for one-dimensional fronts and pulses. They show that the spectrum of the linearization in the co-rotating frame, L∗=DΔ+ω∗∂φ+f′(u∗)L_* = D\Delta + \omega_*\partial_\varphi + f'(u_*), decomposes into: (i) essential spectrum — curves determined by the Floquet dispersion relation of the asymptotic wave train, where the Floquet spectrum is the set of temporal growth rates of the wave train under longitudinal perturbations that are co-periodic in space; (ii) the absolute spectrum — the set of spectral parameters where the spatial (radial) eigenvalues of the far-field operator meet in real part, which controls the far-field shape of eigenfunctions; (iii) isolated extended point spectrum with spatially decaying eigenfunctions; and (iv) boundary spectrum on truncated domains. The memoir proves robustness of spirals under parameter and domain perturbations and convergence, as R→∞R \to \infty, of the spectra of spirals truncated to disks of radius RR with compatible 'boundary sink' boundary conditions.

Stability: what is known and why it is not enough. Call a spiral spectrally stable if it emits a wave train whose Floquet spectrum lies in Re⁡λ<0\operatorname{Re}\lambda < 0 except a simple neutral phase mode at λ=0\lambda = 0, with positive group velocity, if that wave train is in addition transversely stable as a plane wave in the sense of the source — stable under two-dimensional perturbations, i.e. its Bloch spectrum with arbitrary transverse wavenumber ν⊥∈iR\nu_\perp \in i\mathbb{R} lies in Re⁡λ<0\operatorname{Re}\lambda < 0 except neutral modes — and if its absolute and extended point spectra lie in Re⁡λ<0\operatorname{Re}\lambda < 0. The transverse-stability hypothesis cannot be dropped: the memoir shows that the two-dimensional stability of wave trains does not affect the spectra of spiral waves, and proves that the semigroup associated with the linearization at a spiral that is spectrally stable in the one-dimensional Floquet sense but whose asymptotic wave train is transversely unstable exhibits exponential growth with a strictly positive exponential rate. On any fixed large disk, the truncated linearization has compact resolvent and a simple neutral eigenvalue induced by rotational symmetry, so classical semigroup and center-manifold arguments for parabolic problems yield nonlinear asymptotic stability of the rotating wave in that disk. However, the memoir shows that the resolvent norm used to build spectral projections and decay estimates grows exponentially in RR, so the proven basins of attraction shrink exponentially as the disk grows — and the authors show this conclusion is optimal for any argument using only bounded-domain information. On the full plane, the essential spectrum of the emitted wave train reaches the imaginary axis at λ=0\lambda = 0 (the neutral phase mode), so there is no spectral gap: stability can only be diffusive (algebraic decay), and only modulo a persistent far-field phase shift. In one spatial dimension this program was carried out for source defects — time-periodic defects that emit wave trains in the complex Ginzburg–Landau equation, an amplitude equation for oscillatory media — by Beck, Nguyen, Sandstede and Zumbrun, who proved nonlinear stability with algebraic decay of phase-modulated perturbations. The memoir identifies the obstruction to extending this to planar spirals: azimuthal stretching, i.e transported perturbations of the far-field wave trains are sheared as they travel around the rotating core, which prevents the diffusive decay mechanism exploited in one dimension; Roussier's earlier analysis of stability in radially symmetric reaction–diffusion settings exhibits related radial-geometry difficulties.

Problem Statement

Prove or disprove the following, in the generality of the source. Let u∗(r,φ−ω∗t)u_*(r,\varphi-\omega_* t) be a rigidly rotating spiral wave solution of a reaction–diffusion system ut=DΔu+f(u)u_t = D\Delta u + f(u) on R2\mathbb{R}^2 with DD a positive diagonal diffusion matrix, and suppose the spiral is spectrally stable in the sense of Sandstede–Scheel: it emits a spectrally stable wave train with positive group velocity that is in addition transversely stable as a plane wave (its two-dimensional Bloch spectrum, including perturbations with arbitrary transverse wavenumber, lies in Re⁡λ<0\operatorname{Re}\lambda < 0 except neutral modes), and its absolute spectrum and extended point spectrum lie in Re⁡λ<0\operatorname{Re}\lambda < 0. Establish nonlinear asymptotic stability on the full unbounded plane: identify a natural topology for perturbations of the non-decaying far field (admitting at least localized perturbations of the core and perturbations that modulate the far-field phase), and prove that solutions with initial data in a fixed, sizeable class converge as t→∞t \to \infty to the spiral wave, up to the natural Euclidean symmetries (translation and rotation phase) and an allowed bounded far-field phase shift, with quantitative — ideally sharp — decay estimates that reflect the outward transport and diffusive decay encoded in the essential and absolute spectra. The basin of attraction must not shrink with any artificial domain truncation; in particular, arguments relying only on the finite-disk center-manifold route, whose basins shrink exponentially in the disk radius, are insufficient, and the estimates must incorporate the far-field transport. A rigorous negative result would resolve the question equally: for instance, a construction of a spectrally stable spiral wave that is nevertheless nonlinearly unstable on R2\mathbb{R}^2. (This is the leading open problem posed in §12.3 of the source; the further directions listed there — multi-spiral interaction laws, nonlinear bifurcation analysis of meandering, transverse-instability spectral mapping, retracting waves in the vanishing-rotation-frequency limit, and three-dimensional scroll waves — are separate open problems and do not belong to this problem's completion boundary.)

References

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