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Recent Developments in Integrable Systems and Related Topics of Mathematical Physics Kezenoi-Am, Russia, 2016 Springer Proceedings in Mathematics & Statistics Springer Proceedings in Mathematics & Statistics Victor M. Buchstaber Sotiris Konstantinou-Rizos Alexander V. Mikhailov Editors Recent Developments in Integrable Systems and Related Topics of Mathematical Physics Kezenoi-Am, Russia, 2016 Springer Proceedings in Mathematics & Statistics Volume 273 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 Victor M. Buchstaber • Sotiris Konstantinou-Rizos Alexander V. Mikhailov Editors Recent Developments in Integrable Systems and Related Topics of Mathematical Physics Kezenoi-Am, Russia, 2016 123 Editors Victor M. Buchstaber Alexander V. Mikhailov Steklov Mathematical Institute School of Mathematics Russian Academy of Sciences University of Leeds Moscow, Russia Leeds, UK Sotiris Konstantinou-Rizos Centre of Integrable Systems P.G. Demidov Yaroslavl State University Yaroslavl, Russia ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-04806-8 ISBN 978-3-030-04807-5 (eBook) https://doi.org/10.1007/978-3-030-04807-5 Library of Congress Control Number: 2018962352 Mathematics Subject Classification (2010): 16T25, 17B37, 17B68, 33E17, 34L25, 34M50, 35Q30, 35Q15, 46L65, 53D55, 65T99 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Our book is a selection of works presented at the Conference of Mathematical Physics “Kezenoi-Am 2016”. The Organising and Programme Committee of the conference tried to create a programme which embraces the variety of research directions inspired by the modern developments in Mathematical Physics and the theory of Integrable Systems. The authors of the included papers are well known mathematicians from several research groups in Europe and Russia. We hope that the book will attract the attention to these areas of research, and will be interesting both to experts and young researchers. The Conference of Mathematical Physics “Kezenoi-Am 2016” is the first in the series of conferences which are being held partly in the city of Grozny, and partly at a beautiful mountain lake “Kezenoi-Am” of the Chechen Republic, Russia. These conferences are generously supported by the Chechen State University (CheSU), which is driven by the goal to support mathematical culture in Chechen republic. It is important to note that the Chairman of the Organizing Committee, Rector of the CheSU, Prof. Zaurbek Saidov, encourages the idea that the organization of inter- national conferences with the participation of world recognized researches is the optimal way to motivate and attract students to research. During the participants’ welcoming, he stressed out that he considers our conference to be a quite important step towards this direction. We are especially grateful to him for his support and help in the organization of this series of successful international conferences. We are also grateful to the Vice-rector of CheSU, Prof. Zaur Kindarov for his support and hospitality. Dr. Dmitry Grinev played an essential role in the organization of these conferences. We would like to thank him for his activity and patience throughout this procedure. Finally, we acknowledge that this work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project 1.12873.2018/12.1. Yaroslavl, Russia Victor M. Buchstaber June 2018 Sotiris Konstantinou-Rizos Alexander V. Mikhailov v Contents Introduction .............................................. 1 Victor M. Buchstaber, Sotiris Konstantinou-Rizos, and Alexander V. Mikhailov Numerical Instability of the Akhmediev Breather and a Finite-Gap Model of It ................................. 3 P. G. Grinevich and P. M. Santini Movable Poles of Painlevé I Transcendents and Singularities of Monodromy Data Manifolds ............................... 24 V. Yu. Novokshenov Elliptic Calogero-Moser Hamiltonians and Compatible Poisson Brackets .......................................... 38 Vladimir Sokolov Bäcklund Transformations and New Integrable Systems on the Plane .............................................. 47 A. V. Tsiganov Quadro-Cubic Cremona Transformations and Feigin-Odesskii-Sklyanin Algebras with 5 Generators .............. 75 V. Rubtsov From Reflection Equation Algebra to Braided Yangians ............ 107 Dimitri Gurevich and Pavel Saponov Survey of the Deformation Quantization of Commutative Families ................................................. 130 G. Sharygin and A. Konyaev Graded Thread Modules Over the Positive Part of the Witt (Virasoro) Algebra ................................ 154 Dmitry Millionshchikov vii viii Contents Localized Solutions of the Schrödinger Equation on Hybrid Spaces. Relation to the Behavior of Geodesics and to Analytic Number Theory ........................................... 183 Andrei Shafarevich Normal Equation Generated from Helmholtz System: Nonlocal Stabilization by Starting Control and Properties of Stabilized Solutions ...................................... 194 A. V. Fursikov Index ...................................................... 213 Contributors Victor M. Buchstaber Steklov Mathematical Institute, Russian Academy of Sciences, Moskva, Russia A. V. Fursikov Faculty of Mechanics and Mathematics, “Lomonosov” Moscow State University, Moscow, Russian Federation P. G. Grinevich L.D. Landau Institute for Theoretical Physics, pr. Akademika Semenova 1a, Chernogolovka, Russia; Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia; Moscow Institute of Physics and Technology, Moscow Region, Russia Dimitri Gurevich Valenciennes University, EA 4015-LAMAV, F-59313 Valenciennes, France Sotiris Konstantinou-Rizos Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia A. Konyaev Faculty of Mechanics and Mathematics, “Lomonosov” Moscow State University, Moscow, Russia Alexander V. Mikhailov School of Mathematics, University of Leeds, Leeds, UK Dmitry Millionshchikov Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia V. Yu. Novokshenov Institute of Mathematics RAS, Ufa, Russia V. Rubtsov Maths Department, University of Angers, Angers, France; Theory Division, Math. Physics Lab, ITEP, Moscow, Russia P. M. Santini Dipartimento di Fisica, Università di Roma “La Sapienza”, and Istituto Nazionale di Fisica Nucleare, Roma, Italy ix x Contributors Pavel Saponov National Research University Higher School of Economics, Moscow, Russian Federation; NRC “Kurchatov Institute”, Institute for High Energy Physics, Protvino, Russian Federation Andrei Shafarevich M.V. Lomonosov Moscow State University, Moscow, Russia G. Sharygin Faculty of Mechanics and Mathematics, “Lomonosov” Moscow State University, Moscow, Russia Vladimir Sokolov Landau Institute for Theoretical Physics, Moscow, Russia; Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia; Federal University of ABC, Santo André, Sao Paulo, Brazil A. V. Tsiganov St.Petersburg State University, St.Petersburg, Russia Acronyms BT Bäcklund transformation MI Modulation instability NLS Nonlinear Schrödinger NPE Normal parabolic equation PDE Partial differential equation PI Painlevé I PII Painlevé II QM Quantum matrix RE Reflection equation RW Rogue wave SSFM Split-step Fourier method xi Introduction B Victor M. Buchstaber1, Sotiris Konstantinou-Rizos2( ),and Alexander V. Mikhailov3 1 Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkina str., 119991 Moskva, Russia [email protected]
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