Gürses and Pekcan's four-soliton conjecture for higher order Hirota forms

Gürses and Pekcan study the Hirota bilinear equations

D_x(D_x³+α₁D_t+α₂D_y)^{2k+1}{f·f}=0, k=0,1,2,…

For soliton parameters p_i = (k_i, ω_i, l_i) the operator has the polynomial P(p) = k(k³+α₁ω+α₂l)^{2k+1}, and the dispersion relation is ω_i = −(k_i³+α₂l_i)/α₁. The paper proves three-soliton solutions for every k and (α₁,α₂) ≠ (0,0), records the four-soliton condition for k = 1, and states, after "we conjecture that the following lemma is valid for all k":

The equation (bilinearProb2) has four-soliton solution only for k=0. For k≥1, it does not satisfy the four-soliton solution condition (4SC) directly.

Issue #10059 fixes the readings: the dispersion relation is k³ + α₁ω + α₂l = 0, which also covers α₁ = 0; the four-soliton condition is the eight-term display of the source's introduction with the last factor of the fifth term read as P(p₁+p₂+p₃−p₄) (the source prints P(p₁−p₂+p₃−p₄), the last factor of the seventh term, and with that factor the condition fails already for k = 0); "has four-soliton solution for k = 0" is read as the condition vanishing at all parameters satisfying the dispersion relation, and "does not satisfy (4SC) directly" as the existence, for each k ≥ 1, of such parameters at which it does not vanish.

The frozen declaration D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.result proves the conjectured lemma for all real (α₁,α₂) ≠ (0,0): the four-soliton condition vanishes identically on the dispersion relation for k = 0, and for every k ≥ 1 it is negative at the wave numbers (1, 3, 4, 5). The k = 0 equation is a Kadomtsev–Petviashvili-type model of shallow-water waves; the result says that raising its dispersive operator to an odd power 2k + 1 ≥ 3 keeps three-soliton solutions but destroys Hietarinta's four-soliton test.

Issue #10059 preregisters this published conjecture and its literature check. MathDB searches for "four-soliton", "Hirota bilinear" and "Gurses" return no entry for it; Crossref records no citation of the journal version; the authors' three later arXiv papers treat nonlocal reductions and exact solutions of AKNS, NLS and MKdV systems; an arXiv abstract search for four-soliton Hirota conditions returns only the source among recent papers. The source checks k = 1, 2, 3.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent proof.

On the dispersion relation α₁ω_i + α₂l_i = −k_i³, and the map (k, ω, l) ↦ (k, α₁ω + α₂l) is additive, so every value of P at a signed sum of parameters equals x(x³ + e)^m with x the signed sum of the wave numbers and e minus the signed sum of their cubes. For m = 1 the condition becomes a polynomial identity in k₁, …, k₄, which holds.

For m = 2k + 1 ≥ 3 take k = (1, 3, 4, 5) with ω_i = −k_i³/α₁, l_i = 0 (or ω_i = 0, l_i = −k_i³/α₂ when α₁ = 0). Each of the eight terms is a product of seven factors x(x³+e)^m, and the condition equals 48 · 1360488960000^m · F(m) with

F(m) = 13·11^m − 55·(−31)^m + 392·56^m + 525·21^m + 162·18^m − 350·14^m − 567·63^m − 120·(−24)^m.

For odd m the positive terms total at most 1267·56^m, and 1267·56^m < 567·63^m for m = 7 and hence, multiplying by 56 < 63, for every m ≥ 7; F(3) = −64774080 and F(5) = −342030427200. So F(m) < 0 for every odd m ≥ 3.

A refutation of the k = 0 part would be parameters on the dispersion relation with a nonzero condition; the polynomial identity excludes them. A refutation of the k ≥ 1 part would need the condition to vanish at every point of the dispersion relation for some k ≥ 1; the witness (1, 3, 4, 5) gives a negative value for every such k.

Exact rational evaluation with the three-component polynomial P: the corrected condition vanishes for m = 1 at 30 random points of the dispersion relation, including points with α₁ = 0; for m = 3 it equals the source's printed k = 1 expression at k = (1,3,4,5), (2,−3,5,7) and (1,2,6,−9); the condition as printed is −380849837506560000 at m = 1, k = (1,3,4,5), α = (1,0), while the corrected one is 0; F(m) < 0 for every odd m with 3 ≤ m ≤ 401 (F(1) = 0). The Lean proof has only the standard axiom closure propext, Classical.choice, and Quot.sound. These finite checks support the reading but do not establish the universal theorem.

First-tier external named open problem: Gürses and Pekcan, arXiv:2511.18466 (2025), the conjectured lemma after Remark 3 in the subsection on D_x(D_x³+α₁D_t+α₂D_y)^{2k+1}{f·f}=0, preregistered in issue #10059. Resolution: proved, for the four-soliton condition with the fifth term corrected.

The public surface is exactly hirotaP, dispersion, fourSC, claim, and result. This is a uniform theorem for every k, not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies.

The journal version, J. Phys.: Conf. Ser. 3264 (2026) 012016, was not read; whether it keeps the conjecture and the printed fifth term is not known. Semantic Scholar was rate-limited during the check. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The Lean kernel does not authenticate the external source or its version history. The reading of "has four-soliton solution" as Hietarinta's four-soliton condition follows the source's own use; the existence of actual four-soliton solutions for k = 0 is not formalized here beyond that condition.