The Kac-Wakimoto Equation Is Not Integrable

TL;DR

This paper investigates a complex nonlinear wave equation linked to exceptional Lie algebra and demonstrates it is not integrable by Hirota's method, as it lacks a three-soliton solution despite having simpler soliton solutions.

Contribution

The study provides the first proof that the Kac-Wakimoto equation is not Hirota integrable, clarifying its soliton solution structure.

Findings

The equation admits one- and two-soliton solutions.

It does not have a three-soliton solution.

The equation is not Hirota integrable.

Abstract

We study the $(3+1)$-dimensional eight-order nonlinear wave equation associated with the principal representation of the exceptional affine Lie algebra $E_6^{(1)}$, which was constructed by Kac and Wakimoto and stated that $N$-soliton solution of the equation can be formulated. We show that the equation is not Hirota integrable since it does not have three-soliton solution, even it has one- and two-soliton solutions.