A Dressing Method in Mathematical Physics pdf

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 in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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 andLiealgebrainterpretation 3 1.2.2 Hermitian ladderoperators 3 1.2.3 Jaynes–Cummingsmodel 5 1.3 Resultsfordifferential operators 6 1.3.1 Commutingordinarydifferentialoperators 7 1.3.2 Direct consequences of intertwining relations inthematrix caseand multidimensions 8 1.4 Hyperspherical coo rdinate systems and ladder operators . . . . . . 10 1.5 Laplacetransformations 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 Elementaryfactorizationofmatrix 16 1.8 Matrix factorizationsandintegrablesystems 18 1.9 Quasideterminants 20 1.9.1 Definitionofquasideterminants 21 1.9.2 NoncommutativeSylvester–Todalattices 22 1.9.3 Noncommutative o rthogonal polynomials . . . . . . . . . . . . . 22 1.10 TheRiemann–Hilbertproblem 23 1.10.1 TheCauchy-typeintegral 23 1.10.2 ScalarRH problem 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 auxiliar y results. Bell polynomials . . . . . . . 32 2.2 GeneralizedBell polynomials 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 Darb oux transformation 40 2.5.1 Generalstatements 40 2.5.2 Positons 43 2.6 Darboux transformations at associative ring with automorphism 45 2.7 Joint covar iance 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-AbelianHirotasystem 60 2.11 Nahmequations 61 2.12 SolutionsofNahm 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 equationwith threeprojectors 73 3.4 Elementary and twofold Darboux transformations. Generalcase 77 3.5 Schlesinger transformation as a sp ecial case of elementary Darbouxtransformation.Chainsandclosures 80 3.6 Twofold Darb oux 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 Laxpair andDarbouxcovariance 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 DefinitionofcompoundelementaryDT 101 3.10.2 Solution of coupled KdV–MKdV system via compoundelementaryDTs 103 4 Dressing chain equations 109 4.1 Instructiveexamples 110 4.2 Miura maps and dressing chain equations for differential operators 112 4.2.1 Linearproblems 112 4.2.2 Laxpairsofdifferentialoperators 115 4.3 Periodicclosureandtime evolution 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 Towardsthespectralcurve 124 4.7 Dubrovinequations.Generalfinite-gappotentials 127 4.8 Darbouxcoordinates 129 4.9 OperatorZakharov–Shabatproblem 130 4.9.1 Sketchofageneralalgorithm 130 4.9.2 Liealgebrarealization 131 4.9.3 ExamplesofNLSequations 133 4.10 General polynomial in T operatorchains 135 4.10.1 Stationary equations as eigenvalue problems andchains 135 4.10.2 Nonlocaloperatorsofthe firstorder 136 4.10.3 Alternativespectralevolutionequation 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 Darb oux–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 MKdVequation 147 5.2 Goursat and binary Goursat transformations . . . . . . . . . . . . . . . . 149 5.3 Moutardtransformation 152 5.4 Iterationsof Moutardtransformations 152 5.5 Two-dimensionalKdV equation 153 5.5.1 Moutardtransformations 154 5.5.2 Asymptotics of multikink solutions oftwo-dimensionalKdVequation 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 quantummechanics 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 ZRPsandDarbouxtransformations 174 6.3.2 DressingofZRPs 177 6.4 Dressinginmulticenterproblem 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 BinaryBell polynomials 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¨acklundtransformationsand Noethertheorem 214 7.3.1 BT and infinitesimal BT . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 7.3.2 NoetheridentityandNoethertheorem 215 7.3.3 CommentonMiuramap 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 RHproblemandgenerationofnewsolutions 226 8.2 Nonlinear Schr¨odingerequation 228 8.2.1 Jostsolutions 228 8.2.2 Analyticsolutions 229 8.2.3 MatrixRHproblem 231 8.2.4 Solitonsolution 234 8.2.5 NLS breather 235 8.3 Modified nonlinear Schr¨odingerequation 236 8.3.1 Jostsolutions 237 8.3.2 Analyticsolutions 238 8.3.3 MatrixRHproblem 239 8.3.4 MNLSsoliton 241 8.4 Ablowitz–Ladikequation 245 8.4.1 Jostsolutions 245 [...]... recent book by Reyman and Semenov-tyan-Shansky [374] xv xvi Preface Generally, the term dressing implies a construction that contains a transformation from a simpler (bare, seed ) state of a system to a more advanced, dressed state In particular cases, dressing transformations, as the purely algebraic construction, are realized in terms of the B¨cklund transformations a which act in the space of solutions... (LT) for (1.37) has the form a → a 1 = a − ∂x ln(b − ay ), b → b−1 = b − ay , ψ → ψ−1 = ψx + a , ψy b and can be taken as a starting point in the theory of soliton equations in 2+1 dimensions [34, 168] The LT is also a kind of a dressing procedure; it leads to a “partial” factorization of the operator of (1.37) and in the case of zero Laplace invariants at some step of the LT iterations allows us to build... for any λ and ψ, the operator A is referred to as an isospectral transformation Remark 1.2 If for some ψ, A = 0, then the eigenvalue λ of A does not belong to the spectrum of A1 Remark 1.3 If the operator L is factorizable, i.e., L = SA, then A intertwines L and L1 = AS (1.5) For Hermitian L we have S = A+ , A+ is a Hermitian conjugate to A, i.e., the intertwining relation takes place automatically... the commutation relations [M, A+ ] = A+ , [M, A ] = A , (1.6) where A+ and A are mutually adjoint operators The link to the factorization method (Chap 2) is immediately seen Rewriting, for example, the first relation in (1.6) as M A+ = A+ (M + 1), one can easily check that M A+ A = A+ A M So, the operators M and A+ A commute; hence, spectral problems for both can be considered together and there... hyperspherical coordinates [153, 154] are used to describe quantum dynamical evolution of atomic and molecular aggregates, ranging from their compact states to fragments Using such a type of coordinates is directly related to a generalization of the ladder operators’ structure to many degrees of freedom In this approach, in contrast to the traditional independent-particle theory [304], a quantum-mechanical... approach We almost do not touch classical one-dimensional integrability discussed in the books of Perelomov [366, 367] We are very grateful to our colleagues Pilar Est´vez, Nadya Matsuka, Yury e Brezhnev, Marek Czachor, Vladimir Gerdjikov, Maciej Kuna, Franklin Lambert, Vassilis Rothos, Mikhail Salle, Valery Shchesnovich, Johann Springael, Nikolai Ustinov, Rafael Vlasov, Jianke Yang, Artem Yurov, and... of dressing in the sense of a procedure that algebraically connects equations of the same form with different coefficients It resembles the Schlesinger transformation (degenerate elementary DT) that we will deal with in Chap 3 1.6 Matrix factorization In this section we establish the link between the matrix factorization and the intertwining relations and recall basic facts of matrix factorization in. .. terms of dressing procedures 1.6.1 Example It was shown that a factorization (1.5) yields the intertwining relation (1.1) automatically Taking the simplest example of 2 × 2 matrices, let us consider a Hermitian matrix L as a product of mutually conjugate matrices A and A+ : L = A+ A = a ∗ c∗ b∗ d∗ ab cd Introducing the column vectors ψ = = a c |a| 2 + |c|2 a b + c∗ d b∗ a + d∗ c |b|2 + |d|2 and φ =... of the generating ZS problem In [276] an application to some operator problem (Liouville– von Neumann equation) is studied Examples of transformations of different kinds and in different contexts were introduced in [317] (see again [324]) under the name “binary.” The binary transformations in [317, 324] are a 2+1 construction based on alternative Lax pairs This is a combination of the classical DTs for... course, we are greatly indebted to our wives Tania and Ania They offered us encouragement and support when we needed it most and never failed to remind us that there is more to life than the dressing method and solitons Evgeny V Doktorov Sergey B Leble 1 Mathematical preliminaries In this chapter we sketch the basic mathematical notions used in this book, starting from general relations and illustrating them . 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. construction that contains a trans- formation from a simpler (bare, seed ) state of a system to a more advanced, dressed state. In particular cases, dressing transformations, as the purely al- gebraic construction,. is based on the dressing method proposed by Za- kharov and Shabat, first in terms of the factorization of integral operators on a line into a product of two Volterra integral operators [474] and