Nonlinear Physics: Proceedings of the International Conference, Shanghai, People’s Rep. of China, April 24–30, 1989 PDF
Preview Nonlinear Physics: Proceedings of the International Conference, Shanghai, People’s Rep. of China, April 24–30, 1989
Research Reports in Physics Research Reports in Physics Nuclear Structure of the Zirconium Region Editors: J. Eberth, R. A. Meyer, and K. Sistemich Ecodynamics Contributions to Theoretical Ecology Editors: w. Wolff, C.-J. Soeder, and F. R. Drepper Nonlinear Waves 1 Dynamics and Evolution Editors: A. V. Gaponov-Grekhov, M.1. Rabinovich, and J. Engelbrecht Nonlinear Waves 2 Dynamics and Evolution Editors: A. V. Gaponov-Grekhov, M.1. Rabinovich, and J. Engelbrecht Nuclear Astrophysics Editors: M. Lozano, M.1. Gallardo, and J. M. Arias Optimized LCAO Method and the Electronic Structure of Extended Systems By H. Eschrig Nonlinear Waves in Active Media Editor: J. Engelbrecht Problems of Modern Quantum Field Theory Editors: A.A. Belavin, A.U. Klimyk, and A.B. Zamolodchikov Fluctuational Superconductivity of Magnetic Systems By M.A. Savchenko and A.V. Stefanovich Nonlinear Evolution Equations and Dynamical Systems Editors: S. Carillo and O. Ragnisco Nonlinear Physics Editors: Gu Chaohao, Li Yishen, and Tu Guizhang Gu Chaohao Li Yishen Tu Guizhang (Eds.) Nonlinear Physics Proceedings of the International Conference, Shanghai, People's Rep. of China, April 24-30, 1989 With 47 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Professor Gu Chaohao Professor Li Yishen University of Science and Technology of China, Hefei, Anhui 230026, People's Rep. of China Professor Tu Guizhang Computing Center of Academia Sinica, Beijing 100080, People's Rep. of China Executive Editor Associate Professor Zeng Yunbo Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People's Rep. of China Advisory Committee Organizing Committee Chairman Chairman Yang, C. N. (USA) Gu Chaohao Members Members Ablowitz, M. J. (USA) Flato, M. (France) Cao Cewen Hu Hesheng Araki, H. (Japan) Gu, C. H. (China) Guo Benyu Li Yishen Calogero, F. (Italy) Kruskal, M. (USA) Ge Molin Ni Guangjiong Degasperis, A. (italy) Sato, M. (Japan) Hou Boyu Tu Guizhang Faddeev, L. D. (USSR) ISBN-13:978-3-540-52389-5 e-ISBN-13:978-3-642-84148-4 001: 10.1007/978-3-642-84148-4 Library of Congress Cataloging-in-Publication Data. Nonlinear physics 1 Gu Chaohao ... [etal.), eds. p. cm. --(Research reports in physics) A selection of refereed papers presented althe International Con ference on Nonlinear Physics held Apr. 24-30, 1989, in ShanghaLlncludes bibliographical references. ISBN· 13:978-3-540-52389-5(U. S.:al k.paper)1.Sol itons--Congresses.2.Nonl inear theories--Congresses. I. Gu, Ch'ao-hao. II. International Conference on Nonlinear Physics (1989: Shanghai, China) III. Series. QC 17 4.26W28N66 1990 530.1' 4--dc20 90-34467 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad casting, reproduction on microfilms or in other ways, and storage in data banks. Duplication ofthispub lication or parts thereof is only permitted under the provisions ofthe German Copyright Law of Septem ber 9,1965, in its current version, and a copyrightfee must always be paid. Violations fail under the pro secution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990 The useof registered names, trademarks, etc. in this publication does not imply, even intheabsenceofa specific statement, that such names are exempt from the relevant protectiv laws and regulations and therefore free for general use. 2157/3150-543210 - Printed on acid-free paper Preface* The illternational Conference on Nonlinear Physics took place in Shanghai during 24-30 April, 1989. About one-fifth of the 108 participants came from regions outside China and Hong Kong. The main topic of the conference was soliton theory. These proceedings contain 30 of the 83 contributions and reflect the most recent and significant developments in the field, including new methods and new applications. The selected contributions have all undergone referee review. Some of the papers are rather broad surveys of the recent research while others are narrow scope research articles. Because of the wide range of the covered topics, the contributions are divided into five groups. The first part is devoted to the general theory of the integrable system, such as Hamiltonian structure, symmetries, Backlund and Darboux trans formations. ill the second part entitled "Finite Dimensional Dynamical Systems", one paper is concerned with the symmetries and integrability of coupled nonlinear oscillators, others concentrate on finite dimensional integrable systems reduced from infinite dimensional integrable Hamiltonian systems. The third part "Quan tum Aspects and Statistical Mechanics" includes knot theory, braid groups and the R-matrix method. The fourth part, "Physical Phenomena", deals with nonlin ear evolution equations, such as the K-P equation applied to water waves, soliton phenomena in porous media, and lattice models. ill addition, findings related to chaos and the cellular automata are presented. The final part comprises contri butions addressing various questions ranging from the most fundamental ones to those of the exciting theory of strings. We are particularly pleased to thank the illternational Union for Pure and Applied Physics, the illternational Center for Theoretical Physics, the National Commission of Education of China and the National Natural Science Foundation of China for providing financial support. We are indebted to the authors for the hard work they invested in their contributions. We take this opportunity to express our thanks to Executive Editor Zeng Yunbo for the hard work and direct help in preparing the proceedings. Hefei GuChaohao January 1990 Li Yishen Tu Guizhang *The complete manuscript was received by Springer-Verlag on February 14, 1990 v Contents Part I Integrable Systems: Hamiltonian Structure, Symmetries, Backlund and Darboux Transformations Liouville Integrability of Zero Curvature Equations By Tu Guizhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2+ 1 Dimensional Integrable Hierarchies, Lax Operators and Relevant Algebraic Structures By Cheng Yi ......................................... 12 Detennination of Nondegenerate Darboux Operators of First Order in 1 + 2 Dimensions By Zhou Zixiang ...................................... 23 A Series of New Exact Solutions to the Nonlinear Equation Yt+Yxxx- 6y2Yx+6>..yx=O By Au Chi (With 1 Figure) ............................... 29 Backlund Transformations for the Isospectral and Non-Isospectral KdV Hierarchies By Tian Chou and Zhang Youjin ........................... 35 Multiple Darboux Transformations and Multiple Pole Solutions for AKNS Hierarchy By Gu Xinshen ....................................... 42 A Lie Algebraic Structure of G.J. and Its Gauge Equivalent Yang Hierarchies By Li Yishen, Cheng Yi, and Zeng Yunbo ..................... 47 Part II Finite Dimensional Dynamical Systems Coupled Nonlinear Oscillators: Symmetries and Integrability By M. Lakshmanan .................................... 54 Classical Integrable Systems Generated Through Nonlinearization of Eigenvalue Problems By Cao Cewen and Geng Xianguo .......................... 68 VII The Confocal Involutive System and the Integrability of the Nonlinearized Lax Systems of AKNS Hierarchy By Ma Wenxiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79 Two Kinds of Finite-Dimensional Systems Related to the Generalized SchrOdinger Equation By Zeng Yunbo and Li Yishen ............................. 85 Nonlinearization of the Lax Pair for the KdV Equation and Integrable Hamiltonian Systems By Zhuang Dawei and Lin Yuanqu ......................... .92 Part III Quantum Aspects and Statistical Mechanics Quantum and Classical Statistical Mechanics of the Integrable Models in 1 + 1 Dimensions By R.K. Bullough, D.J. Pilling, J. Timonen, Yi Cheng, and Yu-Zhong Chen ....................................... 98 Link Polynomials and Exactly Solvable Models By M. Wadati, Y. Akutsu, and T. Deguchi (With 17 Figures) ........ 111 R-Matrices and Higher Poisson Brackets for Integrable Systems By W. Oevel ......................................... 136 Classical R-Matrix and Semi-Simple Lie Algebras By Liu Zhangju and Qian Min ............................. 146 Witten's Approach, Braid Group Representations and X-Deformations By M.L. Ge, F. Piao, L.Y. Wang, and K. Xue .................. 152 PartN Physical Phenomena Nonlinear Evolution Equations, Solitons, Chaos and Cellular Automata By M.J. Ablowitz, B.M. Herbst, and J.M. Keiser (With 14 Figures) " .. 166 Kadomtsev-Petviashvili Equations in the Description of Water Waves By D. Levi (With 5 Figures) .............................. 190 Three-Dimensional Lattice Model Based on Soliton Theory By N. Saitoh ......................................... 205 Soliton Phenomena in a Porous Medium By D. Takahashi, J.R. Sachs, and J. Satsuma (With 5 Figures) 214 Two-Dimensional Chiral Gauge Theories on a Lattice By Ma Zhongshui and Guo Shuohong ........................ 221 Transformation for the Solutions of the Two-Dimensional Toda Lattice By Liu Qiming ....................................... 227 VIII Part V Other Topics Some Ideas on Nonlinear Evolution Equations By F. Calogero ....................................... 232 Some Problems of the Generalized Kuramoto-Sivashinsky Type Equations with Dispersive Effects By Guo Boling ....................................... 236 Standard Nonlinearities Associated with KdV -like Two-Soliton Solutions By F. Lambert and R. Willox .... . . . . . . . . . . . . . . . . . . . . . . . . .. 242 Complex Singularities and the Riemann Surface for the Burgers Equation By D. Bessis and J.D. Fournier (With 5 Figures) ......... . . . . . . .. 252 From Soliton Theory to String Theory By S. Saito and H. Kato ................................. 258 Non-Linear Equations from a String-Theoretical Point of View By H. Kato and S. Saito ................................. 266 Painleve Analysis and Integrability of the Evolution Equation Ut=uxxx+u2uxx+ 3uu~+ 1/3u4ux By M. Daniel and R. Sahadevan .. . . . . . . . . . . . . . . . . . . . . . . . . .. 273 SUbject Index ....................................... 281 List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 283 Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 287 IX Part I Integrable Systems: Hamiltonian Structure, Symmetries, Backlund and Darboux Transformations Liouville Integrability of Zero Curvature Equations Tu Guizhang* Computing Centre of Academia Sinica, Beijing 100080, People's Rep. of China 1. INTRODUCTION Let S = S (00, -00) be the Schwartz space, and = Ut K(u) (I) be a nonlinear evolution equation (NLEE), where U = (u., ... , up) E SP. There are different definitions on integrability of the equation (1), we shall adopt the following two definitions: (A.) We call the equation (1) Lax integrable if it can be written as a zero curvature equation Ut - V", + [U, V] = 0, where U=U(u), V=V(u) are two matrices which contain u as the 'potential', and [U,V]=UV-VU. (B.) We call the equation (1) Liouville integrable if (1) it can be written as a generalized Hamiltonian equation = J5H/5u Ut where J is a Hamiltonian operator; and (2) it possesses an infinite number of conserved densities {Hn} that are involution in pairs: {Hn.Hm} = 0 (mod D), where = {j,g} (J5f/ou).(5g/ou) and £=0 (mod D) means f = (d/dx)h for some hE SP Example. The well-known KdV and AKNS hierarchies of NLEEs are both Lax integrable and Liouville integrable. In the past decades the theory of generalized Hamiltonian ayatems has undergone a rapid development (see [1]-117]). A central and very important problem in the theory of integrable systems is to search for the nonlinear evolution equations that are both Lax and Liouville integrable. The aim of the present report is the following: (1.) First, we propose a ;;cheme for generating Lax integrable hierarchies of equations *Supported by National Natural Science Foundation through Nankai Institute of Math. 2 Research Reports in Physics Nonlinear Physics Editors: Gu Chaohao· Li Yishen· Tu Guizhang © Spcinger-Verlag Berlin, Heidelberg 1990
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