maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2015-6-795-817mais-296Research ArticleОригинальные статьиArticlesФормальная диагонализация схем Лакса–ДарбуFormal Diagonalisation of Lax-Darboux SchemesМихайловА. В.MikhailovA. V.

 доктор физико-математических наук, профессор,

University of Leeds, Leeds, LS2 9JT, UK

Professor, University of Leeds, Leeds, LS2 9JT, UK

A.V.Mikhailov@leeds.ac.ukШкола математики Университета Лидса (Лидс, Великобритания)ВеликобританияUniversity of Leeds, School of Mathematics (Leeds, UK)United Kingdom201520122015226795817Copyright © Михайлов А.В., 20152015Михайлов А.В.Mikhailov A.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/296

В статье мы обсуждаем концепцию схемы Лакса–Дарбу и иллюстрируем ее на хорошо известных примерах, ассоциированных с нелинейным уравнением Шрёдингера. Мы изучаем связи, возникшие благодаря преобразованиям Дарбу, между иерархиями нелинейного уравнения Шрёдингера, модели Гейзенберга, модели главного кирального поля, а также дифференциально-разностными системами (такими как цепочка Тоды и дифференциально-разностная цепочка Гейзенберга) и конечно-разностными интегрируемыми системами. Мы показываем, что существует формальное преобразование, которое одновременно диагонализует все элементы схемы Лакса–Дарбу. Это приводит нас к производящим функциям локальных законов сохранения для всех интегрируемых систем, полученных в рамках данной схемы Лакса–Дарбу. Обсуждаются связи между законами сохранения систем, принадлежащих заданной схеме Лакса–Дарбу.

We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schro¨dinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating functions of local conservation laws for all integrable systems obtained. We discuss the relations between conservation laws for systems belonging to the Lax-Darboux scheme.

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The authors declare that there are no conflicts of interest present.