On Generalized Fourier Transform for Kaup-Kupershmidt Type Equations

Abstract

We develop the Fourier transform interpretation of the inverse scattering method for nonlinear integrable evolution equations associated with a $\mathbb{Z}_3$ reduced Zakharov-Shabat system for the Lie algebra $\mathfrak{sl}(3,\mathbb{C})$. A simple representative of this integrable hierarchy is the well-known Kaup-Kupershmidt equation. Our results admit a natural extention for nonlinear equations connected to a deeply reduced Zakharov-Shabat system related to an arbitrary simple Lie algebra.