Soliton Equations and Their Algebro-Geometric Solutions, Volume I: (1 + 1)- Dimensional Continuous Models Fritz Gesztesy and Helge Holden Frontmatter More Information

Cambridge University Press 0521753074 - Soliton Equations and their Algebro-Geometric Solutions, Volume I: (1 + 1)- Dimensional Continuous Models Fritz Gesztesy and Helge Holden Frontmatter More information

SOLITON EQUATIONS AND THEIR ALGEBRO-GEOMETRIC SOLUTIONS Volume I: (1 + 1)-Dimensional Continuous Models

The focusof thisbook ison algebro-geometric solutionsofcompletely integrable, nonlinear, partial differential equations in (1+1) dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine– Gordon, and Camassa–Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker–Akhiezer functions, and theta function representations of all relevant quantities involved. The book usestechniquesfrom the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained with ample background material in variousappendices.Detailed notesfor each chapter together with an extensive bibliography enhance the presentation offered in the main text.

© Cambridge University Press www.cambridge.org Cambridge University Press 0521753074 - Soliton Equations and their Algebro-Geometric Solutions, Volume I: (1 + 1)- Dimensional Continuous Models Fritz Gesztesy and Helge Holden Frontmatter More information

Cambridge Studiesin Advanced Mathematics Editorial Board: B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak Already published 2 K. Petersen Ergodic theory 3 P. T. Johnstone Stone spaces 5 J.-P. Kahane Some random series of functions, 2nd edition 7 J. Lambek & P. J. Scott Introduction to higher-order categorical logic 8 H. Matsumura Commutative ring theory 9 C. B. Thomas Characteristic classes and the cohomology of finite groups 10 M. Aschbacher Finite group theory 11 J. L. Alperin Local representation theory 12 P. Koosis The logarithmic integral I 13 A. Pietsch Eigenvalues and s-numbers 14 S. J. Patterson An introduction to the theory of the Riemann zeta-function 15 H. J. Baues Algebraic homotopy 16 V. S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups 17 W. Dicks& M. Dunwoody Groups acting on graphs 18 L. J. Corwin & F. P. Greenleaf Representations of nilpotent Lie groups and their applications 19 R. Fritsch & R. Piccinini Cellular structures in topology 20 H. Klingen Introductory lectures on Siegel modular forms 21 P. Koosis The logarithmic integral II 22 M. J. Collins Representations and characters of finite groups 24 H. Kunita Stochastic flows and stochastic differential equations 25 P. Wojtaszczyk Banach spaces for analysts 26 J. E. Gilbert & M. A. M. Murray Clifford algebras and Dirac operators in harmonic analysis 27 A. Frohlich & M. J. Taylor Algebraic number theory 28 K. Goebel & W. A. Kirk Topics in metric fixed point theory 29 J. F. Humphreys Reflection groups and Coxeter groups 30 D. J. Benson Representations and cohomology I 31 D. J. Benson Representations and cohomology II 32 C. Allday & V. Puppe Cohomological methods in transformation groups 33 C. Soule et al. Lectures on Arakelov geometry 34 A. Ambrosetti & G. Prodi A primer of nonlinear analysis 35 J. Palis& F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations 37 Y. Meyer Wavelets and operators 1 38 C. Weibel An introduction to homological algebra 39 W. Bruns& J. Herzog Cohen-Macaulay rings 40 V. Snaith Explicit Brauer induction 41 G. Laumon Cohomology of Drinfeld modular varieties I 42 E. B. Davies Spectral theory and differential operators 43 J. Diestel, H. Jarchow, & A. Tonge Absolutely summing operators 44 P. Mattila Geometry of sets and measures in Euclidean spaces 45 R. Pinsky Positive harmonic functions and diffusion 46 G. Tenenbaum Introduction to analytic and probabilistic number theory 47 C. Peskine An algebraic introduction to complex projective geometry 48 Y. Meyer & R. Coifman Wavelets 49 R. Stanley Enumerative combinatorics I 50 I. Porteous Clifford algebras and the classical groups 51 M. Audin Spinning tops 52 V. Jurdjevic Geometric control theory 53 H. Volklein Groups as Galois groups 54 J. Le Potier Lectures on vector bundles 55 D. Bump Automorphic forms and representations 56 G. Laumon Cohomology of Drinfeld modular varieties II 57 D. M. Clark & B. A. Davey Natural dualities for the working algebraist 58 J. McCleary A user’s guide to spectral sequences II 59 P. Taylor Practical foundations of mathematics 60 M. P. Brodmann & R. Y. Sharp Local cohomology 61 J. D. Dixon et al. Analytic pro-P groups 62 R. Stanley Enumerative combinatorics II 63 R. M. Dudley Uniform central limit theorems 64 J. Jost & X. Li-Jost Calculus of variations 65 A. J. Berrick & M. E. Keating An introduction to rings and modules 66 S. Morosawa Holomorphic dynamics 67 A. J. Berrick & M. E. Keating Categories and modules with K-theory in view 68 K. Sato Levy processes and infinitely divisible distributions 69 H. Hida Modular forms and Galois cohomology 70 R. Iorio & V. Iorio Fourier analysis and partial differential equations 71 R. Blei Analysis in integer and fractional dimensions 72 F. Borceux & G. Janelidze Galois theories 73 B. Bollobas Random graphs

SOLITON EQUATIONS AND THEIR ALGEBRO-GEOMETRIC SOLUTIONS Volume I: (1 + 1)-Dimensional Continuous Models

FRITZ GESZTESY HELGE HOLDEN University of Missouri Norwegian University of Columbia, Missouri Science and Technology USA Trondheim, Norway

published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom

cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org

C Fritz Gesztesy, Helge Holden 2003

Thisbook isin copyright. Subject to statutoryexception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2003

Printed in the United Kingdom at the University Press, Cambridge

Typeface Times10/13 pt. System LATEX2ε [TB]

A catalog record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data Gesztesy, Fritz, 1953– Soliton equations and their algebro-geometric solutions / Fritz Gesztesy, Helge Holden. p. cm. – (Cambridge studiesinadvanced mathematics; 79) Includesbibliographical referencesand index. Contents: v. 1. (1 + 1)-dimensional continuous models ISBN 0-521-75307-4 (v. 1) 1. Differential equations, Nonlinear – Numerical solutions. 2. Solitons. I. Holden, H. (Helge), 1956– II. Title. III. Series. QC20.7.D5 G47 2003 530.155355 – dc21 2002074069

ISBN 0 521 75307 4 hardback

To our parents Friederike and Franz Gesztesy Kirsten Kiellerup (in memoriam) and Finn Holden

Contents

Acknowledgments page xi

Introduction 1

1 The KdV Hierarchy 19 1.1 Contents19 1.2 The KdV Hierarchy, Recursion Relations, and Hyperelliptic Curves20 1.3 The Stationary KdV Formalism 29 1.4 The Time-Dependent KdV Formalism 64 1.5 General Trace Formulas88 1.6 Notes105

2 The Combined sine-Gordon and Modiﬁed KdV Hierarchy 123 2.1 Contents123 2.2 The sGmKdV Hierarchy, Recursion Relations, and Hyperelliptic Curves124 2.3 The Stationary sGmKdV Formalism 134 2.4 The Time-Dependent sGmKdV Formalism 149 2.5 Notes173

3 The AKNS Hierarchy 177 3.1 Contents177 3.2 The AKNS Hierarchy, Recursion Relations, and Hyperelliptic Curves178 3.3 The Stationary AKNS Formalism 190 3.4 The Time-Dependent AKNS Formalism 212 3.5 The Classical Boussinesq Hierarchy 228 3.6 Notes237

x Contents

4The Classical Massive Thirring System 242 4.1 Contents242 4.2 The Classical Massive Thirring System, Recursion Relations, and Hyperelliptic Curves243 4.3 The Basic Algebro-Geometric Formalism 249 4.4 Theta Function Representations of u,v,u∗,v∗ 267 4.5 Notes 283

5 The Camassa–Holm Hierarchy 286 5.1 Contents286 5.2 The CH Hierarchy, Recursion Relations, and Hyperelliptic Curves287 5.3 The Stationary CH Formalism 293 5.4 The Time-Dependent CH Formalism 308 5.5 Notes322

Appendices 327 A Algebraic Curvesand Their Theta Functionsin a Nutshell 327 B Hyperelliptic Curvesof the KdV-Type 355 C Hyperelliptic Curvesof the AKNS-Type 367 D Asymptotic Spectral Parameter Expansions and Nonlinear Recursion Relations 380 E Lagrange Interpolation 396 F Symmetric Functions, Trace Formulas, and Dubrovin-Type Equations401 G KdV and AKNS Darboux-Type Transformations 425 H Elliptic Functions445 I Herglotz Functions450 J Spectral Measures and Weyl–Titchmarsh m-Functions for Schr¨odinger Operators455

List of Symbols 466 Bibliography 469 Index 501

Acknowledgments

... and the manuscript was becoming an albatross about my neck. There were two possibilities: to forget about it completely, or to publish it as it stood; and I preferred the second. Robert P. Langlands1

Thismonograph hasgrown out of work we have done over the past15 years within the area of completely integrable nonlinear partial differential equations. Our starting point has been that of mathematical analysis with emphasis on spectral theoretic techniques. We have made every effort to be explicit and as detailed as possible in the presentation of results. Consequently, this volume has acquired a somewhat technical appearance. However, our experience in the extensive study of an area, especially one in which the notion of exactly solvable models dom- inatesand hence explicit formulasabound, isthat there can never be too many details— mathematicsisnot a spectatorsport. We are indebted to many co-workersin thisendeavor for the joy of collabora- tion, among them, especially, Wolfgang Bulla, Ronnie Dickson, Victor Enol’skii, Ratnam Ratnaseelan, Walter Renger, Barry Simon, Wilhelm Sticka, Gerald Teschl, Karl Unterkofler, Rudi Weikard, and Zhongxin Zhao. Special and heartfelt thanksare due to Harald Hanche-Olsenand Karl Un- terkofler. Their around-the-clock unselfish assistance with all aspects, both mathematical and TEXnical, of an enterprise of this magnitude, has been of invaluable help to us. Thanks a lot, Harald and Karl! We are grateful to all the individualswho read portionsof the manuscriptand supplied us with valuable comments and criticism. In particular, we would like to thank Wolfgang Bulla, Steve Clark, Harald Hanche-Olsen, Emma Previato, Gerald Teschl, Karl Unterkofler, and Rudi Weikard for their efforts to help us improve the presentation.

1 On the Functional Equations Satisﬁed by Eisenstein Series, Lecture Notesin Mathematics, 544, Springer, Berlin, 1976, p. III.

xii Acknowledgments

We have established a Web page with URL

www.math.ntnu.no/˜holden/solitons

where we intend to keep an updated list of typographical errors for the beneﬁt of the reader. We encourage the reader to send comments, corrections, etc., to the authors. F.G. isindebted to Sergio Albeverio, Margarida de Faria, Gis`ele Ruiz Goldstein, Jerry Goldstein, and Ludwig Streit for encouragement and support. Moreover, he thanksGis` ele and Jerry aswell asMargarida and Ludwig for kind oppor- tunitiesto lecture on variouspartsof the material in thisvolume at LSU, Baton Rouge, Louisiana, USA (April 1996) and CCM, Universidade da Madeira, Portugal (August 2001), respectively. Research in this area has been funded in part by the University of Missouri Research Board (grant RB-97-086), the Norwegian Academy of Science and Letters, and last, but certainly not least, the Research Council of Norway.

Fritz Gesztesy July 2002 Department of Mathematics University of Missouri Columbia, MO 65211, USA [email protected] www.math.missouri.edu/people/fgesztesy.html

Helge Holden Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim, Norway [email protected] www.math.ntnu.no/˜holden/