Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0606059

Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit

Lee-Peng Teo
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.

Explore related subjects

nlin.SIhep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lee-Peng Teo. 2006-06-26. Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit. https://doi.org/10.1142/s0129055x06002838

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Large-space and Large-time Asymptotics for the Focusing Nonlinear Schrödinger Soliton Gas

We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schrödinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with the discrete spectrum confined to two segments $Σ_1$ and $Σ_2$. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate $g$-function mechanism, we show that, as $x\to-\infty$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to+\infty$, we assume that the endpoint $F$ lies on the trajectory of $H(ξ)$ with $ξ=\frac{x}{2t}\in(-E_1-\sqrt{2}E_2,-E_1)$, namely, $F=H(\hatξ)$, $\hatξ\in (-E_1-\sqrt{2}E_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $ξ=\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $ξ\in(-E_1,+\infty)$, a modulated elliptic-wave region $ξ\in(\hatξ,-E_1)$, and an unmodulated elliptic-wave region $ξ\in(-\infty,\hatξ)$.

nlin.SI↗
arXiv2026

On singular solitons of the KP equation and the Go-diagrams

It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the Le-diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend Le-diagrams and arise in the Deodhar decomposition of the Grassmannian.

nlin.SI↗
arXiv2026

Volterra-Bogoyavlensky lattices and solutions of $A_{2n}^{(1)}$ invariant Painlevé equations

The objective of this work is to develop a framework that exploits the lattice structure of the $k$-th Volterra--Bogoyavlensky equations ($k\in\mathbb N$, $k>1$) to generate rational solutions of higher symmetric Painlevé equations. For $k=2$, we show that the Volterra lattice, equipped with suitable initial conditions, exactly models the one- and two-dimensional orbits generated by half-translation operators of the $A_2^{(1)}$ symmetric Painlevé IV equations. This correspondence yields explicit closed-form expressions for all solution components in terms of generalized Okamoto polynomials and leads to new algebraic recurrence relations among these polynomials. We present two generalizations of the above Volterra lattice. One is derived from a fractional translation of the $A_{4}^{(1)}$ symmetric Painlevé equations. It generalizes Volterra lattice structure in the multi-component setup of the affine $A_{4}^{(1)}$ group and it is shown to generate solutions of the $A_{4}^{(1)}$ symmetric Painlevé equations from the seed solution invariant under dihedral group $D_{5}$. The other is the $k=3$ Bogoyavlensky lattice structure. It satisfies recurrence relations that naturally extend recurrence relations of the Volterra lattice. These results shed light on connection between Volterra--Bogoyavlensky lattices, dihedral symmetries, and rational solutions of higher Painlevé systems.

nlin.SI↗