Lie-B¨acklund-Darboux Transformations Y. Charles Li Artyom Yurov Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Theoretical Physics Department, Kaliningrad State University, Kaliningrad, Russia Contents Preface xi Chapter 1. Introduction 1 Chapter 2. A Brief Account on B¨acklund Transformations 3 2.1. A Warm-Up Approach 3 2.2. Chen’s Method 4 2.3. Clairin’s Method 5 2.4. Hirota’s Bilinear Operator Method 5 2.5. Wahlquist-Estabrook Procedure 6 Chapter 3. Nonlinear Schr¨odinger Equation 7 3.1. Physical Background 7 3.2. Lax Pair and Floquet Theory 8 3.3. Darboux Transformation and Homoclinic Orbit 9 3.4. Linear Instability 12 3.5. Quadratic Products of Eigenfunctions 13 3.6. Melnikov Vectors 13 3.7. Melnikov Integrals 16 Chapter 4. Sine-Gordon Equation 19 4.1. Background 19 4.2. Lax Pair 19 4.3. Darboux Transformation 20 4.4. Melnikov Vector 20 4.5. Heteroclinic Cycle 23 4.6. Melnikov Vector Along the Heteroclinic Cycle 26 Chapter 5. Heisenberg Ferromagnet Equation 29 5.1. Background 29 5.2. Lax Pair 29 5.3. Darboux Transformation 30 5.4. Figure Eight Connecting to the Domain Wall 31 5.5. Floquet Theory 33 5.6. Melnikov Vectors 35 5.7. Melnikov Vector Along the Figure Eight 37 5.8. A Melnikov Function for Landau-Lifshitz-Gilbert Equation 38 Chapter 6. Vector Nonlinear Schr¨odinger Equations 41 6.1. Physical Background 41 6.2. Lax Pair 41 vii viii CONTENTS 6.3. Linearized Equations 42 6.4. Homoclinic Orbits and Figure Eight Structures 42 6.5. A Melnikov Vector 45 Chapter 7. Derivative Nonlinear Schr¨odinger Equations 47 7.1. Physical Background 47 7.2. Lax Pair 47 7.3. Darboux Transformations 48 7.4. Floquet Theory 49 7.5. Strange Tori 50 2 7.6. Whisker of the Strange T 52 7.7. Whisker of the Circle 53 7.8. Diffusion 53 2 7.9. Diffusion Along the Strange T 56 7.10. Diffusion Along the Whisker of the Circle 57 Chapter 8. Discrete Nonlinear Schr¨odinger Equation 59 8.1. Background 59 8.2. Hamiltonian Structure 59 8.3. Lax Pair and Floquet Theory 60 8.4. Examples of Floquet Spectra 61 8.5. Melnikov Vectors 62 8.6. Darboux Transformations 64 8.7. Homoclinic Orbits and Melnikov Vectors 66 Chapter 9. Davey-Stewartson II Equation 69 9.1. Background 69 9.2. Linear Stability 70 9.3. Lax Pairs and Darboux Transformation 70 9.4. Homoclinic Orbits 72 9.5. Melnikov Vectors 76 9.6. Extra Comments 80 Chapter 10. Acoustic Spectral Problem 81 10.1. Physical Background 81 10.2. Connection with Linear Schr¨odinger Operator 82 10.3. Discrete Symmetries of the Acoustic Problem 82 10.4. Crum Formulae and Dressing Chains for the Acoustic Problem 83 10.5. Harry-Dym Equation 86 10.6. Modified Harry-Dym Equation 88 10.7. Moutard Transformations 89 Chapter 11. SUSY and Spectrum Reconstructions 93 11.1. SUSY in Two Dimensions 94 11.2. The Level Addition 95 11.3. Potentials with Cylindrical Symmetry 96 11.4. Extended Supersymmetry 97 Chapter 12. Darboux Transformation for Dirac Equation 99 12.1. Dirac Equation 99 CONTENTS ix 12.2. Crum Laws 101 Chapter 13. Moutard Transformations for the 2D and 3D Schr¨odinger Equations 105 13.1. A 2D Moutard Transformation 105 13.2. A 3D Moutard Transformation 106 Chapter 14. BLP Equation 109 14.1. The Darboux Transformation of the BLP Equation 109 14.2. Crum Law 110 14.3. Exact Solutions 113 14.4. Dressing From Burgers Equation 115 Chapter 15. Goursat Equation 117 15.1. The Reduction Restriction 118 15.2. Binary Darboux Transformation 121 15.3. Moutard Transformation for 2D-MKdV Equation 122 Chapter 16. Links Among Integrable Systems 125 16.1. Borisov-Zykov’s Method 125 16.2. Higher Dimensional Systems 126 16.3. Modified Nonlinear Schr¨odinger Equations 128 16.4. NLS and Toda Lattice 129 Bibliography 131 Index 135 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 Hamil- tonian systems of classical mechanics. Icons of integrable systems are the KdV equation, sine-Gordon equation, nonlinear Schr¨odinger 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 soli- ton, multi-soliton, breather, quasi-periodic orbit, homoclinic orbit (focus of this book) etc.. There are several ways now available for obtaining these explicit so- lutions: B¨acklund 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 inte- grable Hamiltonian system. B¨acklund 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-B¨acklund-Darboux transformations which refer to both B¨acklund 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 math- ematically rigorously. Following the same spirit, one may attempt to prove the existence of chaos for near integrable nonlinear wave equations viewed as near in- tegrable 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 Mel- nikov 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 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 loving support during this work. xi CHAPTER 1 Introduction The so-called B¨acklund transformation originated from studies by S. Lie [80] [81] [82] and A. V. B¨acklund [11] [12] [13] [14] [15] on the Lie’s second question on the existence of invariant multi-valued surface transformations [5]. Lie’s first question was on the well-known Lie’s tangent transformations. The first example of a B¨acklund transformation was studied on the Bianchi’s geometrical construction of surfaces of constant negative curvatures – pseudospheres [18]. The Gauss equation of a pseudosphere can be rewritten as the sine-Gordon equation. The B¨acklund transformation for the sine-Gordon equation is an invariant transformation with a so-called B¨acklund parameter first introduced by B¨acklund. The B¨acklund pa- rameter is particularly important in Bianchi’s diagram of iterating the B¨acklund transformation to generate a so-called nonlinear superposition law [18] [19]. Im- mediate further studies on B¨acklund transformations were conducted by J. Clairin [26] and E. Goursat [40]. Darboux transformation was first introduced by Gaston Darboux [29] for the nowadays well-known one-dimensional linear Schr¨odinger equation – a special form of the Sturm-Liouville equations [84]. Darboux found a covariant transformation for the eigenfunction and the potential. The covariant transformation was built upon a particular eigenfunction at a particular value of the spectral parameter. At the beginning, it seemed that B¨acklund transformation and Darboux trans- formation are irrelevant. The first link of the two came about in 1967 when Gardner, Greene, Kruskal, and Miura related KdV equation to its Lax pair of which the spa- tial part is the one-dimensional linear Schr¨odinger equation [38]. Soon afterwards, the B¨acklund transformation for the KdV equation was found. This was the begin- ning of a renaissance of B¨acklund transformations and Darboux transformations. It turned out that the existence of a Lax pair for a nonlinear wave equation, the solvability of the Cauchy problem for the nonlinear wave equation by the inverse scattering transform [38], the existence of a B¨acklund transformation for the nonlin- ear wave equation, and the existence of a Darboux transformation for the nonlinear wave equation and its Lax pair are closely related (although clear relation is still not known). Up to now, B¨acklund transformations and Darboux transformations for most of the nonlinear wave equations solvable by the inverse scattering transform, have been found [96] [84]. The potential lies at utilizing these transformations. All the earlier books [91] [5] [3] [96] [84] [27] [95] [41] focus on using B¨acklund or Darboux or inverse scattering transformation to construct multi-soliton solutions. Such multi-soliton solutions are defined on the whole spatial space with decaying boundary conditions. When the integrable system is posed with periodic boundary conditions, the solutions are temporally quasi-periodic or periodic or homoclinic. The first part of this book will focus on homoclinic orbit. Chapter 3-9 contain many valuable formulas for homoclinic orbits and Melnikov integrals. Here the Darboux 1 2 1. INTRODUCTION transformations are not only used to generate explicit formulas for the homoclinic orbits but also interlaced with the isospectral theory of the corresponding Lax pairs to generate Melnikov vectors crucial for building the Melnikov integrals. The inte- grable systems studied in Chapter 3-9 are the so-called canonical systems each of which models a variety of different phenomena. The formulas can be directly used by the readers to study their own near integrable systems. In Chapter 2, we briefly summarize various methods for deriving B¨acklund transformations. These are all “experimental” or “trial-correction” methods. For a more detailed account on these methods, we refer the readers to the book [96]. There are not many methods for deriving Darboux transformations. The commonly used one is the dressing method [84]. Sometimes Chen’s method in Chapter 2 can be effective too. Again these are all “trial-correction” methods. Chapter 10-16 contain applications of Darboux transformations in more specific physics problems, and various connections among different systems. Here no specific boundary condition is posed. The future of Lie-B¨acklund-Darboux transformations is very bright. Besides the potential of their important applications and new transformations, it is pos- sible to broaden their notion and still end up with useful transformations. This broadening process had begun long ago e.g. the group notion in [5], the jet bundle in [96], the Moutard transformation in this book. The broadened transformations even reached the Euler equations of fluids [63] [79].