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

Valchev, Tihomir

J. Geom. Symmetry Phys. 19, 73-86 (2010).

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

Keywords:

inverse scattering method; Fourier transform; Zakharov-Shabat system; Lie algebra; Kaup-Kupershmidt equation