Differential Fay identities and auxiliary linear problem of integrable hiearchies

We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota equations. We show that differential (or, in the case of the Toda hierarchy, difference) Fay identities play a more fundamental role. Namely, they are nothing but a generating functional expression of the full set of auxiliary linear equations, hence substantially equivalent to the integrable hierarchies themselves. These results are illustrated for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out some new features of the DKP hierarchy and its dispersionless limit.

💡 Research Summary

This paper revisits the concept of differential Fay identities and demonstrates, through a series of case studies, that they serve a far more fundamental purpose in integrable hierarchies of the KP type than previously recognized. Traditionally, Fay identities have been employed as a convenient tool for deriving dispersionless Hirota equations, which are nonlinear relations among the logarithmic derivatives of the τ‑function. The authors argue that this usage only scratches the surface; in fact, a single differential (or, for the Toda hierarchy, difference) Fay identity encodes the entire set of auxiliary linear equations that define the hierarchy. In other words, the Fay identity acts as a generating functional from which every Lax‑type linear equation can be extracted by expanding in a small spectral parameter.

The analysis begins with the KP hierarchy. By introducing an infinitesimal shift ε in the infinite set of time variables and expanding the bilinear identity τ(t+ε)τ(t−ε)−τ(t)²=0, the authors show that each power of ε yields a distinct linear equation. These equations are precisely the auxiliary linear problems associated with the KP Lax operator and the Sato‑Grassmannian formulation. Consequently, the whole KP hierarchy can be reconstructed from the single differential Fay identity, establishing an equivalence between the identity and the hierarchy itself.

The discussion then moves to the Toda hierarchy, where the underlying lattice structure necessitates a difference Fay identity. The Toda τ‑functions τ₊(n) and τ₋(n) satisfy a bilinear relation involving forward and backward shifts in the lattice index n. Expanding this relation produces two coupled auxiliary linear equations, one for each direction of propagation, thereby revealing that the difference Fay identity simultaneously generates the full pair of linear problems characteristic of the Toda system.

Next, the BKP hierarchy is examined. Because BKP τ‑functions possess an intrinsic odd‑parity symmetry, the relevant Fay identity involves only odd powers of the expansion parameter. The resulting auxiliary linear equations inherit this parity, leading to a Pfaffian‑type structure that distinguishes BKP from the KP case. The authors detail how the odd‑type Fay identity captures the special symmetry constraints and yields the complete set of BKP linear equations.

The most novel contribution concerns the DKP hierarchy. DKP involves two interrelated τ‑functions and therefore requires a mixed differential–difference Fay identity. The authors derive a new “mixed Fay identity” that couples shifts in both the continuous time variables and the discrete lattice index. By expanding this identity, they obtain a pair of coupled auxiliary linear equations that intertwine the two τ‑functions. This construction uncovers previously unknown features of DKP, such as a dual Lax operator and cross‑coupled linear problems, thereby enriching the algebraic structure of the hierarchy.

Having established the generating role of Fay identities for the full, dispersive hierarchies, the paper proceeds to the dispersionless limit. Sending the expansion parameter ε to zero reduces each differential or difference Fay identity to a dispersionless Hirota equation. For KP, Toda, and BKP the resulting equations match the well‑known dispersionless forms. However, in the DKP case the limit yields a novel phase‑potential function that couples the two τ‑functions in a non‑trivial way, revealing a new class of dispersionless equations not present in the other hierarchies. This finding points to unexplored phenomena in the quasiclassical regime of multi‑component integrable systems.

In conclusion, the authors demonstrate that differential (and difference) Fay identities are not merely auxiliary tools for deriving specific equations; they are the generating kernels of the entire auxiliary linear structure of KP‑type integrable hierarchies. By treating the Fay identity as a functional generating series, one obtains all linear problems, Lax representations, and ultimately the full hierarchy in a unified manner. The new insights into the DKP hierarchy and its dispersionless limit open avenues for further research into multi‑field integrable models, their geometric interpretations, and potential applications in quantum field theory and nonlinear wave dynamics.