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Proquest Dissertations MULTIPHASE AVERAGING OF PERIODIC SOLITON EQUATIONS Item Type text; Dissertation-Reproduction (electronic) Authors Forest, M. Gregory Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 25/09/2021 10:07:39 Link to Item http://hdl.handle.net/10150/298590 INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction. 1. 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Ml .18106 (3131 761-4700 MULTIPHASE AVERAGING OF PERIODIC SOLITON EQUATIONS by Mark Gregory Forest A Dissertation Submitted to the Faculty of the DEPARTMENT OF MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UlTIVERSITY OF ARIZONA 19 7 9 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE I hereby recommend that this dissertation prepared under my direction by M. Gregory Forest entitled "Multiphase Averaging of Periodic Soliton Equations" be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy h/iAnl td ykhujAUM^ AIM, X ^ H79 Dissertation Director / Date Q J As members of the Final Examination Committee, we certify that we have read this dissertation and agree that it may be presented for final defense. najn.! li) Hc-k-MAki jA iLAX. Date ^ A , lo , I 7979 Date Final approval and acceptance of this dissertation is contingent on the candidate's adec[uate performance and defense thereof at the final oral examination. 11/78 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under the rules of the library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotations from or reproduction of the manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of this material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED 1^- 0 I This dissertation is dedicated to the woman of my life, Barbara, my son, Scott, Mom and Dad, Lee, Robin and her growing family, the Perraults, the Phillips, and the Harahan Gang. iii ACKNOWLEDGMENTS My sincere thanks goes to my adviser, De.v^ McLaughlin. As a person, his patience helped me suirvive the bad times, and as a scholar, he shared with me his thirst and wealth of knowledge. I also express gratitude to Hermann Flaschka and George Lamb, from whom I have gained much more than they realize. iv TABLE OF CONTENTS Page LIST OF TABLES viii LIST OF ILLUSTRATIONS ix ABSTRACT xi CHAPTER I. SPECTRAL THEORY FOR THE PERIODIC SINE-GORDON EQUATION 1 l.I Introduction 1 1.II Preliminaries 2 1.11.1 Sine-Gordon Spectral Theory; Analogies Between Periodic and Whole-Line Problems 2 1.11.2 Transfer Matrix, Floquet Discriminant, and Floquet Solutions 6 1.11.3 Generic Properties of the Spectrum 0 9 l.III Qualitative Insight into the Spectrim a 14 l.IV Finite Degrees of Freedom 27 l.IV.l 2N Invariant Simple Spectra {E,}; Determination of a Riemann Surface 27 l.IV.2 i^-Representation of the N-Phase Wave Train. 30 l.IV.3 0-function Representation of u(x,t); N-Phase Wave Train 37 l.V Traveling-Wave Solutions (Single-Phase Wave Trains). 41 l.V.l Reduction to N = 1 41 l.V.2 Direct Ansatz Method to Display Traveling Waves.41 l.V.3 p-Representation of u(x,t) for N = 1; Catalog and Physical Characteristics of the Traveling Waves In Terms of = {E^,E2} 47 l.V.4 Infinite-Product Representation of 6-Functions and Simmiation Representation of the Traveling Waves 57 l.VI Separable Solutions (N = 2) 73 l.VI.l Definitions and Motivation 73 l.VI.2 Even Initial Data (u(x),II(x)) <=> "Open Circuit" Boundary Conditions 75 l.VI.3 Spatial Symmetry <=> Spectral Symmetry 76 V vi TABLE OF CONTENTS—Continued 1.VI.4 Consequences of the Spectral Synmetry (N = 2); Separability, Standing Waves, and Elliptic Functions 78 2. CANONICAL VARIABLES FOR THE PERIODIC SINE-GORDON EQUATION AND A METHOD OF AVERAGING 86 2.1 Introduction 86 2.II A Prescription for N-Phase Modulational Equations of Completely Integrable Hamiltonian Systems 88 2.Ill Remarks on the Validity of the Modulational Equations. 95 2.IV Canonical Variables for the Periodic Sine-Gordon Equation 103 2.V Reduction to Finite (N) Degrees of Freedom 106 2.VI Action-Angle Description for N Degrees of Freedom. 110 2.VII Method of Averaging : N = 1 Case 11A 2.VIII Future Considerations 121 3. MULTIPHASE AVERAGING AND THE INVERSE SPECTRAL SOLUTION OF THE KORTEWEG-DEVRIES EQUATION 122 3.1 Introduction 122 3.II Whitham's Single-Phase KdV Results 122 3.Ill N-Phase KdV Waves and the 1ST Interpretation of Whitham's Single-Phase Results 133 3.IV 1ST Approach to Multiphase Averaging of the KdV Equation 139 3.IV.I KdV as a Completely Integrable Hamiltonian System and Conservation Laws 139 3.IV.2 Simplification of the Averaging Integrals; Ergodic Flow on an N-Torus 142 3.IV.3 The Role of Abelian Differentials in the Modulational Equations; Summary of Results. .144 4 CONSEQUENCES OF THE INVARIANT REPRESENTATION Ji = 0 148 4.1 Introduction 148 4.II = 0: Frequencies, Wave Numbers, and Conservation of Waves 148 4.Ill ^ and the Action Integrals 150 4.IV = 0 and the Canonical Variables Approach to N-Phase Averag ing 153 4.V Concluding Remarks 159 vii TABLE OF CONTENTS—Continued Page APPENDIX A: SCATTERING MOTIVATION 161 APPENDIX B: DERIVATION OF THE DECOMPOSITION FORMULAS 168 APPENDIX C: PROOF: SPATIAL SYMMETRY <=> SPECTRAL SYMMETRY 170 APPENDIX D: PROOF OF THEOREM 2.IV.1 175 APPENDIX E: CONNECTION BETWEEN DATE y VARIABLES AND THE "y SPECTRUM" ^ 183 APPENDIX F: N DEGREES OF FREEDON; DERIVATIONS 187 APPENDIX G: RIEMANN BILINEAR IDENTITIES. 201 APPENDIX H; MULTIPHASE AVERAGING AND THE INVERSE SPECTRAL SOLUTION OF K.dV 208 REFERENCES. 278 LIST OF TABLES Table Page 1. General Structure of the Loop Integrals 1(a), 1(b), Normalization Constant C, and Period Matrix B. ........68 2. Elliptic Integral Formulas for the Normalization Constant C and Period Matrix B 69 3. Soliton Limit (E^ = E2 < 0) Formulas 70 4.
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