A dressing method in mathematical physics / by Evgeny V. Doktorov and Sergey B. Leble.

Format:

Book

Author/Creator:

Doktorov, Evgeny V.

Contributor:

Leble, S. B. (Sergeĭ Borisovich)

Series:

Mathematical physics studies 0921-3767 ; v. 28.

Mathematical physics studies, 0921-3767 ; v. 28

Language:

English

Subjects (All):

Mathematical physics--Methods.

Mathematical physics.

Differential equations.

Physical Description:

xxiv, 383 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht : Springer, 2007.

Summary:

This monograph systematically develops and considers the so-called "dressing method" for solving differential equations (both linear and nonlinear), a means to generate new non-trivial solutions for a given equation from the (perhaps trivial) solution of the same or related equation.

The primary topics of the dressing method covered here are:

the Moutard and Darboux transformations discovered in XIX century as applied to linear equations;

the Backlund transformation in differential geometry of surfaces;

the factorization method; and

the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for solition equations, plus its extension in terms of the d-bar formalism.

Throughout, the text exploits the "linear experience" of presentation, with special attention given to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization, are also discussed in detail. What's more, the applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated via various nonlinear equations.

Contents:

1 Mathematical preliminaries 1

1.1 Intertwining relation 2

1.2 Ladder operators 2

1.2.1 Definitions and Lie algebra interpretation 3

1.2.2 Hermitian ladder operators 3

1.2.3 Jaynes-Cummings model 5

1.3 Results for differential operators 6

1.3.1 Commuting ordinary differential operators 7

1.3.2 Direct consequences of intertwining relations in the matrix case and multidimensions 8

1.4 Hyperspherical coordinate systems and ladder operators 10

1.5 Laplace transformations 11

1.6 Matrix factorization 14

1.6.1 Example 14

1.6.2 QR algorithm 15

1.6.3 Factorization of the λ matrix 15

1.7 Elementary factorization of matrix 16

1.8 Matrix factorizations and integrable systems 18

1.9 Quasideterminants 20

1.9.1 Definition of quasideterminants 21

1.9.2 Noncommutative Sylvester-Toda lattices 22

1.9.3 Noncommutative orthogonal polynomials 22

1.10 The Riemann-Hilbert problem 23

1.10.1 The Cauchy-type integral 23

1.10.2 Scalar RH problem 26

1.10.3 Matrix RH problem 27

1.11 ∂ Problem 28

2 Factorization and classical Darboux transformations 31

2.1 Basic notations and auxiliary results. Bell polynomials 32

2.2 Generalized Bell polynomials 33

2.3 Division and factorization of differential operators. Generalized Miura equations 35

2.4 Darboux transformation. Generalized Burgers equations 38

2.5 Iterations and quasideterminants via Darboux transformation 40

2.5.1 General statements 40

2.5.2 Positons 43

2.6 Darboux transformations at associative ring with automorphism 45

2.7 Joint covariance of equations and nonlinear problems. Necessity conditions of covariance 48

2.7.1 Towards the classification scheme: joint covariance of one-field Lax pairs 48

2.7.2 Covariance equations 53

2.7.3 Compatibility condition 56

2.8 Non-Abelian case. Zakharov-Shabat problem 56

2.8.1 Joint covariance conditions for general Zakharov-Shabat equations 57

2.8.2 Covariant combinations of symmetric polynomials 58

2.9 A pair of difference operators 59

2.10 Non-Abelian Hirota system 60

2.11 Nahm equations 61

2.12 Solutions of Nahm equations 64

3 From elementary to twofold elementary Darboux transformation 67

3.1 Gauge transformations and general definition of Darboux transformation 68

3.2 Zakharov-Shabat equations for two projectors 69

3.3 Elementary and twofold Darboux transformations for ZS equation with three Projectors 73

3.4 Elementary and twofold Darboux transformations. General case 77

3.5 Schlesinger transformation as a special case of elementary Darboux transformation. Chains and closures 80

3.6 Twofold Darboux transformation and Bianchi-Lie formula 83

3.7 N-wave equations: example 84

3.7.1 Twofold DT of N-wave equations with linear term 84

3.7.2 Inclined soliton by twofold DT dressing of the "zero seed solution" 85

3.7.3 Application of classical DT to three-wave system 86

3.8 Infinitesimal transforms for interated Darboux transformations 88

3.9 Darboux integration of iρ = [H, f(ρ)] 91

3.9.1 General remarks 91

3.9.2 Lax pair and Darboux covariance 93

3.9.3 Self-scattering solutions 95

3.9.4 Infinite-dimensional example 97

3.9.5 Comments 100

3.10 Further development. Definition and application of compound elementary DT 101

3.10.1 Definition of compound elementary DT 101

3.10.2 Solution of coupled KdV-MKdV system via compound elementary DTs 103

4 Dressing chain equations 109

4.1 Instructive examples 110

4.2 Miura maps and dressing chain equations for differential operators 112

4.2.1 Linear problems 112

4.2.2 Lax pairs of differential operators 115

4.3 Periodic closure and time evolution 116

4.4 Discrete symmetry 119

4.4.1 General remarks 119

4.4.2 Irreducible subspaces 120

4.5 Explicit formulas for solutions of chain equations (N = 3) 122

4.6 Towards the spectral curve 124

4.7 Dubrovin equations. General finite-gap potentials 127

4.8 Darboux coordinates 129

4.9 Operator Zakharov-Shabat problem 130

4.9.1 Sketch of a general algorithm 130

4.9.2 Lie algebra realization 131

4.9.3 Examples of NLS equations 133

4.10 General polynomial in T operator chains 135

4.10.1 Stationary equations as eigenvalue problems and chains 135

4.10.2 Nonlocal operators of the first order 136

4.10.3 Alternative spectral evolution equation 137

4.11 Hirota equations 138

4.11.1 Hirota equations chain 138

4.11.2 Solutions of chain equation 139

4.12 Comments 140

5 Dressing in 2+1 dimensions 141

5.1 Combined Darboux-Laplace transformations 142

5.1.1 Definitions 142

5.1.2 Reduction constraints and reduction equations 143

5.1.3 Goursat equation, geometry, and two-dimensional MKdV equation 147

5.2 Goursat and binary Goursat transformations 149

5.3 Moutard transformation 152

5.4 Iterations of Moutard transformations 152

5.5 Two-dimensional KdV equation 153

5.5.1 Moutard transformations 154

5.5.2 Asymptotics of multikink solutions of two-dimensional KdV equation 154

5.6 Generalized Moutard transformation for two-dimensional MKdV equations 158

5.6.1 Definition of generalized Moutard transformation and covariance statement 158

5.6.2 Solutions of two-dimensional MKdV (BLMP1) equations 159

6 Applications of dressing to linear problems 161

6.1 General statements 162

6.1.1 Gauge-Darboux and auto-gauge-Darboux transformations 163

6.1.2 Chains of shape-invariant superpotentials 164

6.2 Integrable potentials in quantum mechanics 166

6.2.1 Peculiarities 166

6.2.2 Nonsingular potentials 167

6.2.3 Coulomb potential as a representative of singular potentials 171

6.2.4 Matrix shape-invariant potentials 173

6.3 Zero-range potentials, dressing, and electron-molecule scattering 174

6.3.1 ZRPs and Darboux transformations 174

6.3.2 Dressing of ZRPs 177

6.4 Dressing in multicenter problem 179

6.5 Applications to Xn and YXn structures 181

6.5.1 Electron-Xn scattering problem 182

6.5.2 Electron-YXn scattering problem 183

6.5.3 Dressing and Ramsauer-Taunsend minimum 184

6.6 Green functions in multidimensions 186

6.6.1 Initial problem for heat equation with a reflectionless potential 186

6.6.2 Resolvent of Schrodinger equation with reflectionless potential and Green functions 188

6.6.3 Dirac equations 191

6.7 Remarks on d = 1 and d = 2 supersymmetry theory within the dressing scheme 191

6.7.1 General remarks on supersymmetric Hamiltonian/quantum mechanics 191

6.7.2 Symmetry and supersymmetry via dressing chains 193

6.7.3 d = 2 Supersymmetry example 193

6.7.4 Level addition 195

6.7.5 Potentials with cylindrical symmetry 197

7 Important links 199

7.1 Bilinear formalism.

The Hirota method 199

7.1.1 Binary Bell polynomials 200

7.1.2 γ-systems associates with "sech2" soliton equations 202

7.2 Darboux-covariant Lax pairs in terms of γ-functions 206

7.3 Backlund transformations and Noether theorem 214

7.3.1 BT and infinitesimal BT 214

7.3.2 Noether identity and Noether theorem 215

7.3.3 Comment on Miura map 217

7.4 From singular manifold method to Moutard transformation 217

7.5 Zakharov-Shabat dressing method via operator factorization 218

7.5.1 Sketch of IST method 218

7.5.2 Dressible operators 219

7.5.3 Example 222

8 Dressing via local Riemann-Hilbert problem 225

8.1 RH problem and generation of new solutions 226

8.2 Nonlinear Schrondinger equation 228

8.2.1 Jost solutions 228

8.2.2 Analytic solutions 229

8.2.3 Matrix RH problem 231

8.2.4 Soliton solution 234

8.2.5 NLS breather 235

8.3 Modified nonlinear Schrodinger equation 236

8.3.1 Jost solutions 237

8.3.2 Analytic solutions 238

8.3.3 Matrix RH problem 239

8.3.4 MNLS soliton 241

8.4 Ablowitz-Ladik equation 245

8.4.1 Jost solutions 245

8.4.2 Analytic solutions 248

8.4.3 RH problem 250

8.4.4 Ablowitz-Ladik soliton 252

8.5 Three-wave resonant interaction equations 254

8.5.1 Jost solutions 255

8.5.2 Analytic solutions 256

8.5.3 RH problem 257

8.5.4 Solition of three-wave equations 258

8.6 Homoclinic orbits via dressing method 261

8.6.1 Homoclinic orbit for NLS equation 261

8.6.2 MNLS equation: Floquet spectrum and Bloch solutions 264

8.6.3 MNLS equation: dressing of plane wave 266

8.6.4 MNLS equation: homoclinic solutions 267

8.7 KdV equation 269

8.7.1 Jost solutions 269

8.7.2 Scattering equation and RH problem 271

8.7.3 Inverse problem 272

8.7.4 Evolution of RH data 274

8.7.5 Soliton solution 274

9 Dressing via nonlocal Riemann-Hilbert problem 277

9.1 Benjamin-One equation 277

9.1.1 Jost solutions 278

9.1.2 Scattering equation and symmetry relations 280

9.1.3 Adjoint spectral problem and asymptotics 283

9.1.4 RH problem 286

9.1.5 Evolution of spectral data 288

9.1.6 Solitions of BO equation 288

9.2 Kadomtsev-Petviashvili I equation-lump solutions 290

9.2.1 Lax representation 291

9.2.2 Eigenfunctions and eigenvalues 292

9.2.3 Scattering equation and closure relations 296

9.2.4 RH problem 297

9.2.5 Evolution of RH data 298

9.2.6 Soliton solution 299

9.2.7 KP I equation-multiple poles 300

9.3 Davey-Stewartson I equation 306

9.3.1 Spectral problem and analytic eigenfunctions 308

9.3.2 Spectral data and RH problem 310

9.3.3 Time evolution of spectral data and boundaries 311

9.3.4 Reconstruction of potential q(ξ, η, t) 315

9.3.5 (1,1) Dromion solution 317

10 Generating solutions via ∂ problem 319

10.1 Nonlinear equations with singular dispersion relations: 1+1 dimensions 319

10.1.1 Spectral transform and Lax pair 320

10.1.2 Recursion operator 324

10.1.3 NLS-Maxwell-Bloch soliton 326

10.1.4 Gauge equivalence 327

10.1.5 Recursion operator for Heisenberg spin chain equation with SDR 328

10.2 Nonlinear evolutions with singular dispersion relation for quadratic bundle 331

10.2.1 ∂ Problem and recursion operator 331

10.2.2 Gauge transformation 334

10.3 Nonlinear equations with singular dispersion relation: 2+1 dimensions 335

10.3.1 Nonlocal ∂ problem 336

10.3.2 Dual function 339

10.3.3 Recursion operator 340

10.4 Kadomtsev-Petviashvili II equation 342

10.4.1 Eigenfunctions and scattering equation 342

10.4.2 Inverse spectral problem 344

10.5 Davey-Stewartson II equation 345

10.5.1 Eigenfunctions and scattering equation 346

10.5.2 Discrete spectrum and inverse problem solution 349

10.5.3 Lump solutions 351.

Notes:

Includes bibliographical references (pages 355-378) and index.

ISBN:

9781402061387

1402061382

OCLC:

143608555