NLS on the Lumpy Torus

Nonlinear Schrödinger evolution on a surface of revolution with metric ds² = dx² + A(x)²dθ², A(x)=√((1+cos²x)/2). Mass-conserving Laplace–Beltrami assembly, Crank–Nicolson time-stepping, Picard nonlinear solve. A Python companion to the lumpy-torus study. All 3-D frames are rendered on the metric-faithful crescent immersion.

The question. This surface is engineered to ask one thing, posed in the source post: can a focusing nonlinearity, working with the curvature, concentrate a wave hard enough along a stable geodesic to blow up in finite time? It sits at the meeting of three threads — dispersive PDE, spectral geometry, and semiclassical analysis — and the geometry makes them visible: a belly where the equatorial geodesic is stable (elliptic) and necks where it is not (hyperbolic).

The arc. The linear geometry already traps waves — whispering-gallery quasimodes bound in the centrifugal well (§02). Turn on focusing and those become the nonlinear quasimodes that Albin–Christianson–Marzuola–Thomann proved stay localized near an elliptic geodesic (§08). Push the mass up and the trapped ring undergoes mass-critical collapse — the blow-up itself, which Godet made rigorous on a curve for rotationally-symmetric surfaces at the log-log rate. Then §10 turns the geometry into a control field and goes past the original question, into chaos, lattices, parametric amplification, an expanding-universe analogue, topological edge channels, and device design. Then §11 hands the loop itself to an agent — the solver as a verified tool — and reports what it found when it, not a human, chose the next experiment. §12 then breaks the surface-of-revolution mould entirely, carrying the physics onto a genus-2 mesh and an open, radiating horn.

01Geometry & curvature

One period x∈[−π/2, π/2] (ends identified) has exactly one elliptic geodesic (belly, K=+0.5) and one hyperbolic (neck, K=−1).

curvature view
Left: the isometric neck–belly–neck immersion. Middle: bent closed into the crescent torus used for every field render, colored by Gaussian curvature K=−A″/A. Right: the Laplace–Beltrami inputs A, 1/A² (peaks at the necks), K.

02Why it works — the centrifugal-barrier reduction

Separating u = e^{ikθ}φ(x) turns Laplace–Beltrami into a 1-D Schrödinger operator with effective potential V_k(x)=k²/A² — the same centrifugal barrier as the atomic radial equation. Both the quantum modes and the classical geodesics live on this well/barrier structure.

Why geodesics at all? Because they are the characteristics of the operator, not a metaphor. The linear part i∂_t u = −Δ_g u has principal symbol |ξ|²_g, whose Hamiltonian flow is exactly the geodesic flow. So in the high-frequency (semiclassical) limit a wave packet's centre rides a geodesic and its transverse width obeys the Jacobi equation J″ + K J = 0 along it — geometric optics on a manifold. Geodesic stability then fixes the packet's fate: bounded (oscillatory) Jacobi fields on a stable orbit keep it concentrated (§05, §07); exponentially growing ones on an unstable orbit spread it. The nonlinearity σ|u|²u is lower-order — it rides on top of this geodesic transport, so a stable geodesic is where a focusing beam stays put long enough to concentrate and blow up (§08). That is the whole reason to ask, first, which geodesic, and is it stable? — and why the classical phase portrait below and the quantum ladder are the same object seen two ways.

How the geometry enters the operator. The metric ds² = dx² + A(x)²dθ² has volume element √|g| = A and inverse metric g^{xx}=1, g^{θθ}=1/A², so the Laplace–Beltrami operator is the flat Laplacian dressed by the metric:

Δ_g u = (1/A) ∂_x(A ∂_x u) + (1/A²) ∂²_θ u  =  u_xx + (A'/A) u_x + (1/A²) u_θθ

Weak form (what the solver assembles) — test against v, integrate by parts on the closed surface (no boundary term):

stiffness:  ∫⟨∇u,∇v⟩_g dV = ∬ [ A · u_x v̄_x  +  (1/A) · u_θ v̄_θ ] dx dθ
mass:                 ∫ u v̄ dV = ∬  A · u v̄  dx dθ

So the geometry is just a set of weights: A weights the x-stiffness and the mass, 1/A the θ-stiffness. Discretely these become a Hermitian stiffness K and a positive lumped-diagonal mass M; because both are symmetric the scheme conserves ∫|u|² = UᴴMU to machine precision — the metric is baked into the conserved quantity.

The local geometry, in two Taylor expansions. Everything follows from how the generating radius A behaves at its critical points — near the belly (x=0) and a neck (x = π/2 − ξ):

belly:  A(x) ≈ 1 − x²/4        (concave — a maximum),   1/A² ≈ 1 + x²/2
neck:   A(ξ) ≈ (1/√2)(1 + ξ²/2) (convex  — a minimum),   1/A² ≈ 2(1 − ξ²)

Fed into the angular-momentum potential V_k = k²/A²:

belly:  V_k ≈ k² + ½ k² x²   →  a harmonic WELL, spring constant k²  (frequency ω = k)
neck:   V_k ≈ 2k²(1 − ξ²)     →  an inverted parabola: a BARRIER

That is the whole story in two lines: the belly is a harmonic trap of frequency k, so its bound quasimodes have width ~k^{-1/2} and breathe at frequency k (the semiclassical beam), and its depth ~k² sets how many bind; the neck is a barrier that expels.

Why the NLS cares. The linear operator sets the stage — well at the belly, barrier at the neck — and σ|u|²u adds self-attraction. Dynamics is a contest between dispersion (spreading, governed by Δ_g) and self-focusing. Where the geometry both traps and has positive curvature (the belly), the two conspire: the wave is held in place and pulls itself together — concentration, and above a threshold, blow-up. Where it expels and the curvature is negative (the neck), it spreads. The surface doesn't merely bend the space the wave lives in; it spatially biases the dispersion-vs-focusing balance — which is precisely why a stable geodesic is the place to look for collapse.

A Fourier-analytic lens. Almost every panel below is clarified by looking in Fourier space — the dual of the real-space well/barrier picture.

• θ-Fourier diagonalizes the linear flow. With u = Σ_k û_k(x)e^{ikθ} the axisymmetric operator is block-diagonal — each angular momentum evolves alone, and k²/A² is exactly the θ-Fourier symbol of the angular Laplacian, weighted by the metric. The whispering-gallery ladder is "one Fourier mode at a time."

• The nonlinearity is convolution — it couples the modes. |u|²u = û ∗ û̄ ∗ û mixes angular momenta, and that coupling is the engine of every instability here: it feeds a pump mode into sidebands. The necklace collapse, Benjamin–Feir, and the Faraday waves are the same phenomenon in Fourier space — exponential growth of sideband modes — differing only in their gain spectrum γ(m). The ring "fragmenting into hot spots" is an azimuthal Fourier cascade.

• Revivals are Fourier rephasing. The Talbot carpet is the phases e^{-in²t} realigning: at t=2π every n²t is a multiple of 2π (full revival); rational fractions give Gauss-sum partial revivals. The recurrence is the Fourier phases coming back.

• Lattices are Bloch/Floquet. A periodic lump makes the x-Fourier (Bloch) label q a good quantum number → the whispering-gallery bands ω(q); disorder destroys it, and Anderson localization is the collapse of that Fourier description.

• Beams are coherent states, and dispersion & horizons are about the symbol ω_k. The semiclassical beam is a minimal-uncertainty wavepacket whose breathing is its phase-space cell rotating in the transverse well. The comb-flattening prototype literally shapes the dispersion relation ω_k=E_0(k) (its curvature d²ω/dk² is the engineered dispersion), and the sonic horizon is where the flow Doppler-shifts ω_k until an upstream mode's group velocity vanishes — a degeneracy read straight off the Fourier symbol.

In one sentence: the geometry writes the dispersion relation ω_k(x), the nonlinearity convolves the modes, and every instability, revival, band, and horizon is a statement about how those Fourier modes phase, couple, or degenerate.

A pseudodifferential lens — the master lens the others fall out of. Everything above is one statement: the operators here, −Δ_g and its angular slices −∂_x² + V_k, are elliptic, self-adjoint pseudodifferential operators (ΨDOs) of order 2 on the closed surface, and their principal symbol σ(−Δ_g)=|ξ|²_g=ξ_x²+k²/A(x)² carries all the geometry. On the torus there are no charts to patch: the ΨDO is a Fourier multiplier dressed by A(x) — the toroidal quantization of Ruzhansky–Turunen — which is exactly why the Fourier lens above is global and rigorous, not a heuristic.

• Geodesics are the bicharacteristics. The Hamiltonian flow of the principal symbol on T*M is the geodesic flow (Clairaut) — the classical portrait of §02 is the symbol's flow — and Egorov's theorem / propagation of singularities promote it to a theorem: conjugating an observable by the propagator transports its symbol along that flow, so the wavefront set of a solution rides those trajectories and a packet microlocalised over the elliptic belly orbit stays trapped (§05, §08).

• The propagator is a Fourier integral operator. e^{itΔ_g} is an FIO whose canonical relation is that flow; the Gaussian-beam / WKB construction (§07) is its parametrix — phase solving the eikonal (Hamilton–Jacobi) equation, amplitude the transport equation, whose leading term is precisely the Jacobi law J″+KJ=0. The whispering-gallery quasimodes are the quantization of the elliptic orbit (a harmonic well of frequency k); the nonlinearity σ|u|²u is a lower-order rider on this ΨDO transport — not a ΨDO itself, but folded in microlocally by paradifferential calculus (Bony), tamed by the same symbol machinery.

• Functional calculus ties off the spectral panels. f(−Δ_g) is again a ΨDO — which is what makes the heat trace, the Weyl count ("hear the shape"), and the Schrödinger propagator legitimate objects; the quasimode ladder and the revival (Talbot) carpet are two calls to the same calculus. And when the geometry moves (next lens) the FIO's canonical relation moves with it: pair production is the propagator failing to preserve the vacuum's positive-frequency splitting as the symbol's characteristic set sweeps. Framework: M. Taylor, Lectures on pseudodifferential operators; Ruzhansky–Turunen, Quantization of ΨDOs on the torus.

A dynamical lens — static geometry → time-varying geometry. Every lens so far fixes A(x). Let it move, A=A(x,t) (equivalently ramp the ring radius / scale factor a(t)), and the ΨDO becomes a time-dependent family: the dispersion relation ω_k(t) it writes now sweeps, and a mode that was an eigenstate is pumped into its partners. One new control axis with three faces — all runnable from the same solver, all in §10:

• Periodic drive → parametric Faraday resonance: a belly breathing at Ω=2ω_n amplifies a Bogoliubov mode out of noise.
• Monotonic ramp → cosmological particle production: an expanding ring redshifts its modes, ω_k=ck/a(t), and the adiabatic vacuum can't follow — phonon pairs appear with a scale-invariant plateau and Sakharov oscillations, the analogue platform of Viermann et al. (Nature, 2022).
• Stationary flow through the fixed throat → an analog horizon: a transonic superfluid traps sound at the neck, a geometry-made black hole.

Static geometry bends where modes go; dynamic geometry decides which modes exist — the metric becomes a source that mixes the vacuum, and it can act back on the nonlinearity itself: breathing the belly resonantly delays or hastens the fast necklace collapse and fully arrests the slow single-hump one into a stable breather (§10), the control-knob face of this axis. And a spatial twist is not even needed for topology — a two-step time-drive alone makes Floquet edge modes at quasienergy 0 and π (§10).

quasimode ladder
Quantum. V_k is a well at the belly (A=1) with barriers at the necks (A=0.707). Its bound states are the whispering-gallery quasimodes; the ground state sharpens ~k^{-1/2} — exactly the semiclassical beam width.
geodesic phase space
Classical. The integrable geodesic flow (Clairaut) has the identical L²/A² potential: an elliptic island of librating orbits around the belly, a separatrix at the neck, circulating orbits beyond — the skeleton the quasimodes sit on.

03Solver validation

A focusing equation near blow-up is unforgiving: a scheme that leaks or injects energy will fake a collapse that isn't there, or smother one that is. So the licence for every later claim sits here — the structure-preserving Laplace–Beltrami assembly conserves ∫A|u|² to machine precision and energy to ~10⁻⁵ over a thousand steps, so when the ring does collapse (§08) we know it is the physics, not the discretisation.

conservation
Mass conserved to ~10⁻¹³ and energy to ~10⁻⁵ over 1000 Crank–Nicolson steps — the diagnostics the reference FreeFEM script omitted.

04The field on the lumpy torus

Two views of the same field, and it pays to read both. The crescent immersion is what the surface looks like; the (x,θ) chart is where the operator lives — where θ-Fourier diagonalises the linear flow and the centrifugal potential k²/A² is plain to see. Every animation below is one view or the other.

wavepacket on torus
3-D|u|² on the crescent immersion.
wavepacket on chart
chartThe same field on the (x,θ) chart.

05Geodesic stability (linear)

Before any nonlinearity, put the geometry's own question (§02) to the test: launch a beam along a parallel geodesic and watch its transverse width. On the elliptic belly the Jacobi field stays bounded and the beam holds together; on the hyperbolic neck it grows and the beam sheds off the pinch. Stable vs unstable orbit, side by side — the linear skeleton every later panel hangs on.

elliptic beam
Elliptic (belly). A confined bright band — the stable geodesic.
hyperbolic beam
Hyperbolic (neck). Disperses off the pinch — the unstable geodesic.
geodesic comparison
Transverse width stays bounded on the elliptic equator; swings ~2× larger on the neck.

06Running over the bumps: the meridian beam

The meridians are the other geodesic family — and unlike the equator they are not stable orbits: each one crosses the K>0 belly and the K<0 necks, so its transverse width solves a Hill/Floquet equation, not a harmonic one. The prediction: focus at the necks, defocus over the belly — exactly as meridians converge at a sphere's poles.

meridian torus
3-DA beam concentrated along a meridian, crossing belly and necks.
meridian diagnostic
Physical width tracks the Jacobi field J(x)=A(x): the necks focus, the belly defocuses — as on a sphere, where meridians converge at the poles.

07Semiclassical beam

Push the angular momentum high — the semiclassical limit — and the ray/wave correspondence sharpens into view: the packet becomes a true Gaussian beam (the FIO parametrix of §02) riding the geodesic, its width breathing as A around the loop. Geometric optics on the manifold, made literal.

semiclassical surf
q=20The beam surfs the crescent loop, through the neck and over the belly.
breathing
Transverse width tracks A(beam) — compress at necks, expand over belly.

08Self-trapping via focusing nonlinearity

On the stable equator a θ-symmetric ring reduces to 1-D — subcritical, so focusing can self-trap it into a soliton (the meridian, lacking a transverse trap, only disperses or collapses).

focusing ring
Focusing self-traps it tighter and brighter.
selftrap comparison
Focusing holds the ring at smaller width and higher peak than the linear mode.
collapse
amp = 5The width locks into a soliton, then fragments into hot spots.
collapse diagnostic
Azimuthal symmetry-breaking grows exponentially from machine roundoff (10⁻¹⁵→10⁻¹) → mass-critical collapse at t≈1.8 (a "necklace" instability).

09Regimes, instabilities & bifurcation

What the geometry and the nonlinearity open up beyond the main experiments.

bifurcation
Elliptic↔hyperbolic bifurcation — the geometry knob. With the tunable family A(x;ε)=√((1+ε cos²x)/(1+ε)) the curvatures are exactly K_belly=ε/(1+ε), K_neck=−ε: belly and neck exchange stability at ε=0. The centrifugal well inverts, and the whispering-gallery bound states switch on as the belly turns elliptic. Our torus is ε=1.
defocusing
defocusingRepulsive σ=+1: a hump expands and depletes its core — dispersive-shock / rarefaction dynamics, the opposite of self-trapping.
modulational instability and dark solitons
Modulational instability & dark solitons on the belly ring. Left: a focusing uniform state is Benjamin–Feir unstable — it breaks into a soliton and recurs (Fermi–Pasta–Ulam). Right: in the defocusing regime, two black solitons (stationary density notches with π phase jumps) are stable.
collapse threshold
Mass-critical collapse threshold. Below M_c≈43 the focusing ring never collapses; above it the necklace collapse sets in, its onset time falling off with mass.
Talbot carpet
Quantum revivals (Talbot carpet). A packet released on the belly ring reforms at t=2π (full revival) and rational fractions — the mechanism behind the dispersive recurrence seen in the full runs.

10Research directions — geometry as a control field

The same tunable metric is a laboratory. Each panel is a proof-of-concept for a direction where surface geometry steers nonlinear wave dynamics — some clean results, some sketches of open questions, all runnable from the same solver.

collapse threshold vs geometry
Geometry tunes the blow-up. Collapse-onset time vs ring mass for four lump depths — the curves separate: deeper necks (larger ε) collapse sooner and at lower mass. So the mass-critical (necklace) collapse — the blow-up the post set out to induce — is a knob the geometry turns, not a fixed number.
geodesic chaos
Break the axisymmetry, break integrability. Adding a θ-lump shatters the nested KAM curves into a chaotic sea with a surviving belly island — the classical stage for quantum scarring of the eigenfunctions, and for nonlinear transport in a mixed phase space.
quantum chaos: level statistics and eigenstates
The quantum face of that chaos. Diagonalising −Δ_g + λV(x,θ) with a generic θ-lump: as the lump strengthens, the nearest-neighbour level-spacing statistics cross from clustering (the symmetric surface's ±k degeneracy, left) to the Wigner–GOE distribution with hard level repulsion (middle) — the universal fingerprint of quantum chaos — and the eigenstates turn irregular (right). The same story as the classical Poincaré section above, read straight off the spectrum.
geometric lattice and Anderson
Corrugation is an intrinsic lattice. A periodic lump chain makes V_k=k²/A² a periodic potential → whispering-gallery Bloch bands with gaps; random lumps → Anderson-localized modes. No external potential — the geometry is the lattice (and the disorder).
topological whispering-gallery edge modes
A chiral lump twist makes topological edge channels. Twist the lumps helically, A(x−cθ): the WG on-site energy k²/A²(x−cθ) becomes the Harper term 2λcos(2παn+k_θ), so the twisted-lump strip is the Hofstadter model. Chiral edge modes traverse every gap (left, colored by which edge they hug), carried by gap states pinned to opposite edges (right) — a purely geometric route to topological photonics, whose time analogue is the Floquet drive in the next panel.
Floquet-topological edge modes from periodic driving
Floquet engineering — a time route to the edge modes. Breathe the lump chain periodically instead of twisting it in space: a two-step hopping cycle (intra-lump bonds on, then inter-lump) makes the one-period evolution operator carry quasienergy bands whose gaps host chiral edge modes — at quasienergy ε=0 (a static-like mode) and at ε=π, an anomalous mode with no static analogue (left), the two living on opposite edges (right). The time-domain sibling of the chiral-twist bands above: drive, don't twist. Real-world: Floquet photonics and periodically-driven cold atoms.
Thouless adiabatic pump from a sliding lump lattice
Thouless pump — quantized transport from a sliding lattice. Slide and breathe the lump chain around a cycle and the filled band's Wannier centre winds by an integer: exactly one unit cell pumped across the strip per period (left; a trivial loop pumps zero), independent of the drive's details. On an open strip the same topology drags a single edge state across the gap from one edge to the other (right). Robust, geometry-driven transport — the adiabatic cousin of the Floquet drive above. Real-world: topological pumps in photonic waveguide arrays and cold atoms.
soliton trapping vs escape
A soliton in the geometric well. A small kick leaves a bright soliton librating in the belly; a larger one sends it circulating over the necks — the nonlinear echo of the geodesic island/separatrix (§02).
Faraday parametric amplification
A breathing torus amplifies (periodic drive). Periodically modulating the geometry parametrically pumps a Bogoliubov mode out of noise (Faraday waves, driven at Ω=2ω_n). Run as a monotonic expansion instead → the next panel.
expanding torus cosmological particle production
Prototype — the expanding torus makes particles from the vacuum (monotonic ramp). Ramp the scale factor a(t) and every belly phonon redshifts ω_k=ck/a(t) (left) until the adiabatic vacuum can no longer follow, so the quench pumps phonon pairs out of nothing: a scale-invariant plateau |β_k|²→(a_f−a_i)²/4a_ia_f (the inflationary analogue) whose density structure factor rings with Sakharov oscillations (right) — the acoustic analogue of the CMB peaks. Exact linear Bogoliubov theory, two-mode-squeeze Wronskian held to 10⁻¹³; the NLS realisation of Viermann et al.'s expanding-universe BEC (Nature, 2022).
dynamical control of the focusing collapse
Prototype — dynamical control of the collapse (geometry vs the nonlinearity). Breathing the belly makes the effective coupling g_eff∝1/A²(t) oscillate — "nonlinearity management." It is a resonant knob on the necklace collapse: in a frequency window (Ω≈40) deep breathing delays the mass-critical blow-up ~5× (left; the delay grows with depth δ, middle); too slow (Ω=16) and the drive parametrically feeds the very sidebands that fragment the ring, so it gives no help or even hastens collapse (right). A geometric Kapitza / Feshbach-management effect (Saito–Ueda) — the same idea, dynamic stabilization of a wave instability, that plasma actuators use, and reduced-model kin to Langmuir-wave collapse. It only delays this fast azimuthal collapse; the next panel shows the collapse it fully arrests.
arresting the Townes collapse by fast breathing
Prototype — arresting the collapse (the management the necklace resisted). Where the blow-up is a single localized hump just above the mass-critical (Townes) threshold, the same knob works cleanly. Bare, it self-focuses and blows up (t_c≈0.11, left, pink); breathing the belly fast and hard (δ=0.8, Ω=100) turns it into a stable breather whose peak settles to a bounded level (blue) and survives indefinitely — a genuine arrest, above a depth threshold (middle) and a frequency threshold (right; too slow a drive instead hastens it). Nonlinearity management (Saito–Ueda); here the belly's curvature and trap supply what pure 2-D cubic management lacks. The contrast with the necklace is the physics: a drive averages away a slow radial collapse but only delays a fast azimuthal one.
animation: collapse vs arrested breather
animationLeft: the bare hump self-focuses and collapses (frozen once it blows up). Right: the same hump under fast belly-breathing survives as a pulsing soliton — the blow-up dynamically arrested, side by side.
Prior art & where this sits. Arresting NLS collapse by modulating the nonlinearity is Feshbach-resonance / nonlinearity management, established since 2003 (Saito–Ueda; Abdullaev et al.; Kevrekidis et al.). What's ours is the framing: the modulation is a geometric one — breathing the belly, g_eff∝1/A²(t) — and the necklace-vs-hump contrast reproduces a subtle known result: nonlinearity management delays a supercritical collapse but need not prevent it, yet fully arrests the near-critical hump. A 2025 analysis (Li–Ning–Zhao) makes the distinction rigorous — dispersion management can prevent blow-up, nonlinearity management only postpones it. The same physics is a live control knob in attractive BECs (the "Bosenova"), in self-focusing / filamentation of intense laser beams, and in Langmuir-wave collapse in plasmas — the reduced-model bridge to the fusion-adjacent question (§10, dynamical lens). Full cites in §11.
cubic-quintic prevents collapse outright
Prevention, not just delay — the cubic-quintic resolution. The management panels only delayed the collapse; adding a defocusing quintic term (σ₅|u|⁴u, repulsive only at high density) removes the collapse threshold outright. The same hump that Townes-collapses under pure cubic settles into a stable soliton with no drive at all (left), and cubic+quintic stays stable at every mass while pure cubic collapses above M_c (right). The clean way to genuinely prevent 2-D collapse — matching the rigorous delay-vs-prevent result above; connects to condensates with three-body repulsion.
rogue waves and Peregrine breather on the belly ring
Rogue waves on the belly ring. Seed the focusing ring's longest-wavelength unstable mode and modulational instability grows it into an Akhmediev breather — a wave that rises from nowhere, peaks at 3× the background (right; the Peregrine rogue-wave limit, reached to within 0.2%), and recurs (Fermi–Pasta–Ulam), the bright events walking across the ring (left). Geometry is the knob: the effective coupling and A(x) set the MI gain, hence how tall and how often the rogue events come. Real-world: optical and hydrodynamic rogue waves.
Kibble-Zurek scaling from a quench on the ring
Kibble–Zurek — defects from a finite-rate quench. Ramp the ring's focusing coupling up through the modulational-instability onset over a quench time τ_Q. The uniform state can't follow adiabatically: its response freezes near the transition and the instability then imprints a pattern whose wavenumber k* is set by when it froze — faster quench → finer pattern, more defects (left). Averaged over noise realizations, k* follows a clean power law k*∼τ_Q^{−0.23} across a decade (right) — the universal Kibble–Zurek scaling, with the geometry's effective coupling as the control parameter. Real-world: KZ defect-counting in cold atoms, ion chains, and superfluids.
geometric dispersion engineering
Prototype — geometric dispersion engineering. The WG spectrum ω_m = E_0(m) is the resonator's modal dispersion, so the lump profile is a design knob. Inverse-designing A(x) (a few harmonics) flattens the integrated dispersion 4.4× over a mode band — a broadband, near-equidistant comb grid engineered by geometry rather than waveguide cross-section.
neural operator surrogate on the manifold
Prototype — a neural operator closes the ML-for-PDE loop. A small MLP learns the forward map lump profile A(x) → modal dispersion D_int(m) from solver data (R²=0.998), then stands in for the eigensolver to inverse-design a flat-dispersion resonator ~2000× faster per evaluation — flattening the integrated dispersion from a wide parabola to near-zero. The learned surrogate for the exact geometry→spectrum map the post set out to motivate.
analog sonic horizon at the neck
Prototype — the neck as an analog black hole. A transonic superfluid speeds up through the small-A throat until the flow speed crosses the sound speed: a sonic horizon provided by the geometry alone (Hawking temperature T_H set by the throat's surface gravity). The upstream v−c sound rays diverge from the throat and can never cross it — nothing escapes the interior.
analog Hawking: exponential redshift and thermal spectrum
Prototype — the horizon radiates (analog Hawking). Take that horizon's radiation seriously. Near it the upstream characteristic obeys dx/dt=v−c≈κ(x−x_h), so a ray peels off exponentially (left, log axis; slope = the surface gravity κ≈0.7, measured dynamically and matching the static value) — the unbounded redshift that turns the vacuum into a thermal spectrum at T_H=κ/2π≈0.11 (right). Honest scope: this is the kinematic Hawking result (temperature + thermality mechanism); the spontaneous pair spectrum |β_ω|² itself needs the dispersive Bogoliubov–de Gennes scattering with its negative-norm partner mode — a noted next step.
Driven-dissipative (scaffolded, lle.py). A lump should pin a dissipative Kerr soliton — the soliton-microcomb regime, and a microtoroid resonator is a lumpy torus. Robust cavity-soliton nucleation needs a detuning-ramp protocol (a clean next step) rather than a seeded pulse, so this one is left as an honest scaffold, not a finished figure.
Still open (application-grounded). The frontier directions above are now prototyped — dispersion engineering, the analog horizon, the expanding-universe analogue, the chiral-twist edge modes, and the neural operator. Still genuinely open: vortices on a curved superfluid shell (NASA Cold-Atom-Lab bubble BECs) and their curvature pinning — a first pass (vortices.py) launches a vortex–antivortex pair on the belly, but on this small ring the pair orbits and annihilates faster than curvature can steer it, so the phase-singularity tracker is sound-limited; a clean demonstration needs a larger shell and a single pinned vortex, left as honest future work.

11What the agent-toolkit found

The panels above were designed by hand. These were not. The same solver, refactored into a modular library and a self-describing agent tool (nls_torus, an MCP server, and a generative sub-agent), was pointed at open questions and left to design → run → verify → retract on its own — with the trust flags computed by the harness, not asserted by the model. Three probes, three honest outcomes: a discovery, a busted myth, and a dead end.

Why a verification block matters. A plausible-looking number and a correct one are indistinguishable from the outside. So every run returns a verification block — mass drift against the conserved ∫A|u|², a grid check across resolutions, a Wronskian residual, an analytic-plateau match — and a result is trusted only if the harness's own flags pass. That discipline is what caught the confound in the first panel: the toolkit's first-pass answer was wrong, and the check is what said so — the agent could not self-certify past it.

collapse threshold is the Townes mass, geometry-blind
The collapse threshold is a local universal invariant — it ignores the geometry. Left: tuned to the same belly curvature K=0.5 but four different global structures — flat, one lobe, two lobes, a localized bump — the critical mass lands on the Townes value ‖Q‖²≈11.7 every time, to 0.4%. Neither curvature nor topology moves it: mass-critical blow-up is a self-similar concentration at a point, which sees only the local (≈flat) metric, and the Townes mass is a pure number. Right — the retraction: a coarse scan had shown M_c falling with curvature (orange), but that was the A-weighting confound — the mass ∫A|u|² read at a fixed amplitude drifts with geometry even though the true threshold (blue, amplitude bisected) is flat. The harness-owned grid + conservation checks forced the correction.
geometry-controlled tunneling doublet
Geometry as a tunable tunnelling qubit. A two-lobe surface makes the centrifugal reduction V_k=k²/A² a double well on the ring — one well per belly lobe, split by the neck barriers (left). Its two lowest states form a near-degenerate doublet, symmetric and antisymmetric across the lobes, whose splitting is the inter-lobe tunnelling rate. Deepen the necks and that rate collapses exponentially — 5×10⁻¹ → 1×10⁻⁴ across ε: 0.5→8 (right; the single-well torus has no such doublet). A purely geometric two-level system with an exponentially tunable coupling — the static, bound-state cousin of the Thouless pump and Floquet channels of §10.
And an honest negative. A third probe asked whether periodically breathing the belly ring builds a discrete time crystal. It does not: the geometric drive is a sharp parametric (Faraday) resonance — a narrow subharmonic band (≲4% in drive frequency), a response pulled ~10% off Ω/2 by the nonlinearity, and no prethermal plateau (the fluctuation energy keeps climbing). A clean mean-field ring has no disorder / many-body-localization mechanism to rigidify the subharmonic or arrest heating — so the honest answer is parametric amplifier, not time crystal, and the metrics simply declined to certify a rigidity that isn't there.
disorder-stabilized discrete time crystal
…and the negative, followed up: what the clean ring was missing was disorder. The H2 diagnosis was specific — a clean mean-field system has no localization to rigidify the subharmonic — so we supplied exactly that: geometric disorder (random on-site energies, the Anderson-localizing stand-in for many-body localization) on a driven nonlinear lattice, each period an imperfect π-swap that flips a density wave. Left: the demodulated period-2 response I_p(−1)^p — with disorder it holds a rigid, coherent value (a genuine time crystal); the clean lattice swings incoherently through zero (the H2 case, reproduced as the control). Right: the rigidity — the order parameter survives drive imperfection out to e≈0.15 with disorder (O≈0.66) but is dead flat without it. The response locks to exactly Ω/2 (the trust flag). The open-horn leakage is ~neutral (green ≈ blue): disorder, not openness, is what the negative result was pointing at — a negative turned into a positive by taking the diagnosis literally.

12Beyond surfaces of revolution — mesh & non-compact extensions

Everything above lives on the metric ds²=dx²+A(x)²dθ² — which can only express genus-1, rotationally-symmetric shapes. The toolkit's mesh module carries the same physics onto an arbitrary triangle mesh via the cotangent (finite-element) Laplace–Beltrami operator, and onto non-compact domains via a complex absorbing collar. That opens two doors the surface-of-revolution engine cannot: a genuine second handle, and an open, radiating geometry.

Validation before new physics. A discrete Laplacian on a hand-made mesh is exactly where silent errors hide, so the operator is checked against a case with a known answer first: on the round sphere it recovers the spectrum −Δ = l(l+1) = 0, 2, 6, 12 with the right multiplicities 1, 3, 5, 7 to sub-percent. Only then is it turned on new topology. For the open horn the trust signal is different but just as load-bearing: an absorbing boundary spawns a thicket of spurious PML/continuum eigenvalues, so a resonance is accepted only if it is invariant when the absorber strength changes — the physical mode stays put, the artifacts move.

genus-2 which-handle tunnelling doublet and neck sweep
H4 on a true genus-2 surface. Two tori fused into a connected genus-2 shape (verified χ=−2 at every step). Its first excited Laplace–Beltrami mode is the which-handle doublet — one sign on each handle, a node at the connecting neck (left) — so λ₁ is the inter-handle tunnelling splitting, the higher-genus echo of the two-lobe doublet in §11. Thin the neck (separate the tori) and the splitting falls monotonically, 0.30→0.18 (right), while the doublet grows more isolated (λ₂/λ₁: 3.0→5.5) as the handles decouple toward two independent zero modes. Tunnelling across a handle, set by geometry.
Townes collapse threshold blind to topology
H1 universality survives topology. The same genus-2 mesh lets us push §11's result to its sharpest test: is the mass-critical collapse threshold still the Townes mass ‖Q‖²≈11.7 when the topology itself changes? A fixed mesh grid-arrests true blow-up, so we read the focusing onset (a bump grows above M_c, disperses below). The threshold lands on 11.6–11.9 on a genus-0 sphere and on genus-2 at every neck width — a 1.5% spread, grid-converged, mass conserved to 10⁻¹⁴. Collapse is a local point-concentration; it sees neither the curvature nor the handles.
non-compact horn leaky resonator, geometry-set Q
A non-compact horn: geometry sets the Q of a leaky mode. The meridian runs to a semi-infinite horn; a throat (a dip in A, a barrier in V_k=k²/A²) traps a cavity mode that slowly radiates out the open end, absorbed by a PML collar. The operator is non-Hermitian, so the mode is a complex resonance E_r−iΓ/2 with a finite lifetime (left; |ψ|² cavity-bound, leaking past the throat). Its width is tunnelling-limited, so deepening the throat drives the quality factor Q=E_r/Γ up exponentially — a straight line on a log axis, Q: 284→85138 (right). Every point survived the absorber-independence check that killed dozens of numerical artifacts.
analog Hawking radiation on the open horn
The same open horn is also an analog black hole. Drive a superfluid through the throat and it becomes a de Laval nozzle: the flow accelerates from a subsonic reservoir, crosses the sound speed exactly at the throat — a sonic horizon — and radiates supersonically out the open end, where (unlike the closed torus neck of §10) the phonons genuinely escape into the absorbing collar. Left: flow speed v and sound speed c crossing at the horizon; the inset traces rays peeling off exponentially, |x−x_h|∼e^{κt}, at the surface gravity κ — the mechanism that renders the vacuum thermal at T_H=κ/2π. Right: a sharper throat makes a hotter horizon — geometry sets the Hawking temperature (T_H: 0.09→0.22). The trust flag is stringent: κ is measured two independent ways — a static fit of v−c and the dynamical ray-peeling rate — and they agree to <1%. (Kinematic scope; the full Bogoliubov pair spectrum is noted, not attempted.)
nonlinear Kerr on the horn leaky mode
The horn's leaky mode, made nonlinear — a bridge to the §13 Kerr comb. §12 gave the horn a leaky mode whose Q the throat geometry sets; §13 built a microcomb whose threshold rides on Q. Here a focusing Kerr nonlinearity acts on that same radiating mode. We excite it and ring it down; as the intracavity power decays, the instantaneous frequency and Q trace out their power dependence (one run sweeps all powers; the g=0 control is flat to 10⁻¹³ at the geometric Q=439). Left: the resonance redshifts linearly with power — Kerr self-phase modulation. Right, the surprise: the nonlinearity doesn't spoil the leaky mode, it self-traps it — pulling the frequency deeper below the throat barrier so it radiates less, lifting Q by +66% — until a critical intracavity power triggers a self-focusing instability that collapses it (the redshift saturates at the very same power, confirming a real mode transition, not a fit artifact). Geometry sets the linear Q; the nonlinearity boosts it, then breaks it.
Wired into the toolkit. These extensions are registered experiments (genus2_tunneling, genus2_collapse, horn_resonator, horn_hawking, kerr_horn) — callable through the same run / sweep / compare / MCP surface as everything else, each carrying its own verification block (genus + doublet-isolation; Townes match; resonance found + η-stable; the two κ-methods agreeing). And the full pipeline closes: a focusing NLS wavepacket evolves on the genus-2 mesh with a split-step Crank–Nicolson stepper, mass conserved to 10⁻¹⁴ — the project's physics, carried intact onto topology and boundaries the original metric could never write down.

13Modeling a real system — what geometry can and can't control in a microcomb

Everything above is a reduced-model laboratory. Pointed at a real device — a silica microdisk Kerr resonator at telecom wavelength (n=1.44, λ≈1.55 μm) — the same two findings that recurred across this gallery become a concrete design principle, in real units and validated against measured devices.

A whispering-gallery mode is a tunnelling problem. It is trapped by the index step and leaks by tunnelling radially through the centrifugal barrier (m²−¼)/r² — exactly the leaky-cavity structure of the horn (§12), now in a real dielectric. Solving the complex radial resonance (with the same absorbing-collar + η-independence check) gives a radiation-limited Q that is exponential in the geometry, and its magnitude matches reality: Q ≈ 10² at R=3 μm climbing past 10⁸ by R≈12 μm — the bending-loss regime real silica microdisks live in.

silica microdisk radiation Q and comb threshold vs radius
Geometry is an exponential lever on Q — but only a lever on Q. Left: the radiation Q of the WGM rises six decades over R=3→12 μm (blue), until it meets the material/roughness ceiling 10⁸ (loaded Q, grey) at a crossover radius R*≈11.7 μm: below it the device is geometry-limited, above it material-limited. Right: the Kerr-comb parametric threshold P_th∝1/Q² inherits this exactly — plunging from an impossible tens of kW at R=3 μm to a realistic ~1 μW near the crossover. So the comb threshold is set by geometry only through Q (a leakage property), and only up to R*. The local nonlinear response it rides on — the mass-critical threshold — is Townes-universal and geometry-blind (§11): you can engineer the resonator's Q with its shape, but you cannot shape away the nonlinearity itself. Every point survived the absorber-independence check that certifies the resonance over five decades of Q.
The takeaway. Two roles, cleanly separated by the verification this whole project is built on: geometry → Q is an exponential design knob (tunnelling), while geometry → nonlinear threshold is a null (universality). The reduced-model loop — geometry, one solver, a conserved-quantity or resonance-stability check, an honesty rubric — carried all the way from a lumpy torus to a design rule for a real microcomb. It's registered too: microdisk_Q is callable through run/sweep/MCP like everything else.

14Background & references

The experiment sits on a well-worn thread of geometry and physics — the reduction to a centrifugal-barrier problem is the same structure that appears across these lines of work.