A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice

Tropical spectral curves, Fay's trisecant identity, and generalized ultradiscrete Toda lattice

We study the generalized ultradiscrete periodic Toda lattice T(M,N) which has tropical spectral curve. We introduce a tropical analogue of Fay's trisecant identity, and apply it to construct a general solution to T(M,N).

Menelaus relation and Fay's trisecant formula are associativity equations

It is shown that the celebrated Menelaus relation and Fay's trisecant formula similar to the WDVV equation are associativity conditions for structure constants of certain three-dimensional algebra.

New degeneration of Fay's identity and its application to integrable systems

In this paper we prove a new degenerated version of Fay's trisecant identity. The new identity is applied to construct new algebro-geometric solutions of the multi-component nonlinear Schrödinger equation. This approach also provides an independent derivation of known algebro-geometric solutions to the Davey-Stewartson equations.

Fay恒等式的高阶退化及其在可积方程中的应用

Higher order degenerations of Fay's identities and applications to integrable equations

本文提出Fay三割线恒等式的高阶退化形式，并证明其可构造Schwarzian KP方程的解。

Computational approach to the Schottky problem

We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus $g\geq 4$. For a given Riemann matrix $\mathbb{B}\in\mathbb{H}^{g}$, the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The theta functions in terms of which these secants are expressed depend on the Abel maps of four arbitrary points on a Riemann surface. However, there is no concept of an Abel map for general $\mathbb{B} \in \mathbb{H}^{g}$. To establish linear …

Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (diff…