Lie-Bäcklund-Darboux Transformations

Transcription

1 Surveys of Modern Mathematics Volume VIII Lie-Bäcklund-Darboux Transformations Y. Charles Li Department of Mathematics, University of Missouri Artyom Yurov Department of Theoretical Physics, Kaliningrad State University, Kaliningrad, Russia International Press HIGHER EDUCATION PRESS

2 Surveys of Modern Mathematics, Volume VIII Lie-Bäcklund-Darboux Transformations Y. Charles Li Department of Mathematics, University of Missouri Artyom Yurov Department of Theoretical Physics, Kaliningrad State University, Kaliningrad, Russia Copyright 2014 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is material in articles to which the author holds the copyright. (If the author holds copyright, notice of this will be given with the article.) In such cases, requests for permission to use or reprint should be addressed directly to the author. ISBN: Printed in the United States of America

3 SURVEYS OF MODERN MATHEMATICS Series Editors Shing-Tung Yau Harvard University Cambridge, Massachusetts U.S.A. Lizhen Ji University of Michigan, Ann Arbor U.S.A. Yat-Sun Poon University of California at Riverside U.S.A. Jie Xiao Tsinghua University Beijing, China Jean-Pierre D ly Institut Fourier Laboratoire de Mathématiques Saint-Martin d Hères, France Eduard J. N. Looijenga Mathematisch Instituut Universiteit Utrecht The Netherlands Neil Trudinger Mathematical Sciences Institute Australian National University Canberra, Australia

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5 Preface One of the mathematical miracles of the 20th century was the discovery of a group of nonlinear wave equations being integrable. These integrable systems are the infinite dimensional counterpart of the finite dimensional integrable Hamiltonian systems of classical mechanics. Icons of integrable systems are the KdV equation, sine-gordon equation, nonlinear Schrödinger equation etc. The beauty of the integrable theory is reflected by the explicit formulas of nontrivial solutions to the integrable systems. These explicit solutions bear the iconic names of soliton, multi-soliton, breather, quasi-periodic orbit, homoclinic orbit (the focus of this book) etc. There are several ways now available for obtaining these explicit solutions: Bäcklund transformation, Darboux transformation, and inverse scattering transform. The clear connection among these transforms is still an open question although they are certainly closely related. These transformations can be regarded as the counterpart of the canonical transformation of the finite dimensional integrable Hamiltonian system. Bäcklund transformation originated from a quest for Lie s second type invariant transformation rather than his tangent transformation. That brings the title of this book: Lie-Bäcklund-Darboux Transformations which refer to both Bäcklund transformations and Darboux transformations. The most famous mathematical miracle of the 20th century was probably the discovery of chaos. When the finite dimensional integrable Hamiltonian systems are under perturbations, their regular solutions can turn into chaotic solutions. For such near integrable systems, existence of chaos can sometimes be proved mathematically rigorously. Following the same spirit, one may attempt to prove the existence of chaos for near integrable nonlinear wave equations viewed as near integrable Hamiltonian partial differential equations. This has been accomplished as summarized in the book [69]. The key ingredients in this theory of chaos in partial differential equations are the explicit formulas for the homoclinic orbit and Melnikov integral. The first author s taste is to use Darboux transformation to obtain the homoclinic orbit and Melnikov integral. This will be the focus of the first part of this book. The second author s taste is to use Darboux transformation in a diversity of applications especially in higher spatial dimensions. The range of applications crosses many different fields of physics. This will be the focus of the second part of

6 vi Preface this book. This book is a result of the second author s several visits at University of Missouri as a Miller scholar. The first author would like to thank his wife Sherry and his son Brandon, and the second author would like to thank his wife Alla and his son Valerian, for their loving support during this work.

7 Contents Chapter 1 Introduction 1 Chapter 2 A Brief Account on Bäcklund Transformations A Warm-Up Approach Chen s Method Clairin s Method Hirota s Bilinear Operator Method Wahlquist-Estabrook Procedure 7 Chapter 3 Nonlinear Schrödinger Equation Physical Background Lax Pair and Floquet Theory Darboux Transformations and Homoclinic Orbit Linear Instability Quadratic Products of Eigenfunctions Melnikov Vectors Melnikov Integrals 20 Chapter 4 Sine-Gordon Equation Background Lax Pair Darboux Transformations Melnikov Vectors Heteroclinic Cycle Melnikov Vectors Along the Heteroclinic Cycle 31 Chapter 5 Heisenberg Ferromagnet Equation Background Lax Pair Darboux Transformations Figure Eight Structures Connecting to the Domain Wall Floquet Theory Melnikov Vectors 42

8 viii Contents 5.7 Melnikov Vectors Along the Figure Eight Structures A Melnikov Function for Landau-Lifshitz-Gilbert Equation 46 Chapter 6 Vector Nonlinear Schrödinger Equations Physical Background Lax Pair Linearized Equations Homoclinic Orbits and Figure Eight Structures A Melnikov Vector 51 Chapter 7 Derivative Nonlinear Schrödinger Equations Physical Background Lax Pair Darboux Transformations Floquet Theory Strange Tori Whisker of the Strange T Whisker of the Circle Diffusion Diffusion Along the Strange T Diffusion Along the Whisker of the Circle 64 Chapter 8 Discrete Nonlinear Schrödinger Equation Background Hamiltonian Structure Lax Pair and Floquet Theory Examples of Floquet Spectra Melnikov Vectors Darboux Transformations Homoclinic Orbits and Melnikov Vectors 75 Chapter 9 Davey-Stewartson II Equation Background Linear Stability Lax Pair and Darboux Transformations Homoclinic Orbits Melnikov Vectors Melnikov Integrals An Example Extra Comments 91 Chapter 10 Acoustic Spectral Problem Physical Background Connection with Linear Schrödinger Operator Discrete Symmetries of the Acoustic Problem 95

9 Contents ix 10.4 Crum Formulae and Dressing Chains for the Acoustic Problem Harry-Dym Equation Modified Harry-Dym Equation Moutard Transformations 103 Chapter 11 SUSY and Spectrum Reconstructions SUSY in Two Dimensions The Level Addition Potentials with Cylindrical Symmetry Extended Supersymmetry 112 Chapter 12 Darboux Transformations for Dirac Equation Dirac Equation Crum Law 117 Chapter 13 Moutard Transformations for the 2D and 3D Schrödinger Equations A 2D Moutard Transformation A 3D Moutard Transformation 124 Chapter 14 BLP Equation The Darboux Transformations for the BLP Equation Crum Law Exact Solutions Dressing From Burgers Equation 134 Chapter 15 Goursat Equation The Reduction Restriction Binary Darboux Transformations Moutard Transformations for 2D-MKdV Equation 143 Chapter 16 Links Among Integrable Systems Borisov-Zykov s Method Higher Dimensional Systems Modified Nonlinear Schrödinger Equations NLS and Toda Lattice 152 Bibliography 155 Index 161

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11 About the Authors Y. Charles Li works in the areas of chaos in partial differential equations, Navier-Stokes equations and turbulence, nano-technology, biological mathematics, and complex systems. His honors include a Guggenheim Fellowship, an AMS Centennial Fellowship, and the Princeton University Merit Prize. His published books include Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations, volume 128 of the Applied Mathematical Sciences series (Springer-Verlag, 1997), and Chaos in Partial Differential Equations (International Press of Boston, 2004). Artyom Yurov works in the area of mathematical physics, with expertise in applying Darboux transformations to various physical problems.