On generating functions in the AKNS hierarchy

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. Generating function of Whitham modulation equations in the AKNS hierarchy is obtained. © 2002 Elsevier Science B.V. All rights reserved. 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 x 2 matrices U and V have matrix elements polynomial in the spectral parameter X 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 sine-Gordon equation and its connection with the mKdV hierarchy has been a 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 the 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. The AKNS hierarchy [1] is based on the Zakharov and Shabat [14] spectral problem Let us consider the first negative flow with the pole at X — t, in the complex plane of the spectral parameter X: 1 (a b \ />, V," (2) Corresponding author. E-mail address: [email protected] (A.M. Kamchatnov). 0375-9601/02/S see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: 80375-9601(02)00935-0 270 A.M. Kamchatnov, M. V Pavlov /Physics Letters A 301 (2002) 269-274 The compatibility condition of systems (1) and (2) yields equations ax=cq-br, bx-2^b = -2aq, cx+2$c-2ar, qr = -2b, rT=2c, (3) which can be reduced to well-known SIT equations with £ playing the role of the "detuning" parameter [8,9]. Then, introduction of a hierarchy of times tn labelled by inverse powers of f , «=0 and of expansions of a, b, c in inverse powers of f , "cn, (5) n=0 n=0 n=0 together with geometric series expansion of 1/(A — £) in powers of A./f leads at once to well-known recurrence relations for the positive AKNS hierarchy [2],