Integrable lattices
type: Item
instance of: scholarly article
title: Integrable lattices
zbMATH Open document ID: 0984.37093
DOI: 10.1007/BF02557310
author: V. G. Marikhin
author: A. B. Shabat
published in: Theoretical and Mathematical Physics
publication date: 2002-05-01T00:00:00Z
review text: In this paper integrable lattices corresponding to the Hamiltonians \[ H= p_xq_x+ V(p,q_x, q)\tag{1} \] are constructed, where \(V\) is a quadratic polynomial in \(p\) and \(q_x\). The authors are interested in only the two cases \[ H= p_xq_x+ q^2_x(\varepsilon p^2+\alpha p+ \beta)+ p^2(\gamma q_x+ \delta),\tag{2} \] where \(\varepsilon\), \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are arbitrary parameters, and \[ H= p_x q_x- q^2_x p^2- p^2 r(q)- \textstyle{{1\over 2}} pr'(q)- \textstyle{{1\over 12}} r''(q),\tag{3} \] where \(r(q)\) is an arbitrary fourth-order polynomial in \(q\). Different choices of the parameters in (2) correspond to numerous integrable generalizations of the nonlinear Schrödinger equation. The authors discuss the Toda, Volterra and Heisenberg models in detail and obtain totally discrete Lagrangians. Moreover, they discuss the relation of this systems to the Hirota equations.
Mathematics Subject Classification ID: 37K60
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 1466051
zbMATH Keywords: Hamiltonians
zbMATH Keywords: nonlinear Schrödinger equations
zbMATH Keywords: integrable lattices
zbMATH Keywords: Bäcklund transformations
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02557310
OpenAlex ID: W3188670290
description: scientific article; zbMATH DE number 1466051