bibkey: gurses2025higher authors: Metin Gürses, Aslı Pekcan year: 2025 title: “Higher order Hirota bilinear forms” doi: 10.48550/arXiv.2511.18466 url: https://arxiv.org/abs/2511.18466v1 claim: “The paper studies Hirota bilinear forms with higher powers of the D-operators, proves that D_x(D_x^3 + a1 D_t + a2 D_y)^(2k+1){f.f} = 0 has three-soliton solutions for every k, and conjectures that it has four-soliton solutions only for k = 0.” strata_touched:

• D5/S3/FluidDynamics/Solitons/GursesPekcanFourSoliton license: citation-only triage: anchor

Higher order Hirota bilinear forms

Gürses and Pekcan test Hirota bilinear equations P(D){f·f} = 0 for multi-soliton solutions with Hietarinta’s conditions. For soliton parameters p_i = (k_i, ω_i, l_i), the four-soliton condition of their introduction is an eight-term identity in the values P(±p₁ ± p₂ ± p₃ ± p₄) and P(p_i ± p_j), to hold on the dispersion relation. In their section on (2+1)-dimensional forms they take

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

prove that it possesses three-soliton solutions for any constants (α₁,α₂) ≠ (0,0) and every k, with the dispersion relation ω_i = −(k_i³+α₂l_i)/α₁, and in Remark 3 record the four-soliton solution for k = 0 and the four-soliton condition for k = 1,

−502096953744 k₁¹¹k₂¹¹k₃¹¹k₄¹¹ ∏_{i<j}(k_i−k_j)²(k_i+k_j)² (k₁²+k₂²+k₃²+k₄²) = 0.

They then write: “We have the similar results for k=2 and k=3 hence we conjecture that the following lemma is valid for all k”, and state

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.

The abstract repeats the conjecture. In the printed four-soliton condition the last factor of the fifth term is P(p₁−p₂+p₃−p₄), the same as that of the seventh term; the pair factors of the fifth term and Hietarinta’s form give P(p₁+p₂+p₃−p₄).

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2511.18466
• URL: https://arxiv.org/abs/2511.18466v1
• Version and location: arXiv:2511.18466v1 (2025-11-23), source file HIROTA-NOV10.tex: the four-soliton condition (4SC) in the introduction; the lemma on three-soliton solutions, Remark 3 and the conjectured lemma in the subsection “Equations of the form D_x(D_x^3+α₁D_t+α₂D_y)^{2k+1}{f·f}=0”. The journal version, J. Phys.: Conf. Ser. 3264 (2026) 012016, DOI 10.1088/1742-6596/3264/1/012016, was not read.