On the Kaup-Kupershmidt equation. Completeness relations for the squared solutions. (English) Zbl 1193.35200

Valchev, Tihomir

Mladenov, Ivaïlo M. (ed.), Proceedings of the 9th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 8–13, 2007. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-42-4/pbk). 308-319 (2008).

Summary: 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.
For the entire collection see [Zbl 1154.17001].

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
22E70	Applications of Lie groups to the sciences; explicit representations
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
35P05	General topics in linear spectral theory for PDEs

Keywords:

Kaup-Kupershmidt equation; Zakharov-Shabat problem; Lie algebras; spectral problem