Abelian solutions of the soliton equations and Riemann-Schottky problems

Krichever, I. M.

doi:10.1070/rm2008v063n06abeh004576

Cited by 6 publications

(10 citation statements)

References 57 publications

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“…However, when the existence of only one trisecant is assumed, all three cases are independent and require its own approach. The approaches used in [36,38] were based on the theories of three main soliton hierarchies (see details in [46]): the KP hierarchy for (i), the 2D Toda hierarchy for (ii) and the Bilinear Discrete Hirota Equations (BDHE) for (iii). Recently, pure algebraic proof of the first two cases of the trisecant conjecture were obtained in [6].…”

Section: Welter's Conjecturementioning

confidence: 99%

Abelian pole systems and Riemann-Schottky type systems

Krichever¹

2022

Preprint

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In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the characterization of Jacobians as ppav whose Kummer variety admits a trisecant line and the Pryms as ppav whose Kummer variety admits a pair of symmetric quadrisecants can be seen as an abelian version of pole systems arising in the theory of elliptic solutions to the basic soliton hierarchies. We present also recent results in this direction on the characterization of Jacobians of curves with involution, which were motivated by the theory of two-dimensional integrable hierarchies with symmetries.

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Section: Welter's Conjecturementioning

confidence: 99%

Abelian pole systems and Riemann-Schottky type systems

Krichever¹

2022

Preprint

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“…For g ≥ 4, one big goal is to eliminate the parameters U, V, W , in order to obtain constraints among the theta constants that define the Schottky locus. That this works in theory is a celebrated theorem of Shiota [20,26], but it has never been carried out in practice. For g = 4, we hope to recover the classical Schottky-Jung relation for the Schottky hypersurface.…”

Section: Genus Four and Beyondmentioning

confidence: 99%

“…For genus four and higher, we run into the Schottky problem: most Riemann matrices B do not arise from algebraic curves. A solution using the KP equation was given by Shiota [20,26]. His characterization of valid matrices B amounts to the existence of the Dubrovin threefold.…”

Section: Introductionmentioning

confidence: 99%

The Dubrovin threefold of an algebraic curve

Agostini,

Çelik,

Sturmfels

2020

Preprint

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The solutions to the Kadomtsev-Petviashvili equation that arise from a fixed complex algebraic curve are parametrized by a threefold in a weighted projective space, which we name after Boris Dubrovin. Current methods from nonlinear algebra are applied to study parametrizations and defining ideals of Dubrovin threefolds. We highlight the dichotomy between transcendental representations and exact algebraic computations. Our main result on the algebraic side is a toric degeneration of the Dubrovin threefold into the product of the underlying canonical curve and a weighted projective plane.

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“…The study of algebro-geometric solutions has opened up a new vista in the analysis of nonlinear partial differential equations. The adopted algebro-geometric techniques brought innovative ideas and led to inspiring results in soliton theory as well as algebraic geometry, for example, a solution of the Riemann-Schottky problem [3,32]. The successful idea in constructing algebro-geometric solutions is to employ the theory of algebraic curves associated with Lax pairs producing soliton hierarchies to represent the Baker-Akhiezer functions [33,34] in terms of the Riemann theta function [35,36].…”

Section: Introductionmentioning

confidence: 99%

Trigonal curves and algebro-geometric solutions to soliton hierarchies II

2017

Proc. R. Soc. A.

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This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker-Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker-Akhiezer functions. We emphasize that we analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

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Abelian solutions of the soliton equations and Riemann-Schottky problems

Cited by 6 publications

References 57 publications

Abelian pole systems and Riemann-Schottky type systems

Abelian pole systems and Riemann-Schottky type systems

The Dubrovin threefold of an algebraic curve

Trigonal curves and algebro-geometric solutions to soliton hierarchies II

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