Classical and Quantum Nonlinear Integrable System Theory and Applications - A.Kundu.pdf - Matematyka - MenelSuperStar

Classical and Quantum Nonlinear Integrable Systems

Theory and Applications

Edited by

A Kundu

Saha Institute of Nuclear Physics

Calcutta, India

Institute of Physics Publishing

Bristol and Philadelphia

IOP Publishing Ltd 2003

a retrieval system or transmitted in any form or by any means, electronic,

mechanical, photocopying, recording or otherwise, without the prior permission

of the publisher. Multiple copying is permitted in accordance with the terms

of licences issued by the Copyright Licensing Agency under the terms of its

agreement with Universities UK (UUK).

British Library Cataloguing-in-Publication Data

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

ISBN 0 7503 0959 8

Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: Tom Spicer

Production Editor: Simon Laurenson

Production Control: Sarah Plenty

Cover Design: Victoria Le Billon

Marketing: Nicola Newey and Verity Cooke

Published by Institute of Physics Publishing, wholly owned by The Institute of

Physics, London

Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK

US Ofﬁce: Institute of Physics Publishing, The Public Ledger Building, Suite 929,

150 South Independence Mall West, Philadelphia, PA 19106, USA

Typeset by Sunrise Setting Ltd, Torquay, Devon, UK

Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

Contents

Preface

PA R T I

Cla ssic a l Sy st e ms

1 A journey through the Korteweg–de Vries equa tion

M Lakshmanan

1.1 Introduction

1.2 Nonlinear dispersive waves: Scott Russell phenomenon and

solitary waves

1.2.1 KdV equation and cnoidal waves and the solitary waves

1.3 The Fermi–Pasta–Ulam (FPU) numerical experiments on

anharmonic lattices

1.3.1 The FPU lattice and recurrence phenomenon

1.4 The KdV equation again!

1.4.1 Asymptotic analysis and the KdV equation

1.5 Numerical experiments of Zabusky and Kruskal: the birth of

solitons

1.5.1 Periodic boundary conditions

1.5.2 Initial condition with just two solitary waves

1.6 Hirota’s bilinearization method: explicit soliton solutions

1.6.1 One-soliton solution

1.6.2 Two-soliton solution

1.6.3 N -soliton solutions

1.6.4 Asymptotic analysis

1.7 The Miura transformation and linearization of KdV: the Lax pair

1.7.1 The Miura transformation

1.7.2 Galilean invariance and the Schr¨odinger eigenvalue

problem

1.7.3 Linearization of the KdV equation

1.7.4 Lax pair

1.8 Lax pair and the method of inverse scattering

1.8.1 The IST method for the KdV equation

1.9 Explicit soliton solutions

1.9.1 One-soliton solution (N

1 )

1.9.2 Two-soliton solution

1.9.3 N -soliton solution

1.9.4 Soliton interaction

1.9.5 Non-reﬂectionless potentials

1.10 Hamiltonian structure of KdV equation: complete integrability

1.10.1 KdV as a Hamiltonian dynamical system

1.10.2 Complete integrability of the KdV equation

1.11 Inﬁnite number of conserved densities

1.12 B¨acklund transformations

1.13 The Painlev´e property for the KdV equations

1.14 Lie and Lie–B¨acklund symmetries

1.15 Conclusion

2 The Pa inlev ´e met ho ds

R Co nte a nd M Musette

2.1 The classical programme of the Painlev´e school and its

achievements

2.2 Integrability and Painlev´e property for partial differential equations

2.3 The Painlev´e test for ODEs and PDEs

2.3.1 The Fuchsian perturbative method

2.3.2 The non-Fuchsian perturbative method

2.4 Singularity-based methods towards integrability

2.4.1 Linearizable equations

2.4.2 Auto-B¨acklund transformation of a PDE: the singular

manifold method

2.4.3 Single-valued solutions of the Bianchi IX cosmological

model

2.4.4 Polynomial ﬁrst integrals of a dynamical system

2.4.5 Solitary waves from truncations

2.4.6 First-degree birational transformations of Painlev´e

equations

2.5 Liouville integrability and Painlev´e integrability

2.6 Discretization and discrete Painlev´e equations

2.7 Conclusion

3 Discret e int eg ra bilit y

K M Tamizhmani, A Ramani, B Grammaticos and T Tamizhmani

3.1 Introduction: who is afraid of discrete systems?

3.2 The detector gallery

3.2.1 Singularity conﬁnement

3.2.2 The perturbative Painlev´e approach to discrete integrability

3.2.3 Algebraic entropy

3.2.4 The Nevanlinna theory approach

3.3 The showcase

3.3.1 The discrete KdV and its de-autonomization

3.3.2 The discrete Painlev´e equations

3.3.3 Linearizable systems

3.4 Beyond the discrete horizon

3.4.1 Differential-difference systems

3.4.2 Ultra-discrete systems

3.5 Parting words

4 The dba r met ho d: a t o o l fo r so lv ing t wo - dimensio na l int eg ra ble

evo l u t i o n P D E s

A S Fokas

4.1 Introduction

4.1.1 The dbar method

4.1.2 Coherent structures

4.1.3 Organization of this chapter

4.2 The KPI equation

4.3 The DSII equation

4.3.1 The defocusing DS equation

4.4 Summary

5 Int ro duct io n t o so lva ble la t t ice mo dels in st a t ist ica l a nd ma t hema t ica l

phy sics

Te t s u o D e g u c h i

5.1 Introduction

5.2 Solvable vertex models

5.2.1 The six-vertex model

5.2.2 The partition function and the transfer matrix

5.2.3 Diagonalization of the transfer matrix

5.2.4 The free energy of the six-vertex model

5.2.5 Critical singularity in the antiferroelectric regime near

the phase boundary

5.2.6 XXZ spin chain and the transfer matrix

5.2.7 Low-lying excited spectrum of the transfer matrix and

conformal ﬁeld theory

5.3 Various integrable models on two-dimensional lattices

5.3.1 Ising model and Potts model

5.3.2 Chiral Potts model

5.3.3 The eight-vertex model

5.3.4 IRF models

5.4 Yang–Baxter equation and the algebraic Bethe ansatz

5.4.1 Solutions to the Yang–Baxter equation

5.4.2 Algebraic Bethe ansatz

5.5 Mathematical structures of integrable lattice models

5.5.1 Braid group

5.5.2 Quantum groups (Hopf algebras)

Appendix. Commuting transfer matrices and the Yang–Baxter equations

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