A Nonlinear Reaction-Diffusion Fold
Every chapter so far in this corpus has been linear: the heat equation's weak form, its discretization, its conditioning are all governed by a single tridiagonal (or Sturm–Liouville) linear operator, and the helix toy model's transverse dynamics are a linear relaxation modulated by a scalar factor. Linear systems can decay, oscillate, or diverge, but they cannot fold: a fold requires two equilibria to collide and annihilate, which needs at least a cubic nonlinearity. This chapter is the first place in Book 6 a genuine nonlinear competing reaction term appears, and the first place a real fold threshold — not an analogy to one — can be computed.
Setup: a tilted bistable reaction-diffusion equation
Take the classical Chafee–Infante reaction-diffusion equation, ut = uxx + λu − u³ on (0,1) with homogeneous Dirichlet boundary conditions, and add a constant-in-time forcing term aligned with the first Dirichlet eigenmode — the “competing reaction” ingredient that breaks the equation's odd symmetry (u → −u) and is exactly what turns a symmetric pitchfork into a genuine, asymmetric fold:
The first Dirichlet eigenvalue of −∂xx on (0,1) is λ₁ = π². Write r := λ − λ₁ for the distance from the onset of instability of the trivial state.
Single-mode reduction
Near onset (r small, forcing h small), the unstable mode is sin(πx) and all other modes are strongly damped; the standard center-manifold heuristic is that the higher modes slave to the unstable one on a fast timescale, leaving a single amplitude a(t) governing u(x,t) ≈ a(t) sin(πx). Substituting this ansatz, multiplying by sin(πx), and integrating over (0,1) — using the standard identities ∫sin²(πx)dx = ½ and ∫sin⁴(πx)dx = ⅜ — gives the reduced amplitude equation:
At h = 0 this is the textbook pitchfork: a single trivial equilibrium for r < 0, splitting into three (a = 0, ±√(4r/3)) once r crosses zero. Turning on h ≠ 0 is what unfolds the pitchfork into a genuine cusp catastrophe — and it is on that unfolded surface that an honest fold (a saddle-node, not a symmetry-breaking bifurcation) exists.
The fold threshold
Equilibria of the reduced equation solve the cubic ¾a³ − ra − h = 0. Writing this as a depressed cubic and setting its discriminant to zero (the condition for a repeated root — two equilibria colliding) gives a closed form for the fold locus in the (r,h) plane:
For |h| < h*(r) the cubic has three real roots (bistability survives, tilted); for |h| > h*(r) two of them have collided and annihilated, leaving one — the fold itself.
Justifying the single-mode reduction
The reduction above kept only u ≈ a(t) sin(πx). Making that rigorous means showing the higher Dirichlet modes are genuinely slaved to a — not merely assumed to vanish. They are, and the order at which they enter is what protects the fold.
Expand u(x,t) = ∑k≥1 ak(t) sin(kπx). The linear growth rate of mode k is μk = λ − k²π² = r − (k²−1)π². Near onset the first mode is critical (μ1 = r ≈ 0) while every higher mode obeys μk ≤ r − 3π² — a spectral gap of at least 3π² below zero. The center-manifold theorem then supplies a one-dimensional invariant manifold ak = hk(a1), k ≥ 2, tangent to the critical mode: the heuristic of the previous section becomes a theorem.
The cubic and the tilt set the slaving order. From sin³(πx) = ¾sin(πx) − ¼sin(3πx), the mode-1 self-interaction drives mode 3 at O(a1³), and the constant tilt drives the odd modes at O(h) (even modes stay zero, since ∫01sin(2πx)dx = 0). So a3 = O(a1³) + O(h), and its feedback onto the mode-1 equation enters only through cross terms ∼ a1²a3 — that is, O(a1⁵).
Near the fold, where h ∼ r3/2 and a1 ∼ r1/2, the leading balance {r a1, a1³, h} is O(r3/2), while the slaved-mode correction is O(r5/2) — one full order smaller. It does not touch the fold locus h* = ±(4/9)r3/2 at leading order. The single-mode amplitude equation is therefore not an ad hoc truncation but the leading-order reduction on the center manifold, with the neglected modes controlled and their correction quantified. (Mode couplings, growth rates, and orders verified symbolically.)
| Statement | Status | Notes |
|---|---|---|
| Reduced amplitude equation | derived | Standard Galerkin projection, mechanical once stated |
| Fold locus h*(r) = ±(4/9)r3/2 | derived + numerically checked | Discriminant of the equilibrium cubic; confirmed against exact roots, not just algebra |
| Rigorous center-manifold reduction | derived | Spectral gap ≥ 3π² (mode 1 critical, modes k≥2 strongly stable); center-manifold theorem gives ak=hk(a1). Slaved modes at O(a³)/O(h); feedback O(a⁵) ∼ r5/2, one order below the fold balance — fold locus unchanged. Verified symbolically |
| Lean / AXLE formalization | scaffold | Algebraic/spectral core in ReactionDiffusionFold.lean — sin³ coupling identity, spectral gap ≥ 3π², fold-locus discriminant. Written to be sound; not yet kernel-checked (compute pending). Center-/inertial-manifold existence invoked, not formalized |
What this does and does not show
The single-mode reduction is now justified to leading order: the section above establishes the spectral-gap estimate (a gap of at least 3π²) under which the higher modes slave to the first, and shows the projected nonlinearity is correct through cubic order — the slaved-mode feedback enters only at O(a⁵), one full order below the fold balance, so the fold locus is unchanged. What is still not done is the fully quantitative inertial-manifold argument with explicit constants and error bounds, and the machine-checked version (see the ledger). Separately, and just as important: the numeric fold threshold found above, h*(3) ≈ 2.309, is what this particular toy instance gives — it is not fitted or adjusted toward the series' standing constants (τ=2, ε₀=1/3, μmax=−2), and no claim is made that it equals or relates to any of them. This chapter does not yet claim the equation above is a dm³ system in the sense of Volume I, only that it is the first place in Book 6 where a genuine fold — the structure that language is about — can actually be computed rather than gestured at.