On generalized Fourier transform for Kaup-Kupershmidt type equations.(English)Zbl 1203.37113

Summary: 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.

MSC:

 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures