Functional representation of the negative DNLS hierarchy

This paper is devoted to the negative flows of the derivative nonlinear Schr"odinger hierarchy (DNLSH). The main result of this work is the functional representation of the extended DNLSH, 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 ones of higher order, derive the generating function of the conservation laws and the N-soliton solutions for the extended DNLSH under non-vanishing boundary conditions.

💡 Research Summary

The paper investigates the derivative nonlinear Schrödinger (DNLS) hierarchy with a focus on its negative flows, and presents a unified functional representation that incorporates both the classical (positive) and the negative members of the hierarchy. The authors adopt Miwa’s shift operators to encode an infinite set of evolution variables ((t_{1},t_{2},\dots,\bar t_{1},\bar t_{2},\dots)) into a single functional framework. By defining shift operators (T_{\xi}) and (T_{\eta}) acting on a pair of dependent fields (U(t,\bar t)) and (V(t,\bar t)), they rewrite the compatibility (holonomy) conditions of the linear auxiliary problem in terms of functional equations (2.6a‑c). The matrices (L(\xi)) and (\bar L(\eta)) are taken linear in a spectral parameter (\zeta); the minimal solution is given by equations (2.7)–(2.10), where the auxiliary scalar functions (G(\xi)) and (g(\eta)) contain the nonlinear dependence on the fields.

The positive sub‑hierarchy is recovered from the functional equations (3.1) and, after expanding the Miwa shifts as power series, yields the familiar Chen‑Lee‑Liu DNLS equation (1.2). The authors also introduce a generating operator (\partial(\xi)=\sum_{j\ge1}\xi^{j}\partial_{j}) that leads to a compact representation (3.7) useful for constructing the generating function of conservation laws.

The negative sub‑hierarchy, derived from (2.6c), turns out to be structurally identical to the positive one when the variables are simply renamed, confirming that the “purely negative” hierarchy does not introduce new dynamics beyond the classical DNLS equations.

The central contribution is the mixed (or extended) DNLS hierarchy obtained by coupling the two copies of the classical hierarchy through auxiliary potentials (\Psi) and (\Phi). Equations (5.1)–(5.4) express the interaction of the Miwa‑shifted fields with these potentials, while (5.10) provides a concise first‑order system that simultaneously encodes all positive and negative flows. The potentials satisfy simple logarithmic relations (5.9), which allow the whole system to be written in a symmetric form.

Several concrete reductions illustrate the richness of the extended hierarchy:

1. Mikhailov‑Fokas‑Lenells equation – By setting (Q=\Phi U) and (R=\Psi V) and taking the limit (\xi,\eta\to0), the authors recover the relativistic two‑dimensional field model (5.13), known as the Fokas‑Lenells system.

2. (1+2)-dimensional Chen‑Lee‑Liu equation – Using the limit (\xi\to0) and defining a new auxiliary field (P(\eta)), they obtain a system (5.18) that reduces to the standard Chen‑Lee‑Liu equation when the spatial variables are identified, thus providing a genuine (1+2)‑dimensional extension.

3. (1+2)-dimensional Kaup‑Newell equation – With a different identification of the dependent variables ((Q=\Psi U, R=\Phi V)), the hierarchy yields equations (5.22) that constitute a two‑dimensional generalisation of the Kaup‑Newell DNLS equation.

4. Adler‑Shabat H5 system – By logarithmically transforming the fields ((\lambda=\ln U, \mu=\ln u)), the mixed hierarchy reproduces the H5 system (5.24), a well‑known integrable model from the Adler‑Shabat classification.

5. Massive Thirring model – Imposing the reduction (V=\mp U^{}, v=\mp u^{}) leads to equations (5.26) that are equivalent to the massive Thirring model. The authors further embed this reduction into a larger matrix‑valued system (5.32)–(5.35), showing that the extended DNLS hierarchy contains the full Thirring hierarchy.

In Section 6 the authors construct the generating function for the infinite set of conserved quantities: \