The Kac-Wakimoto Equation Is Not Integrable

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.

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