The \(n \times n\) KdV hierarchy. (English) Zbl 1251.37070

The authors construct two new soliton hierarchies that are generalisations of the KdV hierarchy, which they call the \(n \times n\) KdV hierarchy and the \(2n \times 2n\) KdV-II hierarchy. These hierarchies are new restrictions of the AKNS \(n \times n\) hierarchy obtained from two unusual splittings of the loop algebra. The splittings arise from automorphisms of the loop algebra rather than from automorphisms of \(\mathrm{sl}(n,\mathbb{C})\). Since the construction is based on a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and Bäcklund transformations for these new systems.
The method how to generate a hierarchy of commuting flows from a Lie algebra splitting is reviewed and basic examples of the theory are presented, which include the AKNS hierarchy, the matrix NLS hierarchy, the Kuperschmidt-Wilson hierarchy, and the \(n \times n\) mKdV hierarchy.


37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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)
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