Pattern dynamics of the nonreciprocal Swift-Hohenberg model

We investigate the pattern dynamics of the one-dimensional nonreciprocal Swift-Hohenberg model. Characteristic spatiotemporal patterns such as disordered, aligned, swap, chiral-swap, and chiral phases emerge depending on the parameters. We classify the characteristic spatiotemporal patterns obtained in numerical simulation by focusing on the spatiotemporal Fourier spectrum of the order parameters. We derive a reduced dynamical system by using the spatial Fourier series expansion. We analyze the bifurcation structure around the fixed points corresponding to the aligned and chiral phases, and explain the transitions between them. The disordered phase is destabilized either to the aligned phase by the Turing bifurcation or to the chiral phase by the wave bifurcation, while the aligned phase and the chiral phase are connected by the pitchfork bifurcation.

💡 Research Summary

This paper investigates the pattern‑forming dynamics of a one‑dimensional nonreciprocal Swift‑Hohenberg (NRSH) model. The model consists of two real, non‑conserved fields ϕ(x,t) and ψ(x,t) that obey coupled partial differential equations containing a linear growth term ε, a fourth‑order spatial operator (the hallmark of the Swift‑Hohenberg equation), cubic saturation, and both reciprocal (χ) and nonreciprocal (α) linear couplings. The nonreciprocal term α cannot be derived from a free‑energy functional and therefore represents genuine nonequilibrium driving. By exploiting the symmetries of the equations, the authors restrict the analysis to χ≥0 and α≥0 without loss of generality.

Through extensive numerical simulations with periodic boundary conditions (system size L = 2π) the authors identify five distinct spatiotemporal regimes, which they label as disordered (D), aligned (A), swap (S), chiral‑swap (CS), and chiral (C). In the D phase the fields remain spatially homogeneous; in the A phase a stationary sinusoidal pattern with a fixed wave number k = 1 persists; the S phase exhibits a standing wave whose amplitude oscillates in time; the CS phase combines amplitude oscillations with unidirectional propagation; and the C phase is a traveling wave of constant amplitude.

To classify these regimes quantitatively, the authors compute the spatiotemporal Fourier spectrum of ϕ (and similarly ψ). Because the system size fixes the fundamental wave number k₀ = 1, the dominant modes are n = ±1. The spectrum is examined in the (k, ω) plane: a single peak at ω = 0 corresponds to the A phase, a symmetric pair of peaks at ω = ±ω₊ with equal amplitudes signals the S phase, an asymmetric pair (ω₊ ≠ |ω₋|) indicates the CS phase, and a single peak at ω > 0 denotes the C phase. By setting quantitative thresholds on peak positions and amplitudes (see Appendix B), the authors map the five regimes onto the ε–α parameter plane, producing a phase diagram that reveals how increasing ε or α drives transitions via Turing, wave, and pitchfork bifurcations.

The theoretical analysis begins with a linear stability calculation. Expanding the fields in Fourier series and retaining only the n = ±1 modes yields eigenvalues λₙ^{±}=ε−(1−n²)² ± √(χ²−α²). When α ≤ χ the eigenvalues are purely real; instability occurs when √(χ²−α²)=−ε, which is a classic Turing bifurcation (red line in the diagram). When α > χ the eigenvalues acquire an imaginary part ±i√(α²−χ²); the real part reduces to ε, so the homogeneous state loses stability at ε = 0, producing a wave (Hopf‑type) bifurcation (green line).

To capture the nonlinear saturation and the emergence of the S, CS, and C phases, the authors perform a mode truncation, keeping only the dominant Fourier amplitudes ϕ₁(t) and ψ₁(t). The resulting complex amplitude equations (11)–(12) contain cubic self‑interaction terms (−3|ϕ₁|²ϕ₁, −3|ψ₁|²ψ₁) and the linear cross‑couplings (χ ± α). By writing ϕ₁=ρ₁e^{iθ₁} and ψ₁=ρ₂e^{iθ₂} and defining the phase difference δ=θ₂−θ₁, the system reduces to three real ODEs for ρ₁, ρ₂, δ (Eqs. 13–15). The equation for δ shows that either sin δ=0 (δ = 0 or π) or a balance condition (χ−α)ρ₁/ρ₂ + (χ+α)ρ₂/ρ₁ = 0 must hold for a fixed point.

The sin δ=0 branch corresponds to the aligned (A) and swap‑type states. Substituting ρ₁, ρ₂ into the steady‑state conditions yields a quartic equation for an auxiliary variable ζ (Eq. 19). The discriminant Δ (Eq. 20) determines the number of real solutions; Δ = 0 defines a saddle‑node curve (black line) that separates regions with one or three fixed points. Within this branch, a Hopf bifurcation (blue line) appears when a pair of complex conjugate eigenvalues of the Jacobian cross the imaginary axis, giving rise to the S and CS oscillatory states.

The alternative branch, (χ−α)ρ₁/ρ₂ + (χ+α)ρ₂/ρ₁ = 0, yields explicit expressions for ρ₁, ρ₂ (Eqs. 21–22) and for cos δ (Eq. 23). These solutions exist only when ε > 0, α > χ, and the auxiliary quantity P = α⁴ − α²χ² − χ²ε² is non‑negative. The condition P = 0 defines a pitchfork bifurcation curve (orange line) that separates the aligned (A) and chiral (C) phases. Crossing this line the system undergoes a continuous symmetry‑breaking transition from a standing wave (δ = 0 or π) to a traveling wave (δ ≠ 0, π).

Numerical integration of the reduced three‑dimensional system confirms the analytical predictions: the locations of the Turing, wave, saddle‑node, pitchfork, and Hopf bifurcations match those observed in the full PDE simulations. Moreover, the reduced model reproduces the sequence of pattern transitions observed when ε or α are varied: D → A (Turing), D → C (wave), A → S (Hopf), S → CS (asymmetry in ω), and A ↔ C (pitchfork).

The authors conclude that the essential bifurcation structure of the NRSH model can be captured by a low‑dimensional amplitude equation involving only the dominant Fourier mode. This insight parallels recent work on nonreciprocal Allen‑Cahn and Cahn‑Hilliard models, as well as on complex Swift‑Hohenberg equations, suggesting a universal framework for describing pattern formation in nonreciprocal, nonequilibrium systems. Potential extensions include higher‑dimensional geometries, stochastic forcing, and experimental realizations in active colloidal or chemical systems where effective nonreciprocal interactions arise.