Integrability of one bilinear equation: singularity analysis and dimension

TL;DR

This paper investigates the integrability of a complex four-dimensional bilinear equation using singularity analysis, revealing its reduction to a three-dimensional form equivalent to the BKP equation when certain conditions are met.

Contribution

It demonstrates that the integrability of a high-dimensional bilinear equation is linked to its reduction to a known integrable form, the BKP equation, via singularity analysis.

Findings

The equation passes the Painlevé test in specific coefficient cases.

The equation reduces to a three-dimensional form equivalent to the BKP equation.

Integrability is connected to the effective dimensional reduction.

Abstract

The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra $D_4^{(1)}$ is studied by means of the singularity analysis. This equation is shown to pass the Painlev\'{e} test in three distinct cases of its coefficients, exactly when the equation is effectively a three-dimensional one, equivalent to the BKP equation.