An integrable hierarchy including the AKNS hierarchy and its strict version (Q681947)

Integrable hierarchies can often be described as evolution equations of deformed generators of a commutative subalgebra of a loop Lie algebra. The Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and its strict version are examples of such hierarchies, as described by the author in [Ad. Math. Phys. 2016, Article ID 3649205, 10 p. (2016; Zbl 1369.37073)]. The relevant Lie algebra is the loop space \({\mathrm{sl}}_2(R)[z,z^{-1}]\) with at most a pole at infinity, where \(R\) is a suitable complex commutative algebra: \[ {\mathrm{sl}}_2(R)[z,z^{-1}] = \Big\{ \sum_{i=- \infty}^N X_i z^i, \quad X_i \in {\mathrm{sl}}_2(R) \Big\}. \] A commutative complex Lie subalgebra of \({\mathrm{sl}}_2(R)[z,z^{-1}]\) is deformed to give rise to Lax equations of the AKNS hierarchy. A similar procedure starting with a different subalgebra gives rise to Lax equations for the strict AKNS hierarchy. The aim of the article is to merge these two systems into a single general integrable hierarchy. It is shown that the combined Lax equations form a compatible set, which means that the projections of products of generators that occur in the Lax equations satisfy a set of zero-curvature relations. These zero-curvature relations also suffice to obtain the Lax equation for both generators. The linearization of the system is described and some special elements, called wave matrices, are characterized. A construction yielding a collection of solutions of the combined AKNS hierarchy is presented.