arXiv subjects
Publications and source records attributed to I. Krichever.
We construct quasi-periodic solutions of the universal hierarchy which includes the multi-component KP and Toda hierarchies and show how they fit into the bilinear formalism. The tau-function is expressed in terms of the Riemann theta-function multiplied by exponential function of a quadratic form in the hierarchical times.
We further develop the approach to many-body systems based on finding conditions of existence of meromorphic solutions to certain linear partial differential and difference equations which serve as auxiliary linear problems for nonlinear integrable equations such as KP, BKP, CKP and different versions of the Toda lattice. These conditions imply equations of the time evolution for poles of singular solutions to the nonlinear equations which are equations of motion for integrable many-body systems of Calogero-Moser and Ruijsenaars-Schneider type. A new many-body system is introduced, which governs dynamics of poles of elliptic solutions to the Toda lattice of type B.
We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic-geometrical solutions in terms of Riemann and Prym theta-functions.
We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^†$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.
A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.
The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Shrödinger equation at the zero energy level is a smooth $M$-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrödinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov--Veselov construction.
We study the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the "superperiodic" operators for which all solutions of the equation $(L+1)ψ=0$ are (anti)periodic. We show that for a superperiodic operator $L$ there exists a unique superperiodic operator ${\cal L}$ of order $(n-k-1)$ which commutes with $L$ and show that the duality $L\leftrightarrow {\cal L}$ coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].
A generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions on an algebraic curve with punctures is proposed. Extremal properties of $M$-curves are proved and connected with the spectral theory of difference operators with positive coefficients.
Using constructions of the Whitham perturbation theory of integrable system we prove a new sharp upper bound of $3g/2-2$ on the dimension of complete subvarieties of $\M_g^{ct}$.
Using meromorphic differentials with real periods, we prove Arbarello's conjecture: any compact complex cycle of dimension $g-n$ in the moduli space $\M_g$ of smooth genus $g$ algebraic curves must intersect the locus of curves having a Weierstrass point of order at most $n$.
Novikov's conjecture on the Riemann-Schottky problem: {\it the Jacobians of smooth algebraic curves are precisely those indecomposable principally polarized abelian varieties (ppavs) whose theta-functions provide solutions to the Kadomtsev-Petviashvili (KP) equation}, was the first evidence of nowadays well-established fact: connections between the algebraic geometry and the modern theory of integrable systems is beneficial for both sides. The purpose of this paper is twofold. Our first goal is to present a proof of the strongest known characterization of a Jacobian variety in this direction: {\it an indecomposable ppav $X$ is the Jacobian of a curve if and only if its Kummer variety $K(X)$ has a trisecant line} and the solution of the characterization problem of principally polarized Prym varieties. The latter problem is almost as old and famous as the Riemann-Schottky problem but is much harder. In some sense the Prym varieties may be geometrically the easiest-to-understand ppavs beyond Jacobians, and studying them may be a first step towards understanding the geometry of more general abelian varieties as well. Our second and primary objective is to take this opportunity to elaborate on motivations underlining the proposed solution of the Riemann-Schottky problem, to introduce a certain circle of ideas and methods, developed in the theory of soliton equations, and to convince the reader that they are algebro-geometric in nature, simple and universal enough to be included in the Handbook of moduli.
We introduce the notion of abelian solutions of the 2D Toda lattice equations and the bilinear discrete Hirota equation and show that all of them are algebro-geometric.
We introduce the notion of abelian solutions of KP equations and show that all of them are algebro-geometric.
We prove Welter's trisecant conjecture: an indecomposable principally polarized abelian variety $X$ is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety $K(X)$.
We prove that Prym varieties of algebraic curves with two smooth fixed points of involution are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit formulae for integrable 2D Schrödinger equation.
We prove that an indecomposable principally polarized abelian variety $X$ is the Jacobain of a curve if and only if there exist vectors $U\neq 0,V$ such that the roots $x_i(y)$ of the theta-functional equation $θ(Ux+Vy+Z)=0$ satisfy the equations of motion of the {\it formal infinite-dimensional Calogero-Moser system}
Given an abelian variety $X$ and a point $a\in X$ we denote by $ $ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $κ: X\to κ(X)\subset\mathbb P^N$ the map from $X$ to its Kummer variety. We prove that an indecomposable abelian variety $X$ is the Jacobian of a curve if and only if there exists a point $a=2b\in X\setminus\{0\}$ such that $ $ is irreducible and $κ(b)$ is a flex of $κ(X)$.
A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of such solutions leads to a notion of local monodromies of difference equations. It is shown that in the continuous limit they converge to the monodromy matrices of differential equations. New type of isomonodromic deformations of difference equations with elliptic coefficients changing the periods of elliptic curves is constructed.