Trilinear Kernel Structure and Its Gravitational Realization

TL;DR

This paper reveals the fundamental role of trilinear kernels in integrable hierarchies and stationary gravity, showing how they generate entire solution spaces and connect to known gravitational equations like Ernst's.

Contribution

It demonstrates that the Yu--Toda--Fukuyama trilinear equation acts as a universal kernel for multidimensional hierarchies and relates stationary axisymmetric gravity to a projective realization of this kernel.

Findings

YTF trilinear equation is a universal generator of hierarchies

Stationary gravity corresponds to a projective realization of the kernel

Gravitational solutions like Ernst and Tomimatsu--Sato are derived from the kernel

Abstract

We clarify the structural role of trilinear kernels in multidimensional integrable hierarchies and in stationary axisymmetric gravity. The Yu--Toda--Fukuyama (YTF) trilinear equation of Ref.~\cite{YuTodaSasaFukuyama:1998hierarchy} is shown to represent not a particular evolution equation but a universal kernel that generates the entire $(3+1)$--dimensional hierarchy by selecting commuting flows. The frequently quoted trilinear equation of Ref.~\cite{YTSF1998} is identified as one such flow of this kernel. We further show that stationary axisymmetric gravity corresponds to a projective realization of the YTF kernel rather than to any single flow. Imposing $\GL(2)$ covariance and homogeneity on the kernel leads uniquely to a gravitational trilinear kernel $\mathcal{Y}(\tau_0,\tau_1)$, whose vanishing reproduces the Ernst equation. The Tomimatsu--Sato family \cite{Tomimatsu1972} and related bilinear solutions are shown to arise as degenerate submanifolds of this projected trilinear structure, in agreement with the multilinear analysis of Ref.~\cite{Fukuyama:2025TS}. These results establish a unified structural framework linking multidimensional trilinear integrability, stationary gravity, and bilinear solution sectors, and clarify why trilinear kernels are both necessary and sufficient for describing soliton dynamics with projective geometry.