JiankeYangNonlinearWavesinIntegrableandNonintegrableSystems推荐.pdf

JiankeYangNonlinearWavesinIntegrableandNonintegrableSystems推荐.pdf

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1、 Nonlinear Waves in Integrable and Nonintegrable Systems MM16_Yang-FM_A.indd 1 9/20/2010 10:45:04 AM About the Series The SIAM series on Mathematical Modeling and Computation draws attention to the wide range of important problems in the physical and life sciences and engineering that are addressed

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5、ogy, epidemiology, morphogenesis, biomedical engineering, reaction-diffusion in chemistry, nonlinear science, interfacial problems, solidification, combustion, transport theory, solid mechanics, nonlinear vibrations, electromagnetic theory, nonlinear optics, wave propagation, coherent structures, sc

6、attering theory, earth science, solid-state physics, and plasma physics. Jianke Yang, Nonlinear Waves in Integrable and Nonintegrable Systems A. J. Roberts, Elementary Calculus of Financial Mathematics James D. Meiss, Differential Dynamical Systems E. van Groesen and Jaap Molenaar, Continuum Modelin

7、g in the Physical Sciences Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Mller, and Birgitt Schnfisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods Ivan Markovsky, Jan C. Willems, Sabine Van Huffel, and Bart De Moor, Exact and Approximate

8、Modeling of Linear Systems: A Behavioral Approach R. M. M. Mattheij, S. W. Rienstra, and J. H. M. ten Thije Boonkkamp, Partial Differential Equations: Modeling, Analysis, Computation Johnny T. Ottesen, Mette S. Olufsen, and Jesper K. Larsen, Applied Mathematical Models in Human Physiology Ingemar Ka

9、j, Stochastic Modeling in Broadband Communications Systems Peter Salamon, Paolo Sibani, and Richard Frost, Facts, Conjectures, and Improvements for Simulated Annealing Lyn C. Thomas, David B. Edelman, and Jonathan N. Crook, Credit Scoring and Its Applications Frank Natterer and Frank Wbbeling, Mathe

10、matical Methods in Image Reconstruction Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion Michael Griebel, Thomas Dornseifer, and Tilman Neunhoeffer, Numerical Simulation in Fluid Dynamics: A Practical Introduction Khosrow Chadan, David Colto

11、n, Lassi Pivrinta, and William Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems Charles K. Chui, Wavelets: A Mathematical Tool for Signal Analysis Editorial Board Alejandro Aceves Southern Methodist University Andrea Bertozzi University of California, Los Angeles Bard Erm

12、entrout University of Pittsburgh Thomas Erneux Universit Libre de Bruxelles Bernie Matkowsky Northwestern University Robert M. Miura New Jersey Institute of Technology Michael Tabor University of Arizona Mathematical Modeling and Computation Editor-in-Chief Richard Haberman Southern Methodist Univer

13、sity MM16_Yang-FM_A.indd 2 9/20/2010 10:45:04 AM Society for Industrial and Applied Mathematics Philadelphia Nonlinear Waves in Integrable and Nonintegrable Systems Jianke Yang University of Vermont Burlington, Vermont MM16_Yang-FM_A.indd 3 9/20/2010 10:45:04 AM Copyright 2010 by the Society for Ind

14、ustrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and A

15、pplied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. MATLAB is a registered trademark of

16、 The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, , . The photo in the preface is reprinted with permission of the Department of Mathematics, Heriot-Watt University, Edinburgh, Scot

17、land. Figure 5.29 is reprinted with permission from A.V. Mamaev, M. Saffman, and A.A. Zozulya, Break-up of two-dimensional bright spatial solitons due to transverse modulation instability, Europhys. Lett. 35, 2530 (1996). Copyright 1996 EDP Sciences; http:/epljournal.edpsciences.org/. Figure 6.20 is

18、 reprinted with permission from D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Spatial solitons in optically induced gratings, Optics Letters 28, 710 (2003). Figure 6.21 is reprinted with permission from B. Eiermann et al., Phys. Rev. Lett. 92, 23401 (2004). Copyright 2004 by the America

19、n Physical Society. http:/link.aps.org/abstract/PRL/v92/e23401 Figure 6.22 is reprinted with permission from Th. Anker et al., Phys. Rev. Lett. 94, 020403 (2005). Copyright 2005 by the American Physical Society. http:/link.aps.org/abstract/PRL/v94/e020403 Figure 6.23 is reprinted with permission fro

20、m H. Martin, E.D. Eugenieva, Z. Chen, and D.N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004). Copyright 2004 by the American Physical Society. http:/link.aps.org/abstract/PRL/v92/e123902 Figure 6.24 is reprinted with permission from Bartal, Manela, Cohen, Fleischer, and Segev, Observation of S

21、econd-Band Vortex Solitons in 2D Photonic Lattices, Phys. Rev. Lett. 95, 053904 (2005). Copyright 2005 by the American Physical Society. http:/link.aps.org/abstract/PRL/v95/e053904 Figure 6.25 is reprinted with permission from R. Fisher et al., Reduced-symmetry 2D solitons, Phys Rev. Lett. 96 023905

22、 (2006). Copyright 2006 by the American Physical Society. http:/link.aps.org/abstract/PRL/v96/e023905 Library of Congress Cataloging-in-Publication Data Yang, Jianke. Nonlinear waves in integrable and nonintegrable systems / Jianke Yang. p. cm. - (Mathematical modeling and computation ; 16) Includes

23、 bibliographical references and index. ISBN 978-0-898717-05-1 (softcover) 1. Nonlinear waves. 2. Wave equation-Numerical solutions. I. Title. QA927.Y365 2010 531.1133-dc22 2010020210 is a registered trademark. MM16_Yang-FM_A.indd 4 9/20/2010 10:45:04 AM 2010/10/8 page v a105 a105 a105 a105 a105 a105

24、 a105 a105 Contents List of Figures ix Preface xxi 1 Derivation of Nonlinear Wave Equations 1 1.1 The Nonlinear Schrdinger Equation for Weakly Nonlinear Wave Packets . 1 1.2 The Generalized Nonlinear Schrdinger Equation for Light Beam Propagation . 5 1.3 Other Nonlinear Wave Equations in Physical Sy

25、stems . 9 2 Integrable Theory for the Nonlinear Schrdinger Equation 15 2.1 Inverse Scattering Transform Method . 16 2.1.1 RiemannHilbert Formulation . 17 2.1.2 Solution of the RiemannHilbert Problem . 23 2.1.3 Time Evolution of Scattering Data . 29 2.1.4 Long-Time Behavior of the Solution . 31 2.2 N

26、-Soliton Solutions . 32 2.3 Infinite Number of Conservation Laws . 37 2.4 Discrete Eigenvalues in the ZakharovShabat System . 38 2.4.1 Eigenvalue Formulae for Special Initial Conditions . 39 2.4.2 General Criteria for Discrete Eigenvalues . 43 2.4.3 Numerical Computations of Eigenvalues . 45 2.5 Clo

27、sure of ZakharovShabat Eigenstates . 50 2.6 Squared Eigenfunctions of the ZakharovShabat System . 55 2.6.1 Variations of Scattering Data via Variation of Potential . . 56 2.6.2 Variation of Potential via Variations of Scattering Data . . 59 v 2010/10/8 page vi a105 a105 a105 a105 a105 a105 a105 a105

28、 vi Contents 2.6.3 Inner Products and Closure Relation of Squared Eigenfunctions . 62 2.6.4 Extension to the General Case . 63 2.7 Squared Eigenfunctions and the Linearization Operator . 65 2.8 Recursion Operator and the AKNS Hierarchy . 68 2.9 Squared Eigenfunctions and the Recursion Operator . 71

29、2.10 Amplitude-Changing and Self-Collapsing Solitons in the AKNS Hierarchy . 73 3 Theories for Integrable Equations with Higher-Order Scattering Operators 79 3.1 Integrable Hierarchy for a Higher-Order Scattering Operator . 80 3.2 Various Reductions of the Hierarchy . 82 3.3 RiemannHilbert Problem f

30、or the Hierarchy . 84 3.4 Time Evolution of Scattering Data . 89 3.5 N-Soliton Solutions . 90 3.6 Infinite Number of Conservation Laws . 91 3.7 Closure of Eigenstates in the Higher-Order Scattering Operator . 93 3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator . . . 95 3.8.1 Square

31、d Eigenfunctions for Generic Potentials . 96 3.8.2 Squared Eigenfunctions under Potential Reductions . . . 102 3.9 Squared Eigenfunctions, the Linearization Operator, and the Recursion Operator .105 3.10 Solutions in the Manakov System .109 3.11 Solutions in a Coupled Focusing-Defocusing NLS System

32、.112 3.12 Solutions in the SasaSatsuma Equation .114 4 Soliton Perturbation Theories and Applications 119 4.1 Direct Soliton Perturbation Theory for the NLS Equation .120 4.1.1 Eigenfunctions and Adjoint Eigenfunctions of the Linearization Operator .122 4.1.2 Solution for the Perturbed Soliton .127

33、4.1.3 Evolution of a Perturbed Soliton in the NLS Equation . . 130 4.2 Higher-Order Effects on Optical Solitons .133 4.2.1 Raman Effect .135 4.2.2 Self-Steepening Effect .139 4.2.3 Third-Order Dispersion Effect .140 4.3 Weak Interactions of NLS Solitons .141 2010/10/8 page vii a105 a105 a105 a105 a1

34、05 a105 a105 a105 Contents vii 4.4 Soliton Perturbation Theory for the Complex Modified KdV Equation .150 5 Theories for Nonintegrable Equations 163 5.1 Solitary Waves in Nonintegrable Equations .163 5.2 Linearization Spectrum of Solitary Waves .165 5.3 VakhitovKolokolov Stability Criterion and Its

35、Generalization .168 5.3.1 VakhitovKolokolov Stability Criterion .169 5.3.2 Generalization of VakhitovKolokolov Criterion .175 5.4 Stability Switching at a Power Extremum .180 5.5 Nonlocal Waves and the Exponential Asymptotics Technique .183 5.6 Embedded Solitons and Their Dynamics .193 5.6.1 Isolate

36、d Embedded Solitons and Their Semistability . . . 194 5.6.2 Continuous Families of Embedded Solitons .206 5.7 Fractal Scattering in Collisions of Solitary Waves .210 5.7.1 PDE Simulation Results for Coupled NLS Equations . . . 212 5.7.2 A Reduced ODE Model and Its Analysis .214 5.8 Fractal Scatterin

37、g in Weak Interactions of Solitary Waves .225 5.8.1 PDE Simulation Results for Generalized NLS Equations . 227 5.8.2 An Asymptotic ODE Model .230 5.8.3 A Universal Separatrix Map .237 5.8.4 Fractal in the Separatrix Map .249 5.8.5 Physical Mechanism for Fractal Scatterings in Weak Interactions .254

38、5.9 Transverse Instability of Solitary Waves .256 5.9.1 Instability of Long Transverse Waves .258 5.9.2 Instability of Short Transverse Waves .260 5.9.3 Experimental Demonstrations of Transverse Instabilities . 264 5.10 Wave Collapse in the Two-Dimensional NLS Equation .265 6 Nonlinear Wave Phenomen

39、a in Periodic Media 269 6.1 One-Dimensional Gap Solitons Bifurcated from Bloch Bands andTheir Stability .271 6.1.1 Bloch Bands and Bandgaps .271 6.1.2 Envelope Equations of Bloch Waves .272 6.1.3 Locations of Envelope Solutions .275 6.1.4 Families of Gap Solitons Bifurcated from Band Edges . . 277 6

40、.1.5 Stability of Gap Solitons Bifurcated from Band Edges . . 279 2010/10/8 page viii a105 a105 a105 a105 a105 a105 a105 a105 viii Contents 6.2 One-Dimensional Gap Solitons Not Bifurcated from Bloch Bands . . . 283 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands .289 6.3.1 2D Bloch Band

41、s and Bandgaps .289 6.3.2 Envelope Equations of 2D Bloch Waves .291 6.3.3 Families of 2D Gap Solitons Bifurcated from Band Edges .294 6.4 Stability of 2D Gap Solitons Bifurcated from Bloch Bands .297 6.4.1 Analytical Calculations of Eigenvalue Bifurcations near Band Edges .298 6.4.2 Numerical Stabil

42、ity Results .310 6.5 Two-Dimensional Gap Solitons Not Bifurcated from Bloch Bands . . . 314 6.6 Experimental Results .317 7 Numerical Methods for Nonlinear Wave Equations 327 7.1 Numerical Methods for Evolution Simulations .328 7.1.1 Pseudospectral Method .328 7.1.2 Split-Step MethodAccuracy and Num

43、erical Stability . . 332 7.1.3 Integrating-Factor Method and Its Numerical Stability . . 353 7.2 Numerical Methods for Computations of Solitary Waves .358 7.2.1 Petviashvili Method .359 7.2.2 Accelerated Imaginary-Time Evolution Method .366 7.2.3 Squared-Operator Iteration Methods .374 7.2.4 Newton

44、Conjugate-Gradient Methods .381 7.3 Numerical Methods for Linear-Stability Eigenvalues of Solitary Waves .389 7.3.1 Fourier Collocation Method for the Whole Spectrum . . . 390 7.3.2 Newton Conjugate-Gradient Method for Individual Eigenvalues .397 Bibliography 405 Index 427 2010/10/8 page ix a105 a10

45、5 a105 a105 a105 a105 a105 a105 List of Figures 1 Recreation of Russells 1834 observation of solitary water waves in a canal. .xxii 1.1 A wave packet in the model equation (1.1). . 2 2.1 Two-soliton solutions |u(x,t)| in the NLS equation (2.1): (a) collision, where Re( 1 ) negationslash= Re( 2 ); (b) bound state, where Re( 1 ) = Re( 2 ). 35 2.2 (a) Output of Program 1, which gives the eigenvalue spectrum of the ZakharovShabat syste