This paper develops and verifies Hirota bilinear forms for a broad catalogue of classical integrable evolution equations by deriving each transformation from explicit differential identities. The presentation separates scalar one-tau equations from coupled tau systems, hierarchy formulations, lattice equations, and potential forms, thereby preventing unsupported use of scalar interaction formulas. For the general fifth-order Korteweg–de Vries (KdV) family, coefficient matching under $u = c(\ln f)_{xx}$ distinguishes the Sawada–Kotera/Caudrey–Dodd–Gibbon, Kaup–Kupershmidt, and Lax branches. The first admits a scalar bilinear equation; the second requires an auxiliary tau function to represent the $q_{xxx}^{2}$ term; and the third requires an auxiliary hierarchy time. We also establish the normalization used for Kadomtsev–Petviashvili (KP)-type equations, derive the full spectral-vector two-soliton coefficient for multidimensional scalar bilinear equations, and clarify why one- and two-soliton solutions do not by themselves prove $N$-soliton integrability for seventh-order KdV-type models. Coupled bilinear structures for modified KdV (mKdV), Gardner, nonlinear Schrödinger (NLS), sine–Gordon, and Manakov systems are treated separately, with representative soliton and breather constructions. Finally, the local clock-fractional reparametrization $T = t^\alpha/\Gamma(1+\alpha)$ is introduced; it preserves the ordinary Hirota algebra and generates exact clock-fractional one-soliton solutions. The resulting catalogue provides a reproducible framework for symbolic verification and normalization control in soliton analysis.
Citation: Alvaro H. Salas, César A. Gómez S., Simeón Casanova Trujillo. Hirota bilinear forms and multisoliton structures: A catalogue of classical integrable evolution equations[J]. AIMS Mathematics, 2026, 11(8): 24582-24649. doi: 10.3934/math.2026991
This paper develops and verifies Hirota bilinear forms for a broad catalogue of classical integrable evolution equations by deriving each transformation from explicit differential identities. The presentation separates scalar one-tau equations from coupled tau systems, hierarchy formulations, lattice equations, and potential forms, thereby preventing unsupported use of scalar interaction formulas. For the general fifth-order Korteweg–de Vries (KdV) family, coefficient matching under $u = c(\ln f)_{xx}$ distinguishes the Sawada–Kotera/Caudrey–Dodd–Gibbon, Kaup–Kupershmidt, and Lax branches. The first admits a scalar bilinear equation; the second requires an auxiliary tau function to represent the $q_{xxx}^{2}$ term; and the third requires an auxiliary hierarchy time. We also establish the normalization used for Kadomtsev–Petviashvili (KP)-type equations, derive the full spectral-vector two-soliton coefficient for multidimensional scalar bilinear equations, and clarify why one- and two-soliton solutions do not by themselves prove $N$-soliton integrability for seventh-order KdV-type models. Coupled bilinear structures for modified KdV (mKdV), Gardner, nonlinear Schrödinger (NLS), sine–Gordon, and Manakov systems are treated separately, with representative soliton and breather constructions. Finally, the local clock-fractional reparametrization $T = t^\alpha/\Gamma(1+\alpha)$ is introduced; it preserves the ordinary Hirota algebra and generates exact clock-fractional one-soliton solutions. The resulting catalogue provides a reproducible framework for symbolic verification and normalization control in soliton analysis.
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Alvaro H. Salas, César A. Gómez S., Simeón Casanova Trujillo. Hirota bilinear forms and multisoliton structures: A catalogue of classical integrable evolution equations[J]. AIMS Mathematics, 2026, 11(8): 24582-24649. doi: 10.3934/math.2026991
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