On the elliptic null-phase solutions of the Kulish–Sklyanin model

We consider Kulish–Sklyanin model (KSM) which is a three-component nonlinear Schrödinger system. Using Dubrovin’s method we derive recurrent relations which allow us to derive all equations in the relevant hierarchy. Thus we derive the first two equations of the Kulish–Sklyanin hierarchy and use them to construct multi-phase solutions of KSM. To do this we first express the monodromy matrix M of the Lax operator L in terms of an elliptic function v(t). In order to derive the solutions of the hierarchies we calculate the corresponding Wronskians which allow us to find also the spectral curves. An important role here plays the matrix Jn in M. Here we have two possibilities: Jn can be either generic block-diagonal or simply diagonal matrix. Thus we are able to derive two classes of solutions of the KSM and demonstrate principal differences between KSM and the other well known integrable models. The first one is that the multi-phase solutions of KSM are more involved than the corresponding solutions of the other hierarchies. The second one is that the algebraic genus of the spectral curves does not always coincide with the number of the phases.