The General Structure of Trilinear Equations

TL;DR

This paper explores the extension of Hirota's bilinear formalism to trilinear equations in integrable systems, revealing a universal kernel structure in the Ernst equations related to Einstein's theory.

Contribution

It introduces a general trilinear kernel criterion and demonstrates its application to stationary axisymmetric Einstein equations and Tomimatsu--Sato solutions, highlighting a universal structure.

Findings

The cubic sector of the Ernst equation decomposes into a YTSF-type trilinear kernel.

The $ ext{delta}=3$ Tomimatsu--Sato solution shares the same trilinear kernel as the $ ext{delta}=2$ case.

The trilinear kernel may govern the highest-derivative sector of the Ernst system.

Abstract

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 Pl\"ucker 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 $\delta=3$ solution possesses the same trilinear kernel structure as the $\delta=2$ 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.