large-wave-effect

Rigorous study of the large-wave effect on periodic discrete chains: a localized impulse on a homogeneous one-dimensional lattice amplifies into a peak that grows with the number of nodes — a genuinely discrete resonance with no continuum counterpart.

One paper (paper/), in two parts built on a shared spectral core (the ring Laplacian L_N, Toeplitz/circulant methods):

• Part I — Linear theory (rigorous core). Amplitude A_N = sup_t max_j |u_j(t)| of the velocity Green's function on the ring; cyclotomic independence, the mod-4 ceiling criterion, the discrete Schrödinger contrast, spectral zeta, operator norms, statistics, and the dimensional threshold.
• Part II — Nonlinear extension. The discrete NLS (Peregrine soliton, modulational instability, rogue-wave statistics), discrete breathers and self-trapping, FPUT recurrence, and the route to chaos (Lyapunov exponent).

Every quantitative claim is reproduced by a standalone, version-controlled script.

Main results

Proved results are kept strictly separate from numerical evidence and the (sharply reduced) open problems.

Linear theory (Part I).

• Order law (theorem, all N). A_N ~ (1/π) ln N for every N — the central result, formerly a conjecture. The lower bound comes from a palindromic argument: the first φ(2N)/2 frequencies 2 sin(πr/N) are ℚ-independent, so a (1−o(1)) fraction of the ceiling can always be Kronecker-aligned. Precisely A_N = (1/π) ln N + O(ln ln ln N).
• Exact amplitude / saturation (theorem). A_N = U_N = (1/π) ln N + (1/π)(γ + ln(2/π)) − π/(72 N²) + ⋯ iff N is prime or a power of two; equivalently every integer frequency relation has coefficient-sum ≡ 0 (mod 4) (antipodal variant for even N). Via an explicit root-of-unity obstruction, {N : A_N = U_N} = {primes} ∪ {2^m} (density zero). The orbit-closure subtorus has dimension exactly φ(2N)/2. The theorem is cross-checked by exact Python/GAP scans (verify_classification_proof.py, exact/gap/scan_classification.g).
• Composite deficit. The order theorem gives the general upper envelope 0 ≤ U_N − A_N = O(ln ln ln N). The strict gap A_N < U_N for every composite non-power-of-two follows from the saturation classification; quantitative small-N gaps are certified by Lipschitz-grid and Lasserre moment–SOS checks. The sharper excess law U_N − A_N = (1/π) ln(m/φ(m)) + O(1) for odd part m remains open beyond prime-power odd part. The per-family limits defect_∞(m) = lim_a (U − A)(m · 2^a) are computed to 10⁻⁸ (0.10368…, 0.03824…) as numerical constants with no closed form.
• Discrete Schrödinger. A localized state is never amplified (unitarity), yet B_N = sup_t ‖e^{−it L_N}‖_{ℓ^∞ → ℓ^∞} = Θ(√N). Here liminf B_N/√N ≥ c_0/√2 = 0.861 is proved (c_0 = (2/π)^{3/2} · B(1/2, 3/4); attained numerically on even N), and the constant splits by parity — numerically even N → c_0/√2, odd N → β_odd = 0.928…, an elliptic-integral constant for which no elementary closed form was found — so limsup ≥ β_odd. For t ≤ N/2 the limiting ℓ¹ profile is the explicit elliptic-integral function F(2t/N) (one-copy term A(s) = ₂F₁ in closed form), and its maximizer max F = F(1) = β_odd is now fully proved — analytically on [1/2, 0.937] (Jensen, sharpened by the tangent to a concave h) and by validated interval arithmetic on [0.937, 1] (F′(s) > 0, worst case 0.051). So the t ≤ N/2 contribution to B_N is settled. For t > N/2 the whole N → ∞ reduction is now rigorous — joint equidistribution of the multi-copy Debye phases and its uniform-in-u transfer to the ℓ¹ constant F(s) (a mollified Koksma–Hlawka / cubic van der Corput estimate; the modulus has divergent Hardy–Krause variation) — leaving a single validated residual, the Gaussian-excess inequality E|Z| − E|G| ≤ K⋆ · Σ_2^{−3/2} Σ_ℓ a_ℓ⁴ (K⋆ = (h(1) − √π/2)/2) for the profile's balanced Debye configurations (the live set is an interval of consecutive integers, so the even/odd channel counts differ by ≤ 1; sharp at the two-copy pair — imbalanced configs violate the unrestricted bound) whose integrated majorant clears β_odd — and whose residual is now validated across the whole band (the θ-Taylor enclosure middle_far_point plus a 130-box rigorous-enclosure tiling, 0 failures), so limsup B_N/√N = β_odd holds modulo this one numerically-validated inequality. On the Dirichlet segment d_j^N ~ (4/π²) ln j for N+1 prime/2^m (the constant C = 4/π² is the published Myshkis–Filimonov 2003 value). Three large-wave laws on one Laplacian.
• Dimension, products & circulants (theorem). The large wave lives exactly at spectral dimension d_s = 1 (O(1) for d_s ≥ 2). A graph-agnostic order criterion plus the embedded ring spectrum give A = Θ(ln N) on every Cartesian product C_N □ H (ladders, prisms, tubes) and, beyond products, on the Möbius ladder C_{2N}({1, N}) via its embedded-ring branch. Multi-jump circulants C_N(S) (|S| ≥ 2) instead saturate (A_N = U_N) off explicit Conway–Jones collision congruences — the nested-radical frequencies destroy the ring's mod-4 obstruction; for the nearest case S = {1, 2} this is proved exactly (Besicovitch–Kummer: no subset product of the α_r is a square in ℚ(ζ_p)) for collision-free primes p ≤ 23 (exact/pari/kjump_kummer.gp). It is an infinite-time ceiling: the finite-time amplitude is Diophantine-limited (recurrence time exponential in N).

Nonlinear extension (Part II), focusing DNLS — now a rigorous chain. The linear large wave persists under weak nonlinearity (Duhamel); the modulational-instability band σ_Q = √(λ_Q (2γ A² − λ_Q)) is read off the L_N spectrum; the on-site breather exists (a finite-dimensional variational argument, no MacKay–Aubry continuation) and is stable (Vakhitov–Kolokolov / Grillakis– Shatah–Strauss), while the bond-centred breather is unstable. The self-trapping threshold is proved energy-sharp at the exact constant γP = 4 = ‖L_N‖: above it, full dispersal of the single-site seed is energetically forbidden (it would force ⟨L_N u, u⟩ → 2P − (γ/2) P² < 0). The remaining strongly nonlinear regime (saturation, rogue-wave focusing), FPUT recurrence and the route to chaos remain computational/exploratory.

Open problems (each reduced to a clean statement, with the proven part marked). (1) The sharp B_N upper bound. The t ≤ N/2 case is fully proved (max F = β_odd, analytic + validated interval arithmetic), and for t > N/2 the entire N → ∞ reduction is now rigorous (joint equidistribution of the Debye phases and its uniform-in-u transfer to F(s) — a mollified Koksma–Hlawka / cubic van der Corput estimate). What remains is a single validated inequality: the Gaussian-excess (expected-modulus) bound E|Z| − E|G| ≤ K⋆ · Σ_2^{−3/2} Σ_ℓ a_ℓ⁴ for the profile's balanced Debye configs (live set = consecutive integers ⇒ even/odd counts differ by ≤ 1; sharp at the two-copy pair — imbalanced configs violate it), whose integrated majorant clears β_odd. The continuum certificate is operational (verify_dagger_continuum.py). The equal-amplitude singleton zone near a_1 = a_3 ≈ 0.73 that the θ-ball middle could not reach (its rigorous radius blows up past t = 12; the true ratio there is only ≈ 0.88, so it was enclosure-limited, not tight) is now covered by a θ-Taylor + t-collocation enclosure (middle_far_point, validated against the reference in verify_middle_far.py): the previously-failing boxes certify (e.g. (0.70, 0.70) → 0.9915; the hardest boxes sit at the high-λ edge a_3 ≈ 0.61 and close under one refinement, worst leaf 0.9999). The full singleton-band cover is now validated across the whole band — the adaptive, node-parallel, checkpointed tiling run_dagger_campaign.py certified all 130 leaf boxes with 0 failures — so limsup B_N/√N = β_odd holds at the dagger certificate's validated-enclosure standard, leaving problem (1) reduced to upgrading that one numerically-validated inequality to a proof. (2) The excess lemma A_N ≤ L_pre + O(1) (equivalent to a rigorous growing deficit bound) — proved when the odd part of N is a prime power (N = 2p sharp at 1/(2N), N = 4p), and unconditional for any fixed ω(m) via the Kronecker sandwich |A_N − L_pre| ≤ U_N − L_pre = (1/π) ln(m/φ(m)). The open content is an absolute constant uniform as ω(m) → ∞. A proved building block is the per-relation bound (Lemma; verify_excess_decomposition.py): a single support-3 relation ω_c = ω_a + ω_b contributes excess g(b_a, b_b, b_c) ≤ b_c/2, from g(α, β, γ) ≤ (γ²/2)(1/α + 1/β) and 1/b_a + 1/b_b = 1/b_c. The obstruction is the uniform coupling of Θ(N) relations (comb generators + shared phases), the sup-side counterpart of McGehee–Pigno–Smith/Konyagin in the Chowla cosine-extremum circle (recently resolved in the opposite direction by Jin–Milojević–Tomon–Zhang / Bedert). (3) A rigorous dispersal proof on the proof gap γP ∈ [0.43, 4): the threshold 4 is now energy-sharp and dispersal is the numerically established behaviour up to ≈ 4, but the rigorous drop is proven only to 0.43. The leading-order drift now vanishes provably (along the linear flow the distance-2 current Im(ā_n a_{n+2}) ≡ 0, so the staggered momentum drifts only at O(γ²) — Maxima + Rocq StaggeredCurrent.v, verify_modified_energy_window.py); what remains is the O(γ²) remainder and the nonperturbative approach to 4. (4) Condition (B) of the order criterion for quasi-1D lattices without an embedded ring (the Möbius ladder and multi-jump circulants are now settled; the next-nearest-neighbour degeneracy law holds for all couplings g via Conway–Jones). Priority note (resolved): Filimonov's 1992 C. R. Acad. Sci. note has been obtained and compared on the ring — the rational-independence dichotomy (peaks for N prime/2^m, dependence for even composite, by his cyclotomic argument) and the unbounded ceiling are Filimonov's; the rate (1/π) ln N, the explicit constant, the mod-4 saturation classification, and the all-N rank φ(2N)/2 are new here.

Reproduce

Uses uv. Each numerics/verify_*.py is a PEP 723 standalone script (inline dependencies), so run it with uv run --script.

uv run pytest                                  # unit tests for the core package
uv run --script numerics/verify_theorem1.py    # any verify_*.py re-checks one claim
uv run --script numerics/make_figures.py       # regenerate paper/figures/*.pdf
cd paper && latexmk -pdf large-wave-effect.tex # build the paper (run twice to resolve refs)

Headline checks: verify_order_theorem.py (the order law + palindromic prefix lemma), verify_classification_proof.py (saturation classification), verify_bn_parity.py / verify_beta_odd.py / verify_bn_profile_max.py (the Schrödinger constant, its parity split, and the explicit t ≤ N/2 profile maximizer), verify_defect_certified.py (certified A_N < U_N), verify_excess_smallcases.py (excess lemma for prime-power odd part), verify_product_order.py and verify_kjump_order.py (C_N □ H, Möbius ladder, k-jump saturation), the DNLS chain verify_dnls_persistence.py / verify_dnls_mi.py / verify_dnls_breather_stability.py, and verify_selftrapping_transition.py (the energy-sharp γP = 4 threshold).

For the Schrödinger continuum cover: verify_middle_far.py validates the θ-Taylor enclosure that closes the a_1 = a_3 ≈ 0.73 zone, prove_box_parallel.py certifies a single box on all cores, and the full band is tiled by the adaptive, checkpointed campaign: uv run --script numerics/run_dagger_campaign.py --jobs 14 (writes each box to dagger_campaign.jsonl).

Exact-arithmetic + formal layer (Fedora: dnf install gap pari-gp fricas rocq-prover). The linear-theory arithmetic is reproduced floating-point-free in GAP and PARI/GP, symbolically in FriCAS, and the mod-4 criterion is machine-checked in Rocq — see exact/ and formal/rocq/. One driver runs the lot and cross-checks against the Python core (missing tools are skipped, not failed):

uv run --script numerics/verify_exact_layer.py   # GAP + PARI + Rocq + Python cross-check
gap -q --nointeract exact/gap/scan_classification.g   # finite exact cross-check to N<=800
rocq compile formal/rocq/Mod4Criterion.v              # machine-checked criterion certificate

Layout

src/large_wave_effect/   shared spectral core (ring, cyclotomic, Schrödinger)
numerics/                verify_*.py (one per claim) + make_figures.py
exact/                   GAP / PARI / FriCAS exact + symbolic cross-checks (gap/, pari/, fricas/)
formal/rocq/             Rocq machine-checked mod-4 criterion certificate
tests/                   pytest suite for the core package
paper/
  large-wave-effect.tex  root: preamble, front matter, \part dividers, bibliography
  sections/linear.tex    Part I  — linear theory
  sections/nonlinear.tex Part II — nonlinear extension
  figures/               generated by make_figures.py
  large-wave-effect.pdf  compiled paper

Provenance

This work continues the author's 2015 undergraduate diploma on the large-wave effect; the original C simulation that first observed the peaks is not included here.

License

Apache License 2.0 — code, paper text, and figures. See LICENSE and NOTICE.

Citation

Ilia Gradina, Large-wave amplification on periodic discrete lattices: linear theory and a nonlinear extension (2026). DOI: 10.5281/zenodo.20795861 (concept DOI — always resolves to the latest version). See CITATION.cff.