A particle-system model of the sine-Gordon hierarchy
type: Item
instance of: scholarly article
title: A particle-system model of the sine-Gordon hierarchy
zbMATH Open document ID: 0595.35095
DOI: 10.1016/0167-2789(86)90193-4
author: Emma Previato
published in: Physica D
publication date: 1986-01-01T00:00:00Z
review text: This is really an extended abstract of author's results, with no proofs of theorems. The author makes the following statement. The first Lax-pair equation to be integrated by constrained harmonic motion (the ''Neumann system'') was KdV. The construction was later: (i) extended to other equations; (ii) viewed as a special case of a class of (constrained) completely integrable systems called rank 2 perturbations; (iii) explained in the more natural context of ''squared eigenfunctions''; (iv) linked with the algebraic geometry that is called for by the angle variables and had been used independently to construct quasiperiodic solutions via Baker-Akhiezer functions. We present an analogous treatment for the sine-Gordon equation; we use a rank-2 perturbation with symmetries to integrate it and spell out the link with algebraic geometry: this involves an unramified double-covering of a hyperelliptic curve, which had appeared in other contexts. The reason for that is that the AKNS eigenfunctions will have essential singularities at two points, 0 and \(\infty\), of the base curve and will be well-defined only when we pass to a double cover. Similar combinations of tangent vectors of the curve at 0 and \(\infty\) may provide a simple way of generating other interesting equations. (AKNS stands for \textit{M. J. Ablowitz}, \textit{D. J. Kaup}, \textit{A. C. Newell}, \textit{H. Segur} [Stud. Appl. Math. 53, 249-315 (1974; Zbl 0408.35068)].) As an application, we give a necessary and sufficient condition for the solution to be real that confirms the preexisting ones. In favorable cases, this can be lifted linearly to the original phase-space \({\mathbb{C}}^{4n}\).
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35B15
zbMATH DE Number: 3957809
zbMATH Keywords: perturbation of particle models
zbMATH Keywords: Lax-pair equation
zbMATH Keywords: Baker-Akhiezer functions
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: AKNS eigenfunctions
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: 0167-2789(86)90193-4
OpenAlex ID: W2001402658
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cites work: Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura
description: scientific article; zbMATH DE number 3957809