New elliptic potentials
type: Item
instance of: scholarly article
title: New elliptic potentials
zbMATH Open document ID: 0824.14023
DOI: 10.1007/BF01001541
author: Armando Treibich
published in: Acta Applicandae Mathematicae
publication date: 1995-11-13T00:00:00Z
review text: The well-known method of finite-gap integration, based on the Krichever dictionary between certain data on Riemann surfaces and algebras of commuting differential operators, allows to construct quasi-periodic solutions of particular classes of nonlinear integrable differential equations in terms of theta functions of Riemann surfaces. One of the most interesting questions, in this framework, is how to find as many as possible solutions of integrable equations in terms of elliptic functions. A great progress, in this direction, was the joint work of J.- L. Verdier and the author of the present paper, the first account of which appeared in 1987 [cf. \textit{J.-L. Verdier}, `New elliptic solitons', in: Algebraic Analysis, dedicated to M. Sato on the occasion of his 60th birthday, Vol. 2, 901-910 (1989; Zbl 0692.35090)]. The author and Verdier succeeded in constructing new elliptic solutions (finite-gap potentials) for the Korteweg-de Vries equation (KdV) and Kadomtsev-Petviashvili equation (KP), which came rather unexpected and revived the interest in this area. Their method of using the so-called tangential covers of elliptic curves (cf. reference above), for which they elaborated an algorithmic procedure, turned out to be extremely effective and led to the discovery of a lot of new elliptic finite-gap potentials of degrees 3, 4, 5 and 6. After the sudden tragic death of J.-L. Verdier, the author of the present paper kept working on this fruitful approach, and the article under review is a report on the present state of the Treibich-Verdier approach to constructing explicitly elliptic finite-gap solutions for the KdV and KP equations. As such, it represents a self-containing and comprehensive, nevertheless down-to earth survey, giving complete proofs in the case of smooth curves [cf. the author, Duke Math. J. 59, No. 3, 611-627 (1989; Zbl 0698.14029); cf. also the author and \textit{J.-L. Verdier}, ``Solitons elliptiques'' in: The Grothendieck Festschrift, Prog. Math.…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 701033
zbMATH Keywords: Weierstrass function
zbMATH Keywords: tau function
zbMATH Keywords: Jacobians
zbMATH Keywords: finite-gap potentials
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: Kadomtsev-Petviashvili equation
zbMATH Keywords: KP equation
zbMATH Keywords: KdV equation
zbMATH Keywords: integrable differential equations
zbMATH Keywords: theta functions of Riemann surfaces
zbMATH Keywords: elliptic finite-gap solutions
zbMATH Keywords: soliton equations
MaRDI profile type: Publication
cites work: On the even-order vanishing of Jacobian theta functions
cites work: Self-similar solutions of equations of Korteweg-deVries type
cites work: Q5628266
cites work: Loop groups and equations of KdV type
cites work: Q3475383
cites work: Tangential polynomials and elliptic solitons
cites work: Q4863653
full work available at URL: https://doi.org/10.1007/bf01001541
OpenAlex ID: W2046227477
Recommended article: Q4680640
Recommended article: On the geometry of elliptic solitons
Recommended article: On the Calculation of Finite-Gap Solutions of the KdV Equation
Recommended article: Q3708401
Recommended article: Elliptic solutions of nonlinear equations
Recommended article: Lamé Potentials and the Stationary (m)KdV Hierarchy
Recommended article: Elliptic solutions of the Korteweg-de Vries equation
Recommended article: 3-tangential polynomials and \(t\)-periodic solutions of the KdV equation
Recommended article: Treibich-Verdier potentials and the stationary (m)KdV hierarchy
description: scientific article; zbMATH DE number 701033