2 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. It is well known that many physically important integrable partial differential equations belong to the AKNS hierarchy [1]. Up to now most attention was paid to its positive flows, where both 2×2 matrices U and V have matrix elements polynomial in the spectral parameter λ what leads to the recursive structure of the hierarchy so that subsequent flows are connected by the recursion operator (see, e.g. [2]). However, negative flows in the AKNS hierarchy have not been considered systematically enough, though the sineGordon equation and its connection with the mKdV hierarchy has been a 1 recurrent theme in the soliton literature (see, e.g., [3]–[7]). Another example of well-studied negative flow is provided by the self-induced transparency (SIT) equations [8, 9] which appeared also in different forms and contexts as Pohlmeyer-Lund-Regge equations [10]–[12]. Their role as a symmetry flow in the AKNS hierarchy has been recently considered in [13] in framework of loop algebra approach to the integrable equations. Here we remark that SIT equations can be interpreted as a generating function of positive and negative flows in the AKNS hierarchy. This leads to simple method of obtaining the integrable equations of the hierarchy and corresponding Lax pairs. Generalization of this point of view leads to new useful connections between integrable equations belonging to as positive so negative flows in the AKNS hierarchy. The AKNS hierarchy [1] is based on the Zakharov and Shabat [14] spectral problem

Cite this paper

@inproceedings{Kamchatnov20082OG, title={2 On generating functions in the AKNS hierarchy}, author={A. M. Kamchatnov}, year={2008} }