On generating functions in the AKNS hierarchy

Abstract

It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leads to other hierarchies of integrable equations. In particular, it is shown that stimulated Raman scattering equations generate the hierarchy of flows which include the Heisenberg model equations. This observation reveals some new relationships between known integrable equations and permits one to construct their new physically important combinations. Reductions of the AKNS hierarchy to ones with complex conjugate and real dependent variables are also discussed and the corresponding generating functions of positive and negative flows are found. Generating function of Whitham modulation equations in the AKNS hierarchy is obtained.