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<!doctype html>
<html lang="en">
<head>
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<title>NLS on the Lumpy Torus — gallery</title>
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<header>
<h1>NLS on the Lumpy Torus</h1>
<p class="lede">Nonlinear Schrödinger evolution on a surface of revolution with metric
<span class="eq">ds² = dx² + A(x)²dθ²</span>, <span class="eq">A(x)=√((1+cos²x)/2)</span>.
Mass-conserving Laplace–Beltrami assembly, Crank–Nicolson time-stepping, Picard nonlinear solve.
A Python companion to <a href="https://smellslike.ml/posts/nls-on-the-lumpy-torus/">the lumpy-torus study</a>.
All 3-D frames are rendered on the metric-faithful <b>crescent immersion</b>.</p>
</header>
<div class="story">
<p><b>The question.</b> This surface is engineered to ask one thing, posed in the
<a href="https://smellslike.ml/posts/nls-on-the-lumpy-torus/">source post</a>: can a focusing
nonlinearity, working <i>with</i> 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 <b>belly</b> where the equatorial
geodesic is stable (elliptic) and <b>necks</b> where it is not (hyperbolic).</p>
<p><b>The arc.</b> The linear geometry already traps waves — whispering-gallery quasimodes bound in the
centrifugal well (§02). Turn on focusing and those become the <i>nonlinear</i> 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
<i>on a curve</i> 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.</p>
</div>
<section>
<h2><span class="num">01</span>Geometry & curvature</h2>
<p class="sub">One period <span class="eq">x∈[−π/2, π/2]</span> (ends identified) has exactly
<b>one elliptic</b> geodesic (belly, <span class="eq">K=+0.5</span>) and <b>one hyperbolic</b>
(neck, <span class="eq">K=−1</span>).</p>
<div class="grid">
<figure class="wide"><img src="curvature_view.png" alt="curvature view" loading="lazy">
<figcaption><b>Left:</b> the isometric neck–belly–neck immersion. <b>Middle:</b> bent closed into the
<b>crescent torus</b> used for every field render, colored by Gaussian curvature
<span class="eq">K=−A″/A</span>. <b>Right:</b> the Laplace–Beltrami inputs
<span class="eq">A</span>, <span class="eq">1/A²</span> (peaks at the necks), <span class="eq">K</span>.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">02</span>Why it works — the centrifugal-barrier reduction</h2>
<p class="sub">Separating <span class="eq">u = e^{ikθ}φ(x)</span> turns Laplace–Beltrami into a 1-D
Schrödinger operator with effective potential <span class="eq">V_k(x)=k²/A²</span> — the same
centrifugal barrier as the atomic radial equation. Both the quantum modes and the classical
geodesics live on this well/barrier structure.</p>
<div class="deriv">
<p><b>Why geodesics at all?</b> Because they are the <i>characteristics of the operator</i>, not a
metaphor. The linear part <span class="eq">i∂_t u = −Δ_g u</span> has principal symbol
<span class="eq">|ξ|²_g</span>, whose Hamiltonian flow is exactly the <b>geodesic flow</b>. So in the
high-frequency (semiclassical) limit a wave packet's <b>centre rides a geodesic</b> and its
<b>transverse width obeys the Jacobi equation</b> <span class="eq">J″ + K J = 0</span> along it —
geometric optics on a manifold. Geodesic <b>stability</b> 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 <span class="eq">σ|u|²u</span> is lower-order — it
rides <i>on top of</i> 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,
<i>which geodesic, and is it stable?</i> — and why the classical phase portrait below and the quantum
ladder are the same object seen two ways.</p>
</div>
<div class="deriv">
<p><b>How the geometry enters the operator.</b> The metric <span class="eq">ds² = dx² + A(x)²dθ²</span>
has volume element <span class="eq">√|g| = A</span> and inverse metric
<span class="eq">g^{xx}=1, g^{θθ}=1/A²</span>, so the Laplace–Beltrami operator is the flat Laplacian
<i>dressed</i> by the metric:</p>
<pre>Δ_g u = (1/A) ∂_x(A ∂_x u) + (1/A²) ∂²_θ u = u_xx + (A'/A) u_x + (1/A²) u_θθ</pre>
<p><b>Weak form</b> (what the solver assembles) — test against v, integrate by parts on the closed
surface (no boundary term):</p>
<pre>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</pre>
<p>So the geometry is just a set of <b>weights</b>: <span class="eq">A</span> weights the x-stiffness and
the mass, <span class="eq">1/A</span> the θ-stiffness. Discretely these become a Hermitian stiffness
<span class="eq">K</span> and a positive lumped-diagonal mass <span class="eq">M</span>; because both are
symmetric the scheme conserves <span class="eq">∫|u|² = UᴴMU</span> to machine precision — the metric is
baked into the conserved quantity.</p>
<p><b>The local geometry, in two Taylor expansions.</b> Everything follows from how the generating radius
<span class="eq">A</span> behaves at its critical points — near the belly (<span class="eq">x=0</span>)
and a neck (<span class="eq">x = π/2 − ξ</span>):</p>
<pre>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)</pre>
<p>Fed into the angular-momentum potential <span class="eq">V_k = k²/A²</span>:</p>
<pre>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</pre>
<p>That is the whole story in two lines: the belly is a harmonic trap of frequency
<span class="eq">k</span>, so its bound quasimodes have width <span class="eq">~k^{-1/2}</span> and
breathe at frequency <span class="eq">k</span> (the semiclassical beam), and its depth
<span class="eq">~k²</span> sets how many bind; the neck is a barrier that expels.</p>
<p><b>Why the NLS cares.</b> The linear operator sets the stage — well at the belly, barrier at the neck
— and <span class="eq">σ|u|²u</span> adds self-attraction. Dynamics is a contest between dispersion
(spreading, governed by <span class="eq">Δ_g</span>) and self-focusing. Where the geometry <i>both</i>
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 <b>biases the dispersion-vs-focusing balance</b> — which is precisely why a stable geodesic is
the place to look for collapse.</p>
</div>
<div class="deriv">
<p><b>A Fourier-analytic lens.</b> Almost every panel below is clarified by looking in Fourier space —
the dual of the real-space well/barrier picture.</p>
<p>• <b>θ-Fourier diagonalizes the linear flow.</b> With <span class="eq">u = Σ_k û_k(x)e^{ikθ}</span> the
axisymmetric operator is block-diagonal — each angular momentum evolves alone, and
<span class="eq">k²/A²</span> 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."</p>
<p>• <b>The nonlinearity is convolution — it couples the modes.</b>
<span class="eq">|u|²u = û ∗ û̄ ∗ û</span> mixes angular momenta, and that coupling is the engine of
every instability here: it feeds a pump mode into sidebands. The <b>necklace collapse</b>,
<b>Benjamin–Feir</b>, and the <b>Faraday</b> waves are the <i>same</i> phenomenon in Fourier space —
exponential growth of sideband modes — differing only in their gain spectrum
<span class="eq">γ(m)</span>. The ring "fragmenting into hot spots" is an azimuthal Fourier cascade.</p>
<p>• <b>Revivals are Fourier rephasing.</b> The Talbot carpet is the phases
<span class="eq">e^{-in²t}</span> realigning: at <span class="eq">t=2π</span> every
<span class="eq">n²t</span> is a multiple of <span class="eq">2π</span> (full revival); rational
fractions give Gauss-sum partial revivals. The recurrence <i>is</i> the Fourier phases coming back.</p>
<p>• <b>Lattices are Bloch/Floquet.</b> A periodic lump makes the x-Fourier (Bloch) label
<span class="eq">q</span> a good quantum number → the whispering-gallery bands
<span class="eq">ω(q)</span>; disorder destroys it, and Anderson localization is the collapse of that
Fourier description.</p>
<p>• <b>Beams are coherent states, and dispersion & horizons are about the symbol
<span class="eq">ω_k</span>.</b> 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 <span class="eq">ω_k=E_0(k)</span> (its curvature
<span class="eq">d²ω/dk²</span> is the engineered dispersion), and the sonic horizon is where the flow
Doppler-shifts <span class="eq">ω_k</span> until an upstream mode's group velocity vanishes — a
degeneracy read straight off the Fourier symbol.</p>
<p><b>In one sentence:</b> the geometry writes the dispersion relation <span class="eq">ω_k(x)</span>, the
nonlinearity convolves the modes, and every instability, revival, band, and horizon is a statement about
how those Fourier modes phase, couple, or degenerate.</p>
</div>
<div class="deriv">
<p><b>A pseudodifferential lens — the master lens the others fall out of.</b> Everything above is
one statement: the operators here, <span class="eq">−Δ_g</span> and its angular slices
<span class="eq">−∂_x² + V_k</span>, are <b>elliptic, self-adjoint pseudodifferential operators
(ΨDOs) of order 2</b> on the closed surface, and their <b>principal symbol</b>
<span class="eq">σ(−Δ_g)=|ξ|²_g=ξ_x²+k²/A(x)²</span> carries all the geometry. On the torus there
are no charts to patch: the ΨDO is a Fourier multiplier <i>dressed</i> by <span class="eq">A(x)</span>
— the <b>toroidal quantization</b> of Ruzhansky–Turunen — which is exactly why the Fourier lens above
is global and rigorous, not a heuristic.</p>
<p>• <b>Geodesics are the bicharacteristics.</b> The Hamiltonian flow of the principal symbol on
<span class="eq">T*M</span> <i>is</i> the geodesic flow (Clairaut) — the classical portrait of §02 is
the symbol's flow — and <b>Egorov's theorem</b> / <b>propagation of singularities</b> 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).</p>
<p>• <b>The propagator is a Fourier integral operator.</b> <span class="eq">e^{itΔ_g}</span> is an FIO
whose canonical relation is that flow; the <b>Gaussian-beam / WKB</b> construction (§07) is its
<b>parametrix</b> — phase solving the eikonal (Hamilton–Jacobi) equation, amplitude the transport
equation, whose leading term is precisely the <b>Jacobi law</b> <span class="eq">J″+KJ=0</span>. The
whispering-gallery quasimodes are the <b>quantization of the elliptic orbit</b> (a harmonic well of
frequency <span class="eq">k</span>); the nonlinearity <span class="eq">σ|u|²u</span> is a lower-order
rider on this ΨDO transport — not a ΨDO itself, but folded in microlocally by <b>paradifferential
calculus</b> (Bony), tamed by the same symbol machinery.</p>
<p>• <b>Functional calculus ties off the spectral panels.</b> <span class="eq">f(−Δ_g)</span> is again a
ΨDO — which is what makes the heat trace, the Weyl count (<i>"hear the shape"</i>), 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 <b>moves</b> (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,
<a href="https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2022/05/psidolect.pdf"><i>Lectures on
pseudodifferential operators</i></a>; Ruzhansky–Turunen,
<a href="https://arxiv.org/abs/0805.2892"><i>Quantization of ΨDOs on the torus</i></a>.</p>
</div>
<div class="deriv">
<p><b>A dynamical lens — static geometry → time-varying geometry.</b> Every lens so far fixes
<span class="eq">A(x)</span>. Let it <b>move</b>, <span class="eq">A=A(x,t)</span> (equivalently ramp the
ring radius / scale factor <span class="eq">a(t)</span>), and the ΨDO becomes a <b>time-dependent
family</b>: the dispersion relation <span class="eq">ω_k(t)</span> 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:</p>
<p>• <b>Periodic</b> drive → parametric <b>Faraday</b> resonance: a belly breathing at
<span class="eq">Ω=2ω_n</span> amplifies a Bogoliubov mode out of noise.<br>
• <b>Monotonic</b> ramp → <b>cosmological particle production</b>: an expanding ring redshifts its modes,
<span class="eq">ω_k=ck/a(t)</span>, and the adiabatic vacuum can't follow — phonon pairs appear with a
scale-invariant plateau and <b>Sakharov oscillations</b>, the analogue platform of Viermann et al.
(<i>Nature</i>, 2022).<br>
• <b>Stationary flow</b> through the fixed throat → an <b>analog horizon</b>: a transonic superfluid traps
sound at the neck, a geometry-made black hole.</p>
<p>Static geometry bends <i>where</i> modes go; dynamic geometry decides <i>which</i> modes exist — the
metric becomes a source that mixes the vacuum, and it can act back on the nonlinearity itself: breathing
the belly resonantly <b>delays or hastens</b> the fast necklace collapse and fully <b>arrests</b> 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 <b>time</b>-drive alone makes <b>Floquet edge modes</b> at
quasienergy 0 and π (§10).</p>
</div>
<div class="grid">
<figure class="wide"><img src="quasimode_ladder.png" alt="quasimode ladder" loading="lazy">
<figcaption><b>Quantum.</b> <span class="eq">V_k</span> is a well at the belly (<span class="eq">A=1</span>)
with barriers at the necks (<span class="eq">A=0.707</span>). Its bound states are the
<b>whispering-gallery quasimodes</b>; the ground state sharpens <span class="eq">~k^{-1/2}</span> —
exactly the semiclassical beam width.</figcaption></figure>
<figure class="wide"><img src="geodesic_phase_space.png" alt="geodesic phase space" loading="lazy">
<figcaption><b>Classical.</b> The integrable geodesic flow (Clairaut) has the identical
<span class="eq">L²/A²</span> potential: an <b>elliptic island</b> of librating orbits around the
belly, a <b>separatrix</b> at the neck, circulating orbits beyond — the skeleton the quasimodes sit on.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">03</span>Solver validation</h2>
<p class="sub">A focusing equation near blow-up is unforgiving: a scheme that leaks or injects energy will
<i>fake</i> 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 <span class="eq">∫A|u|²</span> to
machine precision and energy to <span class="eq">~10⁻⁵</span> over a thousand steps, so when the ring does
collapse (§08) we know it is the physics, not the discretisation.</p>
<div class="grid">
<figure class="wide"><img src="conservation.png" alt="conservation" loading="lazy">
<figcaption><b>Mass conserved to ~10⁻¹³</b> and energy to ~10⁻⁵ over 1000 Crank–Nicolson steps —
the diagnostics the reference FreeFEM script omitted.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">04</span>The field on the lumpy torus</h2>
<p class="sub">Two views of the same field, and it pays to read both. The <b>crescent immersion</b> is what
the surface <i>looks</i> like; the <span class="eq">(x,θ)</span> <b>chart</b> is where the operator
<i>lives</i> — where <span class="eq">θ</span>-Fourier diagonalises the linear flow and the centrifugal
potential <span class="eq">k²/A²</span> is plain to see. Every animation below is one view or the other.</p>
<div class="grid">
<figure><img src="nls_lumpy_torus.gif" alt="wavepacket on torus" loading="lazy">
<figcaption><span class="tag">3-D</span><span class="eq">|u|²</span> on the crescent immersion.</figcaption></figure>
<figure><img src="nls_chart.gif" alt="wavepacket on chart" loading="lazy">
<figcaption><span class="tag">chart</span>The same field on the <span class="eq">(x,θ)</span> chart.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">05</span>Geodesic stability (linear)</h2>
<p class="sub">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 <b>elliptic</b> belly the Jacobi field
stays bounded and the beam holds together; on the <b>hyperbolic</b> 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.</p>
<div class="grid">
<figure><img src="nls_geodesic_elliptic.gif" alt="elliptic beam" loading="lazy">
<figcaption><b>Elliptic (belly).</b> A confined bright band — the stable geodesic.</figcaption></figure>
<figure><img src="nls_geodesic_hyperbolic.gif" alt="hyperbolic beam" loading="lazy">
<figcaption><b>Hyperbolic (neck).</b> Disperses off the pinch — the unstable geodesic.</figcaption></figure>
<figure class="wide"><img src="geodesic_comparison.png" alt="geodesic comparison" loading="lazy">
<figcaption>Transverse width stays bounded on the elliptic equator; swings ~2× larger on the neck.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">06</span>Running over the bumps: the meridian beam</h2>
<p class="sub">The meridians are the <i>other</i> geodesic family — and unlike the equator they are not stable
orbits: each one crosses the <span class="eq">K>0</span> belly <i>and</i> the <span class="eq">K<0</span>
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.</p>
<div class="grid">
<figure><img src="nls_meridian_torus.gif" alt="meridian torus" loading="lazy">
<figcaption><span class="tag">3-D</span>A beam concentrated along a meridian, crossing belly and necks.</figcaption></figure>
<figure class="wide"><img src="meridian_diag.png" alt="meridian diagnostic" loading="lazy">
<figcaption>Physical width tracks the Jacobi field <span class="eq">J(x)=A(x)</span>: the <b>necks focus</b>,
the <b>belly defocuses</b> — as on a sphere, where meridians converge at the poles.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">07</span>Semiclassical beam</h2>
<p class="sub">Push the angular momentum high — the semiclassical limit — and the ray/wave correspondence
sharpens into view: the packet becomes a true <b>Gaussian beam</b> (the FIO parametrix of §02) riding the
geodesic, its width breathing as <span class="eq">A</span> around the loop. Geometric optics on the
manifold, made literal.</p>
<div class="grid">
<figure><img src="nls_semiclassical_torus.gif" alt="semiclassical surf" loading="lazy">
<figcaption><span class="tag">q=20</span>The beam surfs the crescent loop, through the neck and over the belly.</figcaption></figure>
<figure><img src="semiclassical_breathing.png" alt="breathing" loading="lazy">
<figcaption>Transverse width tracks <span class="eq">A(beam)</span> — compress at necks, expand over belly.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">08</span>Self-trapping via focusing nonlinearity</h2>
<p class="sub">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).</p>
<div class="grid">
<figure><img src="nls_selftrap_focusing_torus.gif" alt="focusing ring" loading="lazy">
<figcaption><b>Focusing</b> self-traps it tighter and brighter.</figcaption></figure>
<figure class="wide"><img src="selftrap_comparison.png" alt="selftrap comparison" loading="lazy">
<figcaption>Focusing holds the ring at smaller width and higher peak than the linear mode.</figcaption></figure>
<figure><img src="nls_selftrap_collapse_torus.gif" alt="collapse" loading="lazy">
<figcaption><span class="tag warn">amp = 5</span>The width locks into a soliton, then fragments into hot spots.</figcaption></figure>
<figure><img src="selftrap_collapse.png" alt="collapse diagnostic" loading="lazy">
<figcaption>Azimuthal symmetry-breaking grows <b>exponentially from machine roundoff</b>
(<span class="eq">10⁻¹⁵→10⁻¹</span>) → mass-critical collapse at <span class="eq">t≈1.8</span> (a "necklace" instability).</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">09</span>Regimes, instabilities & bifurcation</h2>
<p class="sub">What the geometry and the nonlinearity open up beyond the main experiments.</p>
<div class="grid">
<figure class="wide"><img src="bifurcation.png" alt="bifurcation" loading="lazy">
<figcaption><b>Elliptic↔hyperbolic bifurcation — the geometry knob.</b> With the tunable family
<span class="eq">A(x;ε)=√((1+ε cos²x)/(1+ε))</span> the curvatures are exactly
<span class="eq">K_belly=ε/(1+ε)</span>, <span class="eq">K_neck=−ε</span>: belly and neck
<b>exchange stability at ε=0</b>. The centrifugal well inverts, and the whispering-gallery
bound states switch on as the belly turns elliptic. Our torus is <span class="eq">ε=1</span>.</figcaption></figure>
<figure><img src="nls_defocusing_torus.gif" alt="defocusing" loading="lazy">
<figcaption><span class="tag">defocusing</span>Repulsive <span class="eq">σ=+1</span>: a hump expands and
depletes its core — dispersive-shock / rarefaction dynamics, the opposite of self-trapping.</figcaption></figure>
<figure class="wide"><img src="ring_modulational.png" alt="modulational instability and dark solitons" loading="lazy">
<figcaption><b>Modulational instability & dark solitons on the belly ring.</b> <b>Left:</b> a focusing
uniform state is <b>Benjamin–Feir unstable</b> — it breaks into a soliton and recurs
(Fermi–Pasta–Ulam). <b>Right:</b> in the defocusing regime, two <b>black solitons</b> (stationary
density notches with π phase jumps) are stable.</figcaption></figure>
<figure class="wide"><img src="threshold.png" alt="collapse threshold" loading="lazy">
<figcaption><b>Mass-critical collapse threshold.</b> Below <span class="eq">M_c≈43</span> the focusing ring
never collapses; above it the necklace collapse sets in, its onset time falling off with mass.</figcaption></figure>
<figure class="wide"><img src="spacetime_carpet.png" alt="Talbot carpet" loading="lazy">
<figcaption><b>Quantum revivals (Talbot carpet).</b> A packet released on the belly ring reforms at
<span class="eq">t=2π</span> (full revival) and rational fractions — the mechanism behind the
dispersive recurrence seen in the full runs.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">10</span>Research directions — geometry as a control field</h2>
<p class="sub">The same tunable metric is a laboratory. Each panel is a proof-of-concept for a direction
where surface geometry <i>steers</i> nonlinear wave dynamics — some clean results, some sketches of open
questions, all runnable from the same solver.</p>
<div class="grid">
<figure class="wide"><img src="mc_geometry.png" alt="collapse threshold vs geometry" loading="lazy">
<figcaption><b>Geometry tunes the blow-up.</b> Collapse-onset time vs ring mass for four lump depths —
the curves <b>separate</b>: deeper necks (larger <span class="eq">ε</span>) 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.</figcaption></figure>
<figure class="wide"><img src="chaos_poincare.png" alt="geodesic chaos" loading="lazy">
<figcaption><b>Break the axisymmetry, break integrability.</b> 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.</figcaption></figure>
<figure class="wide"><img src="quantum_chaos.png" alt="quantum chaos: level statistics and eigenstates" loading="lazy">
<figcaption><b>The quantum face of that chaos.</b> Diagonalising <span class="eq">−Δ_g + λV(x,θ)</span>
with a generic <span class="eq">θ</span>-lump: as the lump strengthens, the nearest-neighbour
level-spacing statistics cross from <b>clustering</b> (the symmetric surface's <span class="eq">±k</span>
degeneracy, left) to the <b>Wigner–GOE</b> distribution with hard <b>level repulsion</b> (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.</figcaption></figure>
<figure class="wide"><img src="lattice.png" alt="geometric lattice and Anderson" loading="lazy">
<figcaption><b>Corrugation is an intrinsic lattice.</b> A periodic lump chain makes
<span class="eq">V_k=k²/A²</span> a periodic potential → whispering-gallery <b>Bloch bands</b> with
gaps; random lumps → <b>Anderson-localized</b> modes. No external potential — the geometry is the
lattice (and the disorder).</figcaption></figure>
<figure class="wide"><img src="topological.png" alt="topological whispering-gallery edge modes" loading="lazy">
<figcaption><b>A chiral lump twist makes topological edge channels.</b> Twist the lumps helically,
<span class="eq">A(x−cθ)</span>: the WG on-site energy <span class="eq">k²/A²(x−cθ)</span> becomes the
Harper term <span class="eq">2λcos(2παn+k_θ)</span>, so the twisted-lump strip <b>is</b> the
Hofstadter model. <b>Chiral edge modes</b> 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.</figcaption></figure>
<figure class="wide"><img src="floquet.png" alt="Floquet-topological edge modes from periodic driving" loading="lazy">
<figcaption><b>Floquet engineering — a <i>time</i> route to the edge modes.</b> 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 <b>quasienergy bands</b> whose gaps host
chiral edge modes — at quasienergy <span class="eq">ε=0</span> (a static-like mode) <b>and</b> at
<span class="eq">ε=π</span>, an <b>anomalous mode with no static analogue</b> (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.</figcaption></figure>
<figure class="wide"><img src="thouless.png" alt="Thouless adiabatic pump from a sliding lump lattice" loading="lazy">
<figcaption><b>Thouless pump — quantized transport from a sliding lattice.</b> Slide and breathe the lump
chain around a cycle and the filled band's Wannier centre winds by an <b>integer</b>: 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 <b>edge state across the gap</b> 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.</figcaption></figure>
<figure class="wide"><img src="soliton_barrier.png" alt="soliton trapping vs escape" loading="lazy">
<figcaption><b>A soliton in the geometric well.</b> 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).</figcaption></figure>
<figure class="wide"><img src="faraday.png" alt="Faraday parametric amplification" loading="lazy">
<figcaption><b>A breathing torus amplifies (periodic drive).</b> Periodically modulating the geometry
parametrically pumps a Bogoliubov mode out of noise (Faraday waves, driven at
<span class="eq">Ω=2ω_n</span>). Run as a monotonic expansion instead → the next panel.</figcaption></figure>
<figure class="wide"><img src="expanding_torus.png" alt="expanding torus cosmological particle production" loading="lazy">
<figcaption><b>Prototype — the expanding torus makes particles from the vacuum (monotonic ramp).</b>
Ramp the scale factor <span class="eq">a(t)</span> and every belly phonon redshifts
<span class="eq">ω_k=ck/a(t)</span> (<b>left</b>) until the adiabatic vacuum can no longer follow, so
the quench pumps phonon <b>pairs</b> out of nothing: a scale-invariant plateau
<span class="eq">|β_k|²→(a_f−a_i)²/4a_ia_f</span> (the inflationary analogue) whose density structure
factor rings with <b>Sakharov oscillations</b> (<b>right</b>) — the acoustic analogue of the CMB peaks.
Exact linear Bogoliubov theory, two-mode-squeeze Wronskian held to <span class="eq">10⁻¹³</span>; the
NLS realisation of Viermann et al.'s expanding-universe BEC (<i>Nature</i>, 2022).</figcaption></figure>
<figure class="wide"><img src="dynamical_stabilization.png" alt="dynamical control of the focusing collapse" loading="lazy">
<figcaption><b>Prototype — dynamical control of the collapse (geometry vs the nonlinearity).</b> Breathing
the belly makes the effective coupling <span class="eq">g_eff∝1/A²(t)</span> oscillate — "nonlinearity
management." It is a <b>resonant</b> knob on the necklace collapse: in a frequency window
(<span class="eq">Ω≈40</span>) deep breathing <b>delays</b> the mass-critical blow-up ~<b>5×</b>
(<b>left</b>; the delay grows with depth <span class="eq">δ</span>, <b>middle</b>); too slow
(<span class="eq">Ω=16</span>) and the drive parametrically feeds the very sidebands that fragment the
ring, so it gives no help or even <b>hastens</b> collapse (<b>right</b>). A geometric Kapitza /
Feshbach-management effect (Saito–Ueda) — the same idea, <b>dynamic stabilization of a wave
instability</b>, that plasma actuators use, and reduced-model kin to Langmuir-wave collapse. It only
<i>delays</i> this fast azimuthal collapse; the next panel shows the collapse it fully
<i>arrests</i>.</figcaption></figure>
<figure class="wide"><img src="collapse_arrest.png" alt="arresting the Townes collapse by fast breathing" loading="lazy">
<figcaption><b>Prototype — <i>arresting</i> the collapse (the management the necklace resisted).</b> Where
the blow-up is a <b>single localized hump</b> just above the mass-critical (Townes) threshold, the same
knob works cleanly. Bare, it self-focuses and blows up (<span class="eq">t_c≈0.11</span>, <b>left,
pink</b>); breathing the belly fast and hard (<span class="eq">δ=0.8, Ω=100</span>) turns it into a
<b>stable breather</b> whose peak settles to a bounded level (<b>blue</b>) and survives indefinitely — a
genuine <b>arrest</b>, above a depth threshold (<b>middle</b>) and a frequency threshold (<b>right</b>;
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 <i>is</i> the
physics: a drive averages away a <b>slow radial</b> collapse but only delays a <b>fast azimuthal</b>
one.</figcaption></figure>
<figure class="wide"><img src="nls_collapse_arrest.gif" alt="animation: collapse vs arrested breather" loading="lazy">
<figcaption><span class="tag">animation</span><b>Left:</b> the bare hump self-focuses and collapses
(frozen once it blows up). <b>Right:</b> the same hump under fast belly-breathing survives as a pulsing
soliton — the blow-up dynamically arrested, side by side.</figcaption></figure>
<div class="note"><b>Prior art & where this sits.</b> Arresting NLS collapse by modulating the
nonlinearity is <b>Feshbach-resonance / nonlinearity management</b>, established since 2003 (Saito–Ueda;
Abdullaev et al.; Kevrekidis et al.). What's ours is the <i>framing</i>: the modulation is a
<b>geometric</b> one — breathing the belly, <span class="eq">g_eff∝1/A²(t)</span> — and the
necklace-vs-hump contrast reproduces a subtle known result: nonlinearity management <b>delays</b> a
supercritical collapse but need not prevent it, yet fully <b>arrests</b> the near-critical hump. A 2025
analysis (Li–Ning–Zhao) makes the distinction rigorous — <i>dispersion</i> management can prevent
blow-up, <i>nonlinearity</i> management only postpones it. The same physics is a live control knob in
attractive BECs (the "Bosenova"), in self-focusing / <b>filamentation</b> 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.</div>
<figure class="wide"><img src="cubic_quintic.png" alt="cubic-quintic prevents collapse outright" loading="lazy">
<figcaption><b>Prevention, not just delay — the cubic-quintic resolution.</b> The management panels only
<i>delayed</i> the collapse; adding a <b>defocusing quintic</b> term (<span class="eq">σ₅|u|⁴u</span>,
repulsive only at high density) removes the collapse threshold outright. The same hump that
Townes-collapses under pure cubic settles into a <b>stable soliton with no drive at all</b> (left), and
cubic+quintic stays stable at <i>every</i> mass while pure cubic collapses above
<span class="eq">M_c</span> (right). The clean way to genuinely <i>prevent</i> 2-D collapse — matching
the rigorous delay-vs-prevent result above; connects to condensates with three-body repulsion.</figcaption></figure>
<figure class="wide"><img src="rogue.png" alt="rogue waves and Peregrine breather on the belly ring" loading="lazy">
<figcaption><b>Rogue waves on the belly ring.</b> Seed the focusing ring's longest-wavelength unstable
mode and modulational instability grows it into an <b>Akhmediev breather</b> — a wave that rises from
nowhere, peaks at <b>3× the background</b> (right; the <b>Peregrine</b> 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 <span class="eq">A(x)</span> set the MI gain, hence how tall and
how often the rogue events come. Real-world: optical and hydrodynamic rogue waves.</figcaption></figure>
<figure class="wide"><img src="kibble_zurek.png" alt="Kibble-Zurek scaling from a quench on the ring" loading="lazy">
<figcaption><b>Kibble–Zurek — defects from a finite-rate quench.</b> Ramp the ring's focusing coupling
up through the modulational-instability onset over a quench time <span class="eq">τ_Q</span>. The
uniform state can't follow adiabatically: its response <b>freezes</b> near the transition and the
instability then imprints a pattern whose wavenumber <span class="eq">k*</span> is set by <i>when</i> it
froze — faster quench → finer pattern, more defects (left). Averaged over noise realizations,
<span class="eq">k*</span> follows a clean <b>power law</b> <span class="eq">k*∼τ_Q^{−0.23}</span> 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.</figcaption></figure>
<figure class="wide"><img src="dispersion_design.png" alt="geometric dispersion engineering" loading="lazy">
<figcaption><b>Prototype — geometric dispersion engineering.</b> The WG spectrum
<span class="eq">ω_m = E_0(m)</span> <i>is</i> the resonator's modal dispersion, so the lump profile is
a design knob. Inverse-designing <span class="eq">A(x)</span> (a few harmonics) flattens the integrated
dispersion <b>4.4×</b> over a mode band — a broadband, near-equidistant comb grid engineered by
geometry rather than waveguide cross-section.</figcaption></figure>
<figure class="wide"><img src="neural_operator.png" alt="neural operator surrogate on the manifold" loading="lazy">
<figcaption><b>Prototype — a neural operator closes the ML-for-PDE loop.</b> A small MLP learns the
forward map lump profile <span class="eq">A(x)</span> → modal dispersion
<span class="eq">D_int(m)</span> from solver data (<span class="eq">R²=0.998</span>), then stands in for
the eigensolver to <b>inverse-design</b> a flat-dispersion resonator ~<b>2000× faster</b> 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.</figcaption></figure>
<figure class="wide"><img src="analog_horizon.png" alt="analog sonic horizon at the neck" loading="lazy">
<figcaption><b>Prototype — the neck as an analog black hole.</b> A transonic superfluid speeds up through
the small-<span class="eq">A</span> throat until the flow speed crosses the sound speed: a
<b>sonic horizon provided by the geometry alone</b> (Hawking temperature <span class="eq">T_H</span>
set by the throat's surface gravity). The upstream <span class="eq">v−c</span> sound rays diverge from
the throat and can never cross it — nothing escapes the interior.</figcaption></figure>
<figure class="wide"><img src="hawking.png" alt="analog Hawking: exponential redshift and thermal spectrum" loading="lazy">
<figcaption><b>Prototype — the horizon radiates (analog Hawking).</b> Take that horizon's radiation
seriously. Near it the upstream characteristic obeys <span class="eq">dx/dt=v−c≈κ(x−x_h)</span>, so a
ray <b>peels off exponentially</b> (left, log axis; slope = the surface gravity
<span class="eq">κ≈0.7</span>, measured dynamically and matching the static value) — the unbounded
redshift that turns the vacuum into a <b>thermal</b> spectrum at <span class="eq">T_H=κ/2π≈0.11</span>
(right). Honest scope: this is the kinematic Hawking result (temperature + thermality mechanism); the
spontaneous pair spectrum <span class="eq">|β_ω|²</span> itself needs the dispersive
Bogoliubov–de Gennes scattering with its negative-norm partner mode — a noted next step.</figcaption></figure>
<div class="note"><b>Driven-dissipative (scaffolded, <code>lle.py</code>).</b> A lump should pin a
dissipative Kerr soliton — the soliton-microcomb regime, and a microtoroid resonator <i>is</i> 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.</div>
<div class="note"><b>Still open (application-grounded).</b> 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: <b>vortices on a curved
superfluid shell</b> (NASA Cold-Atom-Lab bubble BECs) and their curvature pinning — a first pass
(<code>vortices.py</code>) 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.</div>
</div>
</section>
<section>
<h2><span class="num">11</span>What the agent-toolkit found</h2>
<p class="sub">The panels above were designed by hand. These were not. The same solver, refactored into a
<a href="docs/toolkit.md">modular library and a self-describing agent tool</a> (<code>nls_torus</code>, an
MCP server, and a generative sub-agent), was pointed at open questions and left to
<b>design → run → verify → retract</b> on its own — with the trust flags computed by the <i>harness</i>,
not asserted by the model. Three probes, three honest outcomes: a discovery, a busted myth, and a dead end.</p>
<div class="deriv">
<p><b>Why a verification block matters.</b> A plausible-looking number and a correct one are
indistinguishable from the outside. So every run returns a <code>verification</code> block — mass drift
against the conserved <span class="eq">∫A|u|²</span>, 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
<i>wrong</i>, and the check is what said so — the agent could not self-certify past it.</p>
</div>
<div class="grid">
<figure class="wide"><img src="curvature_universality.png" alt="collapse threshold is the Townes mass, geometry-blind" loading="lazy">
<figcaption><b>The collapse threshold is a local universal invariant — it ignores the geometry.</b>
<b>Left:</b> tuned to the <i>same</i> belly curvature <span class="eq">K=0.5</span> but four different
global structures — flat, one lobe, two lobes, a localized bump — the critical mass lands on the
<b>Townes value <span class="eq">‖Q‖²≈11.7</span></b> every time, to 0.4%. Neither curvature nor
topology moves it: mass-critical blow-up is a self-similar concentration at a <i>point</i>, which sees
only the local (≈flat) metric, and the Townes mass is a pure number. <b>Right — the retraction:</b> a
coarse scan had shown <span class="eq">M_c</span> <i>falling</i> with curvature (orange), but that was
the <b>A-weighting confound</b> — the mass <span class="eq">∫A|u|²</span> 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.</figcaption></figure>
<figure class="wide"><img src="tunneling.png" alt="geometry-controlled tunneling doublet" loading="lazy">
<figcaption><b>Geometry as a tunable tunnelling qubit.</b> A <b>two-lobe</b> surface makes the
centrifugal reduction <span class="eq">V_k=k²/A²</span> a <b>double well on the ring</b> — one well per
belly lobe, split by the neck barriers (left). Its two lowest states form a near-degenerate
<b>doublet</b>, symmetric and antisymmetric across the lobes, whose splitting <i>is</i> the inter-lobe
<b>tunnelling rate</b>. Deepen the necks and that rate collapses <b>exponentially</b> —
<span class="eq">5×10⁻¹ → 1×10⁻⁴</span> across <span class="eq">ε: 0.5→8</span> (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.</figcaption></figure>
<div class="note"><b>And an honest negative.</b> A third probe asked whether periodically breathing the
belly ring builds a <b>discrete time crystal</b>. It does not: the geometric drive is a <i>sharp</i>
parametric (Faraday) resonance — a narrow subharmonic band (<span class="eq">≲4%</span> in drive
frequency), a response pulled ~10% off <span class="eq">Ω/2</span> by the nonlinearity, and <b>no
prethermal plateau</b> (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 <i>parametric amplifier, not time crystal</i>, and the metrics simply declined to certify a
rigidity that isn't there.</div>
<figure class="wide"><img src="dtc_horn.png" alt="disorder-stabilized discrete time crystal" loading="lazy">
<figcaption><b>…and the negative, followed up: what the clean ring was missing was disorder.</b> The H2
diagnosis was specific — a clean mean-field system has no localization to rigidify the subharmonic — so
we supplied exactly that: <b>geometric disorder</b> (random on-site energies, the Anderson-localizing
stand-in for many-body localization) on a driven nonlinear lattice, each period an <i>imperfect</i>
π-swap that flips a density wave. <b>Left:</b> the demodulated period-2 response
<span class="eq">I_p(−1)^p</span> — with disorder it holds a <b>rigid, coherent</b> value (a genuine
time crystal); the clean lattice swings <b>incoherently through zero</b> (the H2 case, reproduced as the
control). <b>Right:</b> the rigidity — the order parameter survives drive imperfection out to
<span class="eq">e≈0.15</span> with disorder (<span class="eq">O≈0.66</span>) but is dead flat without it.
The response locks to <b>exactly <span class="eq">Ω/2</span></b> (the trust flag). The open-horn leakage
is ~neutral (green ≈ blue): <i>disorder</i>, not openness, is what the negative result was pointing at —
a negative turned into a positive by taking the diagnosis literally.</figcaption></figure>
</div>
</section>
<section>
<h2><span class="num">12</span>Beyond surfaces of revolution — mesh & non-compact extensions</h2>
<p class="sub">Everything above lives on the metric <span class="eq">ds²=dx²+A(x)²dθ²</span> — which can
only express <b>genus-1, rotationally-symmetric</b> shapes. The toolkit's
<a href="docs/toolkit.md">mesh module</a> carries the same physics onto an <b>arbitrary triangle mesh</b>
via the cotangent (finite-element) Laplace–Beltrami operator, and onto <b>non-compact</b> domains via a
complex absorbing collar. That opens two doors the surface-of-revolution engine cannot: a genuine
<b>second handle</b>, and an <b>open, radiating</b> geometry.</p>
<div class="deriv">
<p><b>Validation before new physics.</b> 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 <i>first</i>: on the round
sphere it recovers the spectrum <span class="eq">−Δ = l(l+1) = 0, 2, 6, 12</span> with the right
multiplicities <span class="eq">1, 3, 5, 7</span> 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 <b>PML/continuum eigenvalues</b>, so a resonance is accepted only if it is
<b>invariant when the absorber strength changes</b> — the physical mode stays put, the artifacts move.</p>
</div>
<div class="grid">
<figure class="wide"><img src="genus2_tunneling.png" alt="genus-2 which-handle tunnelling doublet and neck sweep" loading="lazy">
<figcaption><b>H4 on a true genus-2 surface.</b> Two tori fused into a connected <b>genus-2</b> shape
(verified <span class="eq">χ=−2</span> at every step). Its first excited Laplace–Beltrami mode is the
<b>which-handle doublet</b> — one sign on each handle, a node at the connecting neck (left) — so
<span class="eq">λ₁</span> <i>is</i> 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,
<span class="eq">0.30→0.18</span> (right), while the doublet grows more isolated
(<span class="eq">λ₂/λ₁: 3.0→5.5</span>) as the handles decouple toward two independent zero modes.
Tunnelling across a handle, set by geometry.</figcaption></figure>
<figure class="wide"><img src="genus2_collapse.png" alt="Townes collapse threshold blind to topology" loading="lazy">
<figcaption><b>H1 universality survives topology.</b> The same genus-2 mesh lets us push §11's result to
its sharpest test: is the mass-critical collapse threshold still the <b>Townes mass</b>
<span class="eq">‖Q‖²≈11.7</span> when the topology itself changes? A fixed mesh grid-arrests true
blow-up, so we read the <b>focusing onset</b> (a bump grows above <span class="eq">M_c</span>, disperses
below). The threshold lands on <b>11.6–11.9</b> on a genus-0 sphere and on genus-2 at every neck width —
a <b>1.5% spread</b>, grid-converged, mass conserved to <span class="eq">10⁻¹⁴</span>. Collapse is a local
point-concentration; it sees neither the curvature nor the handles.</figcaption></figure>
<figure class="wide"><img src="horn_resonator.png" alt="non-compact horn leaky resonator, geometry-set Q" loading="lazy">
<figcaption><b>A non-compact horn: geometry sets the Q of a leaky mode.</b> The meridian runs to a
semi-infinite horn; a <b>throat</b> (a dip in <span class="eq">A</span>, a barrier in
<span class="eq">V_k=k²/A²</span>) traps a cavity mode that slowly radiates out the open end, absorbed by
a <b>PML collar</b>. The operator is non-Hermitian, so the mode is a complex <b>resonance</b>
<span class="eq">E_r−iΓ/2</span> with a finite lifetime (left; <span class="eq">|ψ|²</span> cavity-bound,
leaking past the throat). Its width is <b>tunnelling-limited</b>, so deepening the throat drives the
quality factor <span class="eq">Q=E_r/Γ</span> up <b>exponentially</b> — a straight line on a log axis,
<span class="eq">Q: 284→85138</span> (right). Every point survived the absorber-independence check that
killed dozens of numerical artifacts.</figcaption></figure>
<figure class="wide"><img src="horn_hawking.png" alt="analog Hawking radiation on the open horn" loading="lazy">
<figcaption><b>The same open horn is also an analog black hole.</b> Drive a superfluid <i>through</i> the
throat and it becomes a <b>de Laval nozzle</b>: the flow accelerates from a subsonic reservoir, crosses
the sound speed <i>exactly</i> at the throat — a <b>sonic horizon</b> — and radiates supersonically out
the open end, where (unlike the closed torus neck of §10) the phonons genuinely escape into the absorbing
collar. <b>Left:</b> flow speed <span class="eq">v</span> and sound speed <span class="eq">c</span>
crossing at the horizon; the inset traces rays <b>peeling off exponentially</b>,
<span class="eq">|x−x_h|∼e^{κt}</span>, at the surface gravity <span class="eq">κ</span> — the mechanism
that renders the vacuum thermal at <span class="eq">T_H=κ/2π</span>. <b>Right:</b> a <i>sharper</i> throat
makes a <i>hotter</i> horizon — geometry sets the Hawking temperature (<span class="eq">T_H: 0.09→0.22</span>).
The trust flag is stringent: <span class="eq">κ</span> is measured two independent ways — a static fit of
<span class="eq">v−c</span> and the dynamical ray-peeling rate — and they agree to <b><1%</b>. (Kinematic
scope; the full Bogoliubov pair spectrum is noted, not attempted.)</figcaption></figure>
<figure class="wide"><img src="nonlinear_kerr_horn.png" alt="nonlinear Kerr on the horn leaky mode" loading="lazy">
<figcaption><b>The horn's leaky mode, made nonlinear — a bridge to the §13 Kerr comb.</b> §12 gave the
horn a leaky mode whose Q the throat geometry sets; §13 built a microcomb whose threshold rides on Q.
Here a <b>focusing Kerr</b> 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 <span class="eq">g=0</span> control is flat to
<span class="eq">10⁻¹³</span> at the geometric <span class="eq">Q=439</span>). <b>Left:</b> the resonance
<b>redshifts linearly</b> with power — Kerr self-phase modulation. <b>Right, the surprise:</b> the
nonlinearity doesn't spoil the leaky mode, it <b>self-traps</b> it — pulling the frequency deeper below
the throat barrier so it radiates <i>less</i>, lifting <span class="eq">Q</span> by <b>+66%</b> — until a
critical intracavity power triggers a <b>self-focusing instability</b> 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.</figcaption></figure>
<div class="note"><b>Wired into the toolkit.</b> These extensions are registered experiments
(<code>genus2_tunneling</code>, <code>genus2_collapse</code>, <code>horn_resonator</code>,
<code>horn_hawking</code>, <code>kerr_horn</code>) — callable through
the same <code>run</code> / <code>sweep</code> / <code>compare</code> / 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, <b>mass conserved to <span class="eq">10⁻¹⁴</span></b> — the project's physics, carried intact
onto topology and boundaries the original metric could never write down.</div>
</div>
</section>
<section>
<h2><span class="num">13</span>Modeling a real system — what geometry can and can't control in a microcomb</h2>
<p class="sub">Everything above is a reduced-model laboratory. Pointed at a real device — a
<b>silica microdisk Kerr resonator</b> at telecom wavelength (<span class="eq">n=1.44</span>,
<span class="eq">λ≈1.55 μm</span>) — the same two findings that recurred across this gallery become a concrete
<b>design principle</b>, in real units and validated against measured devices.</p>
<div class="deriv">
<p><b>A whispering-gallery mode is a tunnelling problem.</b> It is trapped by the index step and leaks by
tunnelling <i>radially</i> through the centrifugal barrier <span class="eq">(m²−¼)/r²</span> — exactly the
leaky-cavity structure of the horn (§12), now in a real dielectric. Solving the complex radial resonance
(with the same absorbing-collar + <span class="eq">η</span>-independence check) gives a
<b>radiation-limited Q that is exponential in the geometry</b>, and its magnitude matches reality:
<span class="eq">Q ≈ 10²</span> at <span class="eq">R=3 μm</span> climbing past <span class="eq">10⁸</span>
by <span class="eq">R≈12 μm</span> — the bending-loss regime real silica microdisks live in.</p>
</div>
<div class="grid">
<figure class="wide"><img src="microdisk_Q.png" alt="silica microdisk radiation Q and comb threshold vs radius" loading="lazy">
<figcaption><b>Geometry is an exponential lever on Q — but only a lever on Q.</b> <b>Left:</b> the
radiation <span class="eq">Q</span> of the WGM rises six decades over <span class="eq">R=3→12 μm</span>
(blue), until it meets the material/roughness ceiling <span class="eq">10⁸</span> (loaded Q, grey) at a
<b>crossover radius <span class="eq">R*≈11.7 μm</span></b>: below it the device is <i>geometry-limited</i>,
above it <i>material-limited</i>. <b>Right:</b> the Kerr-comb parametric threshold
<span class="eq">P_th∝1/Q²</span> inherits this exactly — plunging from an impossible tens of kW at
<span class="eq">R=3 μm</span> to a realistic <span class="eq">~1 μW</span> near the crossover. So the
comb threshold is set by geometry <b>only through Q</b> (a leakage property), and only up to
<span class="eq">R*</span>. The <i>local</i> nonlinear response it rides on — the mass-critical
threshold — is <b>Townes-universal and geometry-blind</b> (§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.</figcaption></figure>
<div class="note"><b>The takeaway.</b> Two roles, cleanly separated by the verification this whole project
is built on: <b>geometry → Q</b> is an exponential design knob (tunnelling), while <b>geometry → nonlinear
threshold</b> 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: <code>microdisk_Q</code> is callable through
<code>run</code>/<code>sweep</code>/MCP like everything else.</div>
</div>
</section>
<section>
<h2><span class="num">14</span>Background & references</h2>
<p class="sub">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.</p>
<ul class="refs">
<li><b>The direct backbone (what the post is after)</b> — <b>nonlinear quasimodes near a stable geodesic</b>:
Albin, Christianson, Marzuola & Thomann,
<a href="https://arxiv.org/abs/1103.3253"><i>Nonlinear quasimodes near elliptic periodic geodesics</i> (2011)</a>
— the NLS keeps a quasimode localized on the elliptic orbit (our self-trapping, §08); and
<b>rigorous blow-up on a curve</b>: Godet,
<a href="https://arxiv.org/abs/1204.3301"><i>Blow up on a curve for a NLS on Riemannian surfaces</i> (2012)</a>
— focusing NLS blowing up on a curve, at the log-log rate, on rotationally-symmetric surfaces (our
collapse, §08–10). Also Sulem & Sulem, <i>The Nonlinear Schrödinger Equation</i>; Kac,
<i>Can one hear the shape of a drum?</i> (spectral geometry).</li>
<li><b>Whispering-gallery modes</b> — Lord Rayleigh, <i>The Problem of the Whispering Gallery</i> (1910);
realized in ultra-high-Q optical microtoroid resonators: Armani, Kippenberg, Spillane & Vahala,
<a href="https://www.nature.com/articles/nature01371"><i>Nature</i> <b>421</b>, 925 (2003)</a>.</li>
<li><b>Quasimodes / Gaussian beams on stable geodesics</b> — Ralston,
<a href="https://projecteuclid.org/euclid.cmp/1103900389"><i>Comm. Math. Phys.</i> <b>51</b>, 219 (1976)</a>;
Babich & Buldyrev, <i>Asymptotic Methods in Short-Wavelength Diffraction Theory</i>.</li>
<li><b>NLS on compact manifolds</b> — Burq, Gérard & Tzvetkov,
<a href="https://muse.jhu.edu/article/168342"><i>Amer. J. Math.</i> <b>126</b>, 569 (2004)</a>.</li>
<li><b>The pseudodifferential / microlocal framework</b> (the master lens of §02) — M. Taylor,
<a href="https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2022/05/psidolect.pdf"><i>Lectures on
pseudodifferential operators</i></a> (principal symbol, the propagator as an FIO, propagation of
singularities along the geodesic/Hamiltonian flow, functional calculus); and the ΨDO calculus built from
<i>global Fourier series on the torus</i> — Ruzhansky & Turunen,
<a href="https://arxiv.org/abs/0805.2892"><i>Quantization of pseudo-differential operators on the
torus</i> (2010)</a> — the rigorous backing for the Fourier lens here.</li>
<li><b>Self-trapping, necklaces & collapse</b> — Soljačić, Sears & Segev,
<a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.81.4851"><i>Phys. Rev. Lett.</i> <b>81</b>, 4851 (1998)</a>;
mass-critical blow-up universality: Merle & Raphaël,
<a href="https://link.springer.com/article/10.1007/s00222-003-0346-z"><i>Invent. Math.</i> <b>156</b>, 565 (2004)</a>.</li>
<li><b>Attractive BEC analogue</b> (Gross–Pitaevskii; bright soliton; "Bosenova" collapse) — Donley et al.,
<a href="https://www.nature.com/articles/35085500"><i>Nature</i> <b>412</b>, 295 (2001)</a>.</li>
<li><b>Integrable geodesic flow on surfaces of revolution</b> — Clairaut's relation
<span class="eq">A²θ′ = const</span> (classical).</li>
<li><b>What the frontier prototypes connect to</b> — soliton microcombs: Kippenberg, Gaeta, Lipson &
Gorodetsky, <a href="https://www.science.org/doi/10.1126/science.aan8083"><i>Dissipative Kerr solitons
in optical microresonators</i>, Science <b>361</b> (2018)</a>; analog gravity: Barceló, Liberati &
Visser, <a href="https://link.springer.com/article/10.12942/lrr-2011-3"><i>Analogue Gravity</i>,
Living Rev. Relativity <b>14</b>, 3 (2011)</a>; analog cosmology in a BEC: Viermann et al.,
<a href="https://www.nature.com/articles/s41586-022-05313-9"><i>Quantum field simulator for dynamics in
curved spacetime</i>, Nature <b>611</b>, 260 (2022)</a>, with the Sakharov-oscillation observation of
Hung, Gurarie & Chin, <a href="https://www.science.org/doi/10.1126/science.1237557"><i>Science</i>
<b>341</b>, 1213 (2013)</a>; topological bands via Harper–Hofstadter: Hofstadter,
<a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.14.2239"><i>Phys. Rev. B</i> <b>14</b>,
2239 (1976)</a>; anomalous Floquet edge modes: Rudner, Lindner, Berg & Levin,
<a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.3.031005"><i>Phys. Rev. X</i> <b>3</b>,
031005 (2013)</a>; dynamical (Feshbach-resonance) management of collapse: Saito & Ueda,
<a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.90.040403"><i>Phys. Rev. Lett.</i>
<b>90</b>, 040403 (2003)</a>.</li>
<li><b>Controlling NLS finite-time collapse</b> — nonlinearity / Feshbach-resonance management: Saito &
Ueda (above); Abdullaev, Caputo, Kraenkel & Malomed, <i>Controlling collapse … by temporal modulation
of the scattering length</i> (Phys. Rev. A, 2003); Kevrekidis, Theocharis, Frantzeskakis & Malomed,
<i>Feshbach-resonance management for BECs</i> (Phys. Rev. Lett., 2003). The delay-vs-prevent distinction
made rigorous: Li, Ning & Zhao, <a href="https://arxiv.org/abs/2503.23716"><i>On blowup solution in
NLS under dispersion or nonlinearity management</i> (2025)</a>. Spatial control via nonlinear lattices:
Kartashov, Malomed & Torner, <a href="https://arxiv.org/abs/1010.2254"><i>Solitons in nonlinear
lattices</i>, Rev. Mod. Phys. <b>83</b>, 247 (2011)</a>. Geometry <i>localising</i> blow-up on a curve:
Godet (above).</li>
<li><b>Nonequilibrium & topological transport</b> — Kibble–Zurek defect scaling: Zurek,
<a href="https://www.nature.com/articles/317505a0"><i>Nature</i> <b>317</b>, 505 (1985)</a>, and the
review of del Campo & Zurek (<i>Int. J. Mod. Phys. A</i>, 2014); Thouless adiabatic charge pump:
Thouless, <a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.27.6083"><i>Phys. Rev. B</i>
<b>27</b>, 6083 (1983)</a>; rogue waves / Peregrine soliton: Peregrine,
<i>J. Austral. Math. Soc. B</i> <b>25</b>, 16 (1983); quantum chaos & level statistics: Bohigas,
Giannoni & Schmit, <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.52.1"><i>Phys.
Rev. Lett.</i> <b>52</b>, 1 (1984)</a>.</li>
<li><b>The real-device model (§13)</b> — whispering-gallery radiation-Q as tunnelling through the
centrifugal (bending) barrier: Marcatili, <i>Bends in optical dielectric guides</i>
(<i>Bell Syst. Tech. J.</i> <b>48</b>, 2103, 1969); ultra-high-Q silica microtoroids: Armani et al.
(above). Kerr-comb parametric threshold and its <span class="eq">1/Q²</span> scaling: Herr, Kippenberg
et al., <a href="https://www.science.org/doi/10.1126/science.aan8083"><i>Dissipative Kerr solitons in
optical microresonators</i>, Science <b>361</b> (2018)</a>; the mass-critical universality that makes the
nonlinear threshold geometry-blind: Merle & Raphaël (above).</li>
</ul>
</section>
<footer>
Generated from the solver in <code>nls_lumpy_torus.py</code> and the render/analysis scripts
(<code>render*.py</code>, <code>curvature_view.py</code>, <code>quasimode_ladder.py</code>,
<code>geodesic_phase_space.py</code>, <code>bifurcation.py</code>, <code>ring_1d.py</code>,
<code>threshold_scan.py</code>, <code>spacetime_carpet.py</code>, <code>defocusing.py</code>,
<code>hero.py</code>), and the frontier prototypes (<code>faraday.py</code>,
<code>expanding_torus.py</code>, <code>analog_horizon.py</code>, <code>dispersion_design.py</code>,
<code>lattice.py</code>, <code>topological.py</code>, <code>neural_operator.py</code>,
<code>dynamical_stabilization.py</code>, <code>collapse_arrest.py</code>, <code>floquet.py</code>,
<code>thouless.py</code>, <code>rogue.py</code>, <code>cubic_quintic.py</code>,
<code>quantum_chaos.py</code>, <code>kibble_zurek.py</code>, <code>hawking.py</code>), and the
agent-toolkit findings (<code>tunneling.py</code>, <code>curvature_universality.py</code>,
<code>sta_transport.py</code>, <code>dtc_horn.py</code>), and the beyond-surfaces-of-revolution extensions
(<code>nls_torus/mesh.py</code>, <code>mesh_genus2.py</code>, <code>genus2_tunneling.py</code>,
<code>genus2_collapse.py</code>, <code>horn_resonator.py</code>, <code>horn_hawking.py</code>,
<code>nonlinear_kerr_horn.py</code>), and the real-device model
(<code>microdisk_Q.py</code>), all driven by the
<code>nls_torus</code> library — see <code>docs/toolkit.md</code>. Every 3-D frame uses the crescent embedding
(<span class="eq">x</span> → toroidal, <span class="eq">θ</span> → poloidal).
</footer>
</div>
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