The General Structure of Trilinear Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914542135541760 |
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| author | Fukuyama, Takeshi |
| author_facet | Fukuyama, Takeshi |
| contents | We investigate trilinear structures as a natural extension of the Hirota bilinear formalism in integrable systems. While bilinear equations are associated with Grassmannian geometry and Plücker relations, trilinear equations suggest a higher algebraic structure involving three-slot couplings of tau functions.
Focusing on the stationary axisymmetric Einstein equations, we show that when the Ernst potential is written in a tau-ratio form, the nonlinear equation decomposes into a cubic sector containing all second-derivative terms and a quartic gradient envelope. The cubic sector is identified with a YTSF-type trilinear kernel.
We formulate a general trilinear kernel criterion and apply it to the Tomimatsu--Sato solutions. In particular, we demonstrate that the $δ=3$ solution possesses the same trilinear kernel structure as the $δ=2$ case, with a universal normalization up to a constant factor.
These results suggest that the trilinear kernel represents a universal structure governing the highest-derivative sector of the Ernst system, providing a new perspective on integrability beyond the bilinear hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The General Structure of Trilinear Equations Fukuyama, Takeshi Exactly Solvable and Integrable Systems General Relativity and Quantum Cosmology We investigate trilinear structures as a natural extension of the Hirota bilinear formalism in integrable systems. While bilinear equations are associated with Grassmannian geometry and Plücker relations, trilinear equations suggest a higher algebraic structure involving three-slot couplings of tau functions. Focusing on the stationary axisymmetric Einstein equations, we show that when the Ernst potential is written in a tau-ratio form, the nonlinear equation decomposes into a cubic sector containing all second-derivative terms and a quartic gradient envelope. The cubic sector is identified with a YTSF-type trilinear kernel. We formulate a general trilinear kernel criterion and apply it to the Tomimatsu--Sato solutions. In particular, we demonstrate that the $δ=3$ solution possesses the same trilinear kernel structure as the $δ=2$ case, with a universal normalization up to a constant factor. These results suggest that the trilinear kernel represents a universal structure governing the highest-derivative sector of the Ernst system, providing a new perspective on integrability beyond the bilinear hierarchy. |
| title | The General Structure of Trilinear Equations |
| topic | Exactly Solvable and Integrable Systems General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2605.05624 |