On the Kaup-Kupershmidt Equation: On the Polarization of Gravitational Waves

Abstract

We regard a cubic spectral problem associated with the Kaup-Kupershmidt equation. For this spectral problem we prove a completeness of its “squared” solutions and derive the completeness relations which they satisfy. The spectral problem under consideration can be naturally viewed as a $\mathbb{Z}_{3}$-reduced Zakharov-Shabat problem related to the algebra $\mathfrak{sl}(3,\mathbb{C})$. This observation is crucial for our considerations.