Table Of ContentA DressingMethodin MathematicalPhysics
MATHEMATICALPHYSICSSTUDIES
EditorialBoard:
MaximKontsevich,IHES,Bures-sur-Yvette,France
MassimoPorrati,NewYork,University,NewYork,U.S.A.
VladimirMatveev,Universite´Bourgogne,Dijon,France
DanielSternheimer,Universite´Bourgogne,Dijon,France
VOLUME28
A Dressing Method
in Mathematical Physics
by
Evgeny V. Doktorov
InstituteofPhysics,Minsk,Belarus
and
Sergey B. Leble
UniversityofTechnology,Gdansk,Poland
AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN978-1-4020-6138-7(HB)
ISBN978-1-4020-6140-0(e-book)
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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 cio` contemplo.
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
57 for by myself I think on this in vain.
Translated by A. Mandelbaum
Contents
Preface .................................................... xv
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 orthogonalpolynomials............. 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
vii
viii Contents
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–Shabatproblem ............... 56
2.8.1 Joint covariance conditions for general
Zakharov–Shabatequations......................... 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–Shabatequations 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
Contents ix
3.8 Infinitesimal transforms for iterated 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–Shabatproblem .......................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 Solution of chain equation ..........................139
4.12 Comments..............................................140
x Contents
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-dimensionalKdV 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-invariantsuperpotentials.............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–Taunsendminimum..........184
6.6 Green functions in multidimensions ........................186
6.6.1 Initial problem for heat equation with
a reflectionless potential............................186
Contents xi
6.6.2 Resolvent of Schro¨dinger 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 -systems associated with “sech2” soliton equations ...202
Y
7.2 Darboux-covariantLax pairs in terms of -functions .........206
Y
7.3 B¨acklund 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 Schro¨dinger 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 Schro¨dinger 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
