Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics
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A Dressing Method in Mathematical Physics
MATHEMATICAL PHYSICS STUDIES
Editorial Board:
Maxim Kontsevich, IHES, Bures-sur-Yvette, France
Massimo Porrati, New York, University, New York, U.S.A.
Vladimir Matveev, Universit
´
e Bourgogne, Dijon, France
Daniel Sternheimer, Universit
´
e Bourgogne, Dijon, France
VOLUME 28
A Dressing Method
in Mathematical Physics
by
Evgeny V. Doktorov
Institute of Physics, Minsk, Belarus
and
Sergey B. Leble
University of Technology, Gdansk, Poland
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6138-7 (HB)
ISBN 978-1-4020-6140-0 (e-book)
Published by Springer,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
c
2007 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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55 udir convienmi ancor come l’essemplo
56 e l’essemplare non vanno d’un modo,
57 ch´e io per me indarno a ci`ocontemplo.
Dante Alighieri, Divina Commedia
Paradiso, Canto XXVIII
55 then I still have to hear just how the model
56 and copy do not share in one same plan
57forbymyselfIthinkonthisinvain.
Translated by A. Mandelbaum
Contents
Preface .................................................... xv
1 Mathematical preliminaries ................................ 1
1.1 Intertwiningrelation..................................... 2
1.2 Ladderoperators........................................ 2
1.2.1 Definitions andLie algebrainterpretation ............ 3
1.2.2 Hermitianladderoperators ......................... 3
1.2.3 Jaynes–Cummingsmodel........................... 5
1.3 Resultsfordifferentialoperators........................... 6
1.3.1 Commutingordinarydifferentialoperators............ 7
1.3.2 Direct consequences of intertwining relations
in the matrix caseand multidimensions .............. 8
1.4 Hyperspherical coordinate systems and ladder operators . . . . . . 10
1.5 Laplacetransformations.................................. 11
1.6 Matrixfactorization ..................................... 14
1.6.1 Example ......................................... 14
1.6.2 QR algorithm..................................... 15
1.6.3 Factorization of the λ matrix ....................... 15
1.7 Elementaryfactorizationofmatrix......................... 16
1.8 Matrixfactorizationsandintegrablesystems ................ 18
1.9 Quasideterminants....................................... 20
1.9.1 Definition ofquasideterminants ..................... 21
1.9.2 NoncommutativeSylvester–Todalattices ............. 22
1.9.3 Noncommutative orthogonal polynomials . . . . . . . . . . . . . 22
1.10 TheRiemann–Hilbertproblem ............................ 23
1.10.1 TheCauchy-typeintegral........................... 23
1.10.2 ScalarRHproblem ................................ 26
1.10.3 MatrixRHproblem ............................... 27
1.11
¯
∂ Problem.............................................. 28
vii
viii Conten ts
2 Factorization and classical Darboux transformations ....... 31
2.1 Basic notations and auxiliary results. Bell polynomials . . . . . . . 32
2.2 GeneralizedBellpolynomials.............................. 33
2.3 Division and factorization of differential operators.
GeneralizedMiuraequations.............................. 35
2.4 Darboux transformation. Generalized Burgers equations . . . . . . 38
2.5 Iterations and quasideterminants via Darboux
transformation .......................................... 40
2.5.1 Generalstatements ................................ 40
2.5.2 Positons.......................................... 43
2.6 Darboux transformations at associative ring
withautomorphism...................................... 45
2.7 Joint covariance of equations and nonlinear problems.
Necessityconditionsofcovariance ......................... 48
2.7.1 Towards the classification scheme: joint covariance
ofone-fieldLaxpairs .............................. 48
2.7.2 Covarianceequations .............................. 53
2.7.3 Compatibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.8 Non-Abeliancase.Zakharov–Shabatproblem ............... 56
2.8.1 Joint covariance conditions for general
Zakharov–Shabatequations......................... 57
2.8.2 Covariant combinations of symmetric polynomials . . . . . 58
2.9 Apairofdifferenceoperators ............................. 59
2.10 Non-AbelianHirota system .............................. 60
2.11 Nahmequations......................................... 61
2.12 Solutions ofNahm equations.............................. 64
3 From elementary to twofold elementary Darboux
transformation ............................................. 67
3.1 Gauge transformations and general definition
ofDarbouxtransformation ............................... 68
3.2 Zakharov–Shabatequationsfortwoprojectors............... 69
3.3 Elementary and twofold Darboux transformations for ZS
equation with three projectors ............................ 73
3.4 Elementary and twofold Darboux transformations.
Generalcase............................................ 77
3.5 Schlesinger transformation as a special case of elementary
Darbouxtransformation.Chainsandclosures ............... 80
3.6 Twofold Darboux transformation and Bianchi–Lie formula . . . . 83
3.7 N-waveequations: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
seedsolution”..................................... 85
3.7.3 Application of classical DT to three-wave system . . . . . . 86
Conten ts ix
3.8 Infinitesimal transforms for iterated Darboux
transformations ......................................... 88
3.9 Darboux integration of i ˙ρ =[H, f(ρ)] ...................... 91
3.9.1 Generalremarks .................................. 91
3.9.2 LaxpairandDarbouxcovariance.................... 93
3.9.3 Self-scatteringsolutions ............................ 95
3.9.4 Infinite-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . 97
3.9.5 Comments........................................100
3.10 Further development. Definition and application
ofcompoundelementaryDT..............................101
3.10.1 Definitionof compoundelementaryDT ..............101
3.10.2 Solution of coupled KdV–MKdV system
via compound elementary DTs......................103
4 Dressing chain equations ...................................109
4.1 Instructiveexamples .....................................110
4.2 Miura maps and dressing chain equations
for differentialoperators..................................112
4.2.1 Linearproblems...................................112
4.2.2 Laxpairsof differentialoperators....................115
4.3 Periodicclosureandtimeevolution ........................116
4.4 Discretesymmetry.......................................119
4.4.1 Generalremarks ..................................119
4.4.2 Irreduciblesubspaces ..............................120
4.5 Explicit formulas for solutions of chain equations (N =3) ....122
4.6 Towardsthe spectralcurve ...............................124
4.7 Dubrovinequations.Generalfinite-gappotentials............127
4.8 Darbouxcoordinates.....................................129
4.9 OperatorZakharov–Shabatproblem .......................130
4.9.1 Sketch of a generalalgorithm .......................130
4.9.2 Liealgebrarealization .............................131
4.9.3 ExamplesofNLSequations.........................133
4.10 General p olynomial in T operator chains ...................135
4.10.1 Stationary equations as eigenvalue problems
andchains........................................135
4.10.2 Nonlocal operatorsof the first order .................136
4.10.3 Alternative spectralevolutionequation...............137
4.11 Hirotaequations ........................................138
4.11.1 Hirotaequationschain .............................138
4.11.2 Solutionof chainequation..........................139
4.12 Comments..............................................140
xContents
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 Moutardtransformation..................................152
5.4 Iterationsof Moutardtransformations......................152
5.5 Two-dimensionalKdVequation ...........................153
5.5.1 Moutardtransformations...........................154
5.5.2 Asymptotics of multikink solutions
oftwo-dimensionalKdV equation ...................154
5.6 Generalized Moutard transformation for two-dimensional
MKdVequations ........................................158
5.6.1 Definition of generalized Moutard transformation
and covariancestatement...........................158
5.6.2 Solutions of two-dimensional MKdV
(BLMP1)equations................................159
6 Applications of dressing to linear problems ................161
6.1 Generalstatements ......................................162
6.1.1 Gauge–Darboux and auto-gauge–Darboux
transformations ...................................163
6.1.2 Chains of shape-invariant superpotentials . . . . . . . . . . . . . 164
6.2 Integrablepotentialsin quantum mechanics.................166
6.2.1 Peculiarities ......................................166
6.2.2 Nonsingularpotentials .............................167
6.2.3 Coulomb potential as a representative of singular
potentials ........................................171
6.2.4 Matrixshape-invariantpotentials....................173
6.3 Zero-range potentials, dressing, and electron–molecule
scattering ..............................................174
6.3.1 ZRPs andDarboux transformations .................174
6.3.2 DressingofZRPs..................................177
6.4 Dressingin multicenter problem...........................179
6.5 Applications to X
n
and YX
n
structures....................181
6.5.1 Electron–X
n
scatteringproblem.....................182
6.5.2 Electron–YX
n
scatteringproblem ...................183
6.5.3 Dressing and Ramsauer–Taunsend minimum . . . . . . . . . . 184
6.6 Greenfunctionsinmultidimensions........................186
6.6.1 Initial problem for heat equation with
areflectionlesspotential............................186
Contents xi
6.6.2 Resolvent of Schr¨odinger equation with reflectionless
potentialand Greenfunctions.......................188
6.6.3 Diracequations ...................................191
6.7 Remarks on d =1andd = 2 supersymmetry theory within
the dressingscheme......................................191
6.7.1 General remarks on supersymmetric
Hamiltonian/quantummechanics....................191
6.7.2 Symmetry and supersymmetry via dressing chains . . . . . 193
6.7.3 d =2Supersymmetryexample......................193
6.7.4 Leveladdition ....................................195
6.7.5 Potentialswithcylindricalsymmetry.................197
7 Important links ............................................199
7.1 Bilinear formalism. The Hirota method . . . . . . . . . . . . . . . . . . . . . 199
7.1.1 BinaryBellpolynomials............................200
7.1.2 Y-systems associated with “sech
2
” soliton equations . . . 202
7.2 Darboux-covariant Lax pairs in terms of Y-functions.........206
7.3 B¨acklundtransformationsandNoether theorem .............214
7.3.1 BT and infinitesimal BT . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.3.2 NoetheridentityandNoethertheorem ...............215
7.3.3 Commenton Miuramap ...........................217
7.4 From singular manifold method to Moutard transformation . . . 217
7.5 Zakharov–Shabat dressing method
via operatorfactorization.................................218
7.5.1 SketchofISTmethod..............................218
7.5.2 Dressibleoperators ................................219
7.5.3 Example .........................................222
8 Dressing via local Riemann–Hilbert problem ..............225
8.1 RHproblemandgenerationof new solutions................226
8.2 Nonlinear Schr¨odingerequation ...........................228
8.2.1 Jost solutions.....................................228
8.2.2 Analyticsolutions .................................229
8.2.3 MatrixRH problem ...............................231
8.2.4 Solitonsolution ...................................234
8.2.5 NLSbreather .....................................235
8.3 Modified nonlinear Schr¨odingerequation ...................236
8.3.1 Jost solutions.....................................237
8.3.2 Analyticsolutions .................................238
8.3.3 MatrixRH problem ...............................239
8.3.4 MNLS soliton.....................................241
8.4 Ablowitz–Ladikequation .................................245
8.4.1 Jost solutions.....................................245