Functional representation of the negative AKNS hierarchy

This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa’s shifts, which contains all equations of the hierarchy. Using the obtained functional representation we convert the nonlocal equations of the negative subhierarchy into local systems of higher order, derive the generating function of the conservation laws and the N-dark-soliton solutions for the extended AKNS hierarchy. As an additional result we obtain the functional representation of the Landau-Lifshitz hierarchy.

💡 Research Summary

The paper addresses the long‑standing problem of incorporating the negative flows of the AKNS hierarchy into a unified, functional framework. While the positive (classical) AKNS flows—generated by the infinite set of times (t_{1}, t_{2},\dots)—are well understood in terms of Lax pairs, Hamiltonian structures and τ‑functions, the negative flows (t_{-1}, t_{-2},\dots) have traditionally been described by non‑local equations involving inverse derivatives such as (\partial_{x}^{-1}). This non‑locality makes both analytical treatment and the construction of explicit solutions cumbersome.

The authors overcome this difficulty by employing Miwa’s shift formalism. They replace the infinite set of continuous times by a discrete set of complex parameters (\zeta) and introduce two shift operators, (\Delta_{+}(\zeta)) for the positive hierarchy and (\Delta_{-}(\zeta)) for the negative hierarchy. Acting with these operators on the basic AKNS fields (q(x,\mathbf{t})) and (r(x,\mathbf{t})) yields a compact pair of functional equations
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