Darboux Transformations in Integrable Systems Theory and their Appli
The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry. Th
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MATHEMATICAL PHYSICS STUDIES Editorial Board:
Maxim Kontsevich, IHES, Bures-sur-Yvette, France Massimo Porrati, New York University, New York, U.S.A. Vladimir Matveev, Université Bourgogne, Dijon, France Daniel Sternheimer, Université Bourgogne, Dijon, France
VOLUME 26
Darboux Transformations in Integrable Systems Theory and their Applications to Geometry
by
Chaohao Gu Fudan University, Shanghai, China
Hesheng Hu Fudan University, Shanghai, China and
Zixiang Zhou Fudan University, Shanghai, China
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-3087-8 (HB) ISBN 1-4020-3088-6 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Springer, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Springer, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved © 2005 Springer No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in the Netherlands.
Contents
Preface
ix
1. 1+1 DIMENSIONAL INTEGRABLE SYSTEMS 1.1
1
KdV equation, MKdV equation and their Darboux transformations 1.1.1 Original Darboux transformation 1.1.2 Darboux transformation for KdV equation 1.1.3 Darboux transformation for MKdV equation 1.1.4 Examples: single and double soliton solutions 1.1.5 Relation between Darboux transformations for KdV equation and MKdV equation
10
1.2
AKNS system 1.2.1 2 × 2 AKNS system 1.2.2 N × N AKNS system
11 11 16
1.3
Darboux transformation 1.3.1 Darboux transformation for AKNS system 1.3.2 Invariance of equations under Darboux transformations 1.3.3 Darboux transformations of higher degree and the theorem of permutability 1.3.4 More results on the Darboux matrices of degree one
18 18
30
KdV hierarchy, MKdV-SG hierarchy, NLS hierarchy and AKNS system with u(N ) reduction 1.4.1 KdV hierarchy 1.4.2 MKdV-SG hierarchy 1.4.3 NLS hierarchy 1.4.4 AKNS system with u(N ) reduction
34 35 40 46 48
1.4
1 1 2 3 6
23 25
vi
DARBOUX TRANSFORMATIONS IN INTEGRABLE SYSTEMS
1.5
Darboux transformation and scattering, inverse scattering theory 1.5.1 Outline of the scattering and inverse scattering theory for the 2 × 2 AKNS system 1.5.2 Change of scattering data under Darboux transformations for su(2) AKNS system
2. 2+1 DIMENSIONAL INTEGRABLE SYSTEMS
51 51 58 65
2.1
KP equation and its Darboux transformation
65
2.2
2+1 dimensional AKNS system and DS equation
68
2.3
Darboux transformation 2.3.1 General Lax pair 2.3.2 Darboux transformation of degree one 2.3.3 Darboux transformation of higher degree and the theorem of permutability
70 70 71
2.4
Darboux transformation and binary Darboux transformation for DS equation 2.4.1 Darboux transformation for DSII equatio
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