Nonlinear stability, reduced interaction dynamics, and bifurcations of planar spiral waves beyond finite-disk truncation

Spiral waves are the canonical coherent defects of two-dimensional excitable and oscillatory media, and Sandstede and Scheel's Memoir 'Spiral Waves: Linear and Nonlinear Theory' (2023) established a robust linear and far-field theory for them in general reaction-diffusion systems. The authors explicitly flag the complementary nonlinear theory on the unbounded plane as open: (i) nonlinear asymptotic stability of a spectrally stable spiral wave with decay estimates reflecting the outward transport encoded in the essential and absolute spectra, instead of finite-disk arguments whose basins shrink exponentially with disk radius; (ii) a justified finite-dimensional slow manifold describing the exponentially slow drift induced by distant boundaries and the interaction of multiple well-separated spiral cores; and (iii) nonlinear bifurcation analysis, including existence of meandering spirals past a supercritical Hopf bifurcation and the conjectured convergence of disk-truncated center-manifold dynamics to Barkley's skew-product equations. These problems are open, and partial progress exists only in one-dimensional analogues, spectral theory with rank-deficient diffusion, and formal or geometric-flow analogues.

Background

Spiral waves are rotating wave patterns observed in excitable and oscillatory media such as cardiac tissue and the Belousov-Zhabotinsky reaction, and are modeled by reaction-diffusion systems $u_t=D\Delta u+f(u)$ with $u\in\mathbb{R}^m$ and $D$ a positive diagonal diffusion matrix. A planar spiral wave is a rigidly rotating solution that, far from its core, converges to a periodic traveling wave (a 'wave train', a one-dimensional periodic profile translating at constant speed) with a selected wavenumber. Viewed in a frame corotating with the spiral's angular velocity, it becomes an equilibrium. The linearization about this equilibrium has a spectrum that splits into point spectrum near the core and far-field spectra inherited from the asymptotic wave train: the essential spectrum, generated by the wave train's dispersion relation, and the absolute spectrum, the set where the spatial eigenvalues of the far-field radial dynamics change splitting and which controls what finite-domain truncations can see. Sandstede and Scheel's Memoir (Mem. Amer. Math. Soc. 285 (2023), no. 1413; arXiv:2002.10352) built this linear theory for general reaction-diffusion systems using radial spatial dynamics, i.e. treating the radius as an evolutionary variable and using exponential dichotomies (splittings of the radial ODE flow into exponentially decaying and growing parts, the standard tool for one-dimensional fronts and pulses). They proved robustness of spirals under parameter changes and under truncation to large disks $|x|<R$, described eigenfunction shapes, point spectra, and accumulation of truncated spectra, and showed how instabilities of wave trains (convective or absolute, including far-field breakup) determine spiral spectra.

The nonlinear consequences of this linear theory, however, remain bounded by a structural weakness the authors state explicitly. On any large finite disk, spectral stability yields a spectral gap, so standard semigroup theory (decay estimates for the linear evolution operators generated by the linearization) gives nonlinear asymptotic stability of the rotating wave; but the resolvent (the inverse $(\lambda-L)^{-1}$ of the shifted linearized operator, from which those decay estimates are built) has norm growing exponentially in $R$, so the proven basin of attraction shrinks exponentially with the disk radius, and the authors show this is optimal for purely bounded-domain information when far-field breakup occurs. What is missing is a full-plane argument incorporating the essential spectrum: tracking how perturbations are transported away from the spiral core and decay diffusively in the far field. In one spatial dimension the analogous nonlinear stability of sources (defects that emit wave trains) is known - Beck, Nguyen, Sandstede, and Zumbrun proved it for source defects in the complex Ginzburg-Landau equation (a universal amplitude equation for oscillatory media), and Roussier proved nonlinear stability of multidimensional radially symmetric traveling waves in the bistable scalar case - but the planar spiral geometry introduces azimuthal stretching that prevents the same diffusive decay. Beyond stability, the phenomenology of spiral dynamics also lacks nonlinear reduction theorems: meandering (flower-shaped tip trajectories arising from a Hopf bifurcation, in which a complex-conjugate pair of eigenvalues of the linearization crosses the imaginary axis and creates time-periodic dynamics) and drifting spirals are predicted by Barkley's skew-product equations, which exploit the Euclidean symmetry of the plane (translations and rotations acting on the core position and phase), and drift bifurcations of relative equilibria (solutions that are equilibria only modulo that symmetry, such as rigidly rotating or drifting spirals) were classified formally by Ashwin, Melbourne, and Nicol; but no theorem on the unbounded plane derives these reduced dynamics from the reaction-diffusion PDE, and one-dimensional defect interaction is understood only in the classification of Sandstede and Scheel for oscillatory media. The Memoir's closing section, 'Open problems', formulates precisely this nonlinear program; the problem consists of its first three themes - nonlinear stability, boundaries and multi-spiral interaction, and nonlinear bifurcations.

Problem Statement

For a reaction-diffusion system $u_t=D\Delta u+f(u)$ on the plane with positive diagonal diffusion $D$, let $u_*$ be a rigidly rotating spiral wave that is spectrally stable in the sense of Sandstede and Scheel's Memoir (no extended point spectrum, i.e. no isolated eigenvalues emanating from the far-field spatial dynamics rather than the core, to the right of a stability margin; stable absolute spectrum; resonances in the absolute spectrum excluded). Establish the nonlinear theory of $u_*$ on the unbounded plane, going beyond finite-disk truncation, in its three linked components as posed in the Memoir's open-problems section:

1. Nonlinear stability: prove nonlinear asymptotic stability of $u_*$ under sufficiently localized perturbations on the full plane, with decay estimates and basins of attraction that incorporate the outward transport and diffusive decay encoded in the essential and absolute spectra - not basins that shrink exponentially with the radius of an artificial truncation disk; or prove that spectral stability genuinely fails to imply such nonlinear stability.

2. Reduced dynamics: construct and rigorously justify finite-dimensional slow manifolds for (i) a spiral in a large bounded domain, capturing the drift of core position and phase induced by the distant boundary, exponentially slow in the core-boundary distance, and (ii) configurations of multiple well-separated spiral cores, capturing their position and phase interaction dynamics, exponentially slow in the core separation, with wave-train phase matching across the intervening sinks (stationary interfaces where the wave trains emitted by neighboring cores meet and annihilate).

3. Bifurcations: prove existence of meandering spiral waves past a supercritical Hopf bifurcation on the unbounded plane; in particular, prove or refute the Memoir's conjecture that the Taylor jets (truncated Taylor expansions) of the center-manifold vector fields - the reduction of the dynamics onto the invariant manifold tangent to the neutral eigendirections given by translations, rotation, and the critical Hopf pair - on disks of radius $R$ converge exponentially as $R\to\infty$ to Barkley's skew-product equations for tip dynamics, and extend the center-manifold normal-form reduction to the resonant drift transitions that bounded-domain truncation cannot capture because it destroys the Euclidean symmetry.

The three components jointly constitute the nonlinear counterpart of the Memoir's linear theory and share one objective: removing the exponentially small-in-$R$ basins that bounded-domain semigroup and center-manifold arguments force. The three components are linked, and each stated prove-or-refute alternative is itself a resolution of its component: in component 1, proving that spectral stability does not imply nonlinear stability resolves the stability component, and in component 3, refuting the Taylor-jet/skew-product convergence conjecture resolves that conjectural claim. Component 3 moreover requires both proving existence of meandering spirals past a supercritical Hopf bifurcation and extending the center-manifold normal-form reduction to the resonant drift transitions. The same open-problems section lists further directions - transverse wave-train instabilities and semigroup growth versus spectral mapping, the retracting-wave limit of vanishing rotation frequency, vanishing diffusion coefficients, resonances in the absolute spectrum, more ambitious reductions coupling the localized core dynamics to far-field modulation equations such as the viscous eikonal or Kuramoto-Sivashinsky equations, target and multi-armed spirals, and three-dimensional scroll-wave filament dynamics - which are out of scope for this problem: they are separate adjacent problems and are not part of its completion boundary.

References

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