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Four-soliton solutions of D_x(D_x^3 + a1 D_t + a2 D_y)^(2k+1) exist only for k = 0

Abstract

For the Hirota bilinear equation D_x(D_x^3 + a1 D_t + a2 D_y)^(2k+1){f.f} = 0, the four-soliton condition holds identically on the dispersion relation when k = 0 and fails at the wave numbers 1, 3, 4, 5 for every k at least one.

Definition 1.1 (The polynomial of the bilinear operator).

Formalization. D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.hirotaP (✓ std3).

Citation. Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Commentary.

For the soliton parameters p = (k, omega, l) of an exponential exp(kx + omega t + l y), the operator D_x(D_x^3 + a1 D_t + a2 D_y)^m has the polynomial k (k^3 + a1 omega + a2 l)^m.

Definition 1.2 (The dispersion relation).

Formalization. D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.dispersion (✓ std3).

Citation. Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Commentary.

The source writes omega = -(k^3 + a2 l)/a1 for a1 nonzero; the relation k^3 + a1 omega + a2 l = 0 is the same condition and also covers a1 = 0 with a2 nonzero.

Definition 1.3 (The four-soliton condition).

Formalization. D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.fourSC (✓ std3).

Citation. Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Commentary.

The eight terms of the source’s display, Hietarinta’s form with the first sign fixed, for any function P of the parameters and parameters p_1, …, p_4. The last factor of the fifth term is P(p_1 + p_2 + p_3 - p_4): the source prints P(p_1 - p_2 + p_3 - p_4), the last factor of the seventh term, while the pair factors of the fifth term belong to the sign pattern (+, +, +, -).

Definition 1.4 (The conjectured lemma of Gurses and Pekcan).

Formalization. D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.claim (✓ std3).

Citation. Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Commentary.

The first conjunct says that for k = 0 the four-soliton condition vanishes at all parameters satisfying the dispersion relation. The second says that for every k at least one there are such parameters at which it does not vanish, so the condition is not satisfied directly.

Theorem 1.5 (The four-soliton condition holds only for k = 0).

Proof. Machine-checked in Lean as D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.result (✓ std3). ∎

Resolves. Problems/gurses-pekcan-2025-higher-hirota-four-soliton (proved) by D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton.result.

Source. Repository-derived.

Acknowledgement. Metin Gürses, Aslı Pekcan (2025). Higher order Hirota bilinear forms. DOI: 10.48550/arXiv.2511.18466. URL: https://arxiv.org/abs/2511.18466v1.

Commentary.

On the dispersion relation a1 omega + a2 l = -k^3, and the map (k, omega, l) to (k, a1 omega + a2 l) is additive, so every value of P at a signed sum of parameters equals x (x^3 + 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 the four wave numbers, which holds. For m = 2k + 1 with k at least one, the wave numbers 1, 3, 4, 5 with omega_i = -k_i^3/a1 and l_i = 0 (or omega_i = 0 and l_i = -k_i^3/a2 when a1 = 0) satisfy the dispersion relation, and each of the eight terms is a product of seven factors x (x^3 + e)^m, which collects into 48 g^m F(m) with g = 1360488960000 and 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 of F(m) are at most 1267 56^m, and 1267 56^m < 567 63^m from m = 7 on by induction; F(3) and F(5) are negative by direct computation. So the condition is negative for every k at least one.

References