AKNS hierarchy coupled with Tyurin parameters

An elliptic analogue of the ordinary AKNS hierarchy is constructed. This gives an example of Krichever’s construction of Lax and zero-curvature equations on an algebraic curve. In addition to the usual AKNS fields, this hierarchy contains the so called Tyurin parameters as dynamical variables. The construction of the hierarchy resembles that of the ordinary AKNS hierarchy. A half of the Tyurin parameters appear in the matrices of the zero-curvature equations as “movable poles”; the other half are related to a factorized form of the residue of these matrices at these poles. Two distinct solutions of the auxiliary linear system form a Riemann-Hilbert pair, but unlike the case of the ordinary AKNS system, these pair have degeneration points. The Riemann-Hilbert problem can be translated to the language of an infinite dimensional Grassmann variety. This reveals a connection with Sato’s approach to soliton equations. arXiv:nlin.SI/0307030