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Atsushi Nakayashiki

Publications and source records attributed to Atsushi Nakayashiki.

At least 19 recordsLinked to original sources
arXiv2026

Degenerate Addition Formulas of the KP Hierarchy and Applications

It is well known that tau functions of the KP hierarchy satisfy addition formulas. Among them we consider the formula which expresses a tau function with shifted arguments by $2n$ parameters in terms of the same tau function with shifted arguments by two parameters in the form of determinant. We then take the limits of it tending some of parameters to zero. As an important special case we obtain the formula which connects the shifted tau function to the Wronskian of functions obtained by substituting parameter values into the spectral variable of the wave function. As an application, we prove the equivalence of vertex operators and Darboux transformations. As another application, we derive a new addition formula for Riemann's theta functions of Riemann surfaces by considering theta function solutions of the KP hierarchy. It can be considered as a limit of Fay's famous formula (43) in the book [Lecture Notes in Math., Vol. 352, Springer, Berlin, 1973]. But the formula we have derived is a different limit from the limits of (43) considered in that book.

nlin.SI↗
arXiv2023

Vertex Operators of the KP hierarchy and Singular Algebraic Curves

Quasi-periodic solutions of the KP hierarchy acted by vertex operators are studied. We show, with the aid of the Sato Grassmannian, that solutions thus constructed correspond to torsion free rank one sheaves on some singular algebraic curves whose normalizations are the non-singular curves corresponding to the seed quasi-periodic solutions. It means that the action of the vertex operator has an effect of creating singular points on an algebraic curve. We further check, by examples, that solutions obtained here can be considered as solitons on quasi-periodic backgrounds, where the soliton matrices are deterimed by parameters in the vertex operators.

nlin.SI↗
arXiv2020

Tau Functions of (n,1) curves and Soliton Solutions on Non-Zero Constant Backgrounds

We study the tau function of the KP-hierarchy associated with an (n,1) curve $y^n=x-α$. If $α=0$ the corresponding tau function is 1. On the other hand if $α\neq 0$ the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.

nlin.SI↗
arXiv2019

Sato Grassmannian and Degenerate Sigma Function

The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus $g$ in terms of the tau functions corresponding to a hyperelliptic curve of genus $g-1$. Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function.

nlin.SI↗
arXiv2019

On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.

nlin.SI↗
arXiv2017

Degeneration of Trigonal Curves and Solutions of the KP-Hierarchy

It is known that soliton solutions of the KP-hierarchy corresponds to singular rational curves with only ordinary double points. In this paper we study the degeneration of theta function solutions corresponding to certain trigonal curves. We show that, when the curves degenerate to singular rational curves with only ordinary triple points, the solutions tend to some intermediate solutions between solitons and rational solutions. They are considered as cerain limits of solitons. The Sato Grassmannian is extensively used here to study the degeneration of solutions, since it directly connects solutions of the KP-hierarchy to the defining equations of algebraic curves.We define a class of solutions in the Wronskian form which contains soliton solutions as a subclass and prove that, using the Sato Grassmannian, the degenerate trigonal solutions are connected to those solutions by certain gauge transformations

nlin.SI↗
arXiv2017

On the Expansion Coefficients of KP Tau Function

We study the expansion coefficients of the tau function of the KP hierarchy. If the tau function does not vanish at the origin, it is known that the coefficients are given by Giambelli formula and that it characterizes solutions of the KP hierarchy. In this paper, we find a generalization of Giambelli formula to the case when the tau function vanishes at the origin. Again it characterizes solutions of the KP hierarchy.

nlin.SI↗
arXiv2015

Tau Function Approach to Theta Functions

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We describe the initial term of the expansion by the Schur function corresponding to the partition determined by the gap sequence of a certain flat line bundle. The other is the expansion of the theta function and its certain derivatives in one of the variables on the Abel-Jacobi images of k points on a Riemann surface with k less than or equal to g. We determine the initial term of the expansion as certain derivatives of the theta function successively. As byproducts, firstly we obtain a refinement of Riemann's singularity theorem. Secondly we determine normalization constants of higher genus sigma functions of a Riemann surface, defined by Korotkin and Shramchenko, such that they become modular invariant.

math-ph↗
arXiv2013

On Addition Formulae for Sigma Functions of Telescopic Curves

A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. We extend the addition formulae for sigma functions of $(n,s)$ curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.

math.AG↗
arXiv2012

Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images

We study derivatives of Schur and tau functions from the view point of the Abel-Jacobi map. We apply the results to establish several properties of derivatives of the sigma function of an (n,s) curve. As byproducts we have an expression of the prime form in terms of derivatives of the sigma function and addition formulae which generalize those of Onishi for hyperelliptic sigma functions.

math.AG↗
arXiv2012

Sigma Function as A Tau Function

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n,s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the multivariate sigma function. This expression is in turn used to prove fundamental properties on the series expansion of the sigma function established in a previous paper in a different method.

math.AG↗
arXiv2012

On Hyperelliptic Abelian Functions of Genus 3

The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions.

math.AG↗
arXiv2010

Baker-Akhiezer Modules on the Intersections of Shifted Theta Divisors

The restriction, on the spectral variables, of the Baker-Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of differential operators in $k$ variables. The remaining g-k derivations define evolution equations for generators of the BA-module. As a corollary new examples of commutative ring of partial differential operators with matrix coefficients and their non-trivial evolution equations are obtained.

math.AG↗
arXiv2009

On the Space of KdV Fields

The space of functions A over the phase space of KdV-hierarchy is studied as a module over the ring D generated by commuting derivations. A D-free resolution of A is constructed by Babelon, Bernard and Smirnov by taking the classical limit of the construction in quantum integrable models assuming a certain conjecture. We propose another D-free resolution of A by extending the construction in the classical finite dimensional integrable system associated with a certain family of hyperelliptic curves to infinite dimension assuming a similar conjecture. The relation of two constructions is given.

math-ph↗
arXiv2008

Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations

Solutions of the qKZ equation associated with the quantum affine algebra $U_q(\hat{sl}_2)$ and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of $q$-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko.

math.QA↗
arXiv2008

On Algebraic Expressions of Sigma Functions for (n,s) Curves

An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be the Schur function determined from the gap sequence at infinity.

math.AG↗