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. Vladimir Matveev, Université Bourgogne, Dijon, France Daniel Sternheimer, Université Bourgogne, Dijon, France VOLUME 28 www.com A Dressing Method in Mathematical Physics by Evgeny V. Doktorov Institute of Physics, Minsk, Belarus and Sergey B. Leble University of Technology, Gdansk, Poland www.
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. Box 17, 3300 AA Dordrecht, The Netherlands.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.com 55 udir convienmi ancor come l’essemplo 56 e l’essemplare non vanno d’un modo, 57 ché io per me indarno a ciò 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.com Contents Preface.
xv 1 Mathematical preliminaries .1 Definitions and Lie algebra interpretation .2 Hermitian ladder operators .3 Jaynes–Cummings model .3 Results for differential operators .1 Commuting ordinary differential operators .2 Direct consequences of intertwining relations in the matrix case and multidimensions .4 Hyperspherical coordinate systems and ladder operators .3 Factorization of the λ matrix .7 Elementary factorization of matrix .8 Matrix factorizations and integrable systems .1 Definition of quasideterminants .2 Noncommutative Sylvester–Toda lattices .3 Noncommutative orthogonal polynomials .10 The Riemann–Hilbert problem .1 The Cauchy-type integral .2 Scalar RH problem .3 Matrix RH problem .com viii Contents 2 Factorization and classical Darboux transformations .1 Basic notations and auxiliary results.2 Generalized Bell polynomials .3 Division and factorization of differential operators. Generalized Miura equations. Generalized Burgers equations .5 Iterations and quasideterminants via Darboux transformation .6 Darboux transformations at associative ring with automorphism .7 Joint covariance of equations and nonlinear problems. Necessity conditions of covariance .1 Towards the classification scheme: joint covariance of one-field Lax pairs .8 Non-Abelian case.
Zakharov–Shabat problem .1 Joint covariance conditions for general Zakharov–Shabat equations .2 Covariant combinations of symmetric polynomials .9 A pair of difference operators .10 Non-Abelian Hirota system .12 Solutions of Nahm equations. 64 3 From elementary to twofold elementary Darboux transformation .1 Gauge transformations and general definition of Darboux transformation .2 Zakharov–Shabat equations for two projectors.3 Elementary and twofold Darboux transformations for ZS equation with three projectors .4 Elementary and twofold Darboux transformations.5 Schlesinger transformation as a special case of elementary Darboux transformation. Chains and closures .6 Twofold Darboux transformation and Bianchi–Lie formula .7 N -wave equations: example .1 Twofold DT of N -wave equations with linear term .2 Inclined soliton by twofold DT dressing of the “zero seed solution” .3 Application of classical DT to three-wave system .com Contents ix 3.8 Infinitesimal transforms for iterated Darboux transformations .9 Darboux integration of iρ̇ = [H, f (ρ)] .2 Lax pair and Darboux covariance .3 Self-scattering solutions .4 Infinite-dimensional example. Definition and application of compound elementary DT .1 Definition of compound elementary DT .2 Solution of coupled KdV–MKdV system via compound elementary DTs.
103 4 Dressing chain equations .2 Miura maps and dressing chain equations for differential operators .2 Lax pairs of differential operators .3 Periodic closure and time evolution .5 Explicit formulas for solutions of chain equations (N = 3) .6 Towards the spectral curve. General finite-gap potentials .9 Operator Zakharov–Shabat problem .1 Sketch of a general algorithm .2 Lie algebra realization .3 Examples of NLS equations .10 General polynomial in T operator chains .1 Stationary equations as eigenvalue problems and chains .2 Nonlocal operators of the first order .3 Alternative spectral evolution equation .1 Hirota equations chain .2 Solution of chain equation .com x Contents 5 Dressing in 2+1 dimensions .1 Combined Darboux–Laplace transformations .2 Reduction constraints and reduction equations .3 Goursat equation, geometry, and two-dimensional MKdV equation .2 Goursat and binary Goursat transformations .4 Iterations of Moutard transformations .5 Two-dimensional KdV equation .2 Asymptotics of multikink solutions of two-dimensional KdV equation .6 Generalized Moutard transformation for two-dimensional MKdV equations .1 Definition of generalized Moutard transformation and covariance statement .2 Solutions of two-dimensional MKdV (BLMP1) equations. 159 6 Applications of dressing to linear problems .1 Gauge–Darboux and auto-gauge–Darboux transformations .2 Chains of shape-invariant superpotentials .2 Integrable potentials in quantum mechanics .3 Coulomb potential as a representative of singular potentials .4 Matrix shape-invariant potentials .3 Zero-range potentials, dressing, and electron–molecule scattering .1 ZRPs and Darboux transformations .2 Dressing of ZRPs .4 Dressing in multicenter problem .5 Applications to Xn and YXn structures .1 Electron–Xn scattering problem .2 Electron–YXn scattering problem .3 Dressing and Ramsauer–Taunsend minimum .6 Green functions in multidimensions .1 Initial problem for heat equation with a reflectionless potential .com Contents xi 6.2 Resolvent of Schrödinger equation with reflectionless potential and Green functions .7 Remarks on d = 1 and d = 2 supersymmetry theory within the dressing scheme .1 General remarks on supersymmetric Hamiltonian/quantum mechanics .2 Symmetry and supersymmetry via dressing chains .5 Potentials with cylindrical symmetry. The Hirota method .1 Binary Bell polynomials .2 Y-systems associated with “sech2 ” soliton equations .2 Darboux-covariant Lax pairs in terms of Y-functions .3 Bäcklund transformations and Noether theorem .1 BT and infinitesimal BT .2 Noether identity and Noether theorem .3 Comment on Miura map .4 From singular manifold method to Moutard transformation .5 Zakharov–Shabat dressing method via operator factorization .1 Sketch of IST method.
222 8 Dressing via local Riemann–Hilbert problem .1 RH problem and generation of new solutions .2 Nonlinear Schrödinger equation .3 Matrix RH problem .3 Modified nonlinear Schrödinger equation .3 Matrix RH problem .4 Ablowitz–Ladik equation .com xii Contents 8.4 Ablowitz–Ladik soliton .5 Three-wave resonant interaction equations .4 Solitons of three-wave equations .6 Homoclinic orbits via dressing method .1 Homoclinic orbit for NLS equation .2 MNLS equation: Floquet spectrum and Bloch solutions .3 MNLS equation: dressing of plane wave .4 MNLS equation: homoclinic solution .2 Scattering equation and RH problem .4 Evolution of RH data. 274 9 Dressing via nonlocal Riemann–Hilbert problem .1 Benjamin–Ono equation .2 Scattering equation and symmetry relations .3 Adjoint spectral problem and asymptotics .5 Evolution of spectral data .6 Solitons of BO equation .2 Kadomtsev–Petviashvili I equation—lump solutions .2 Eigenfunctions and eigenvalues .3 Scattering equation and closure relations .5 Evolution of RH data .7 KP I equation—multiple poles .3 Davey–Stewartson I equation .1 Spectral problem and analytic eigenfunctions .2 Spectral data and RH problem .3 Time evolution of spectral data and boundaries .4 Reconstruction of potential q(ξ, η, t) .com Contents xiii 10 Generating solutions via ∂¯ problem .1 Nonlinear equations with singular dispersion relations: 1+1 dimensions .1 Spectral transform and Lax pair .3 NLS–Maxwell–Bloch soliton .5 Recursion operator for Heisenberg spin chain equation with SDR .2 Nonlinear evolutions with singular dispersion relation for quadratic bundle .1 ∂¯ Problem and recursion operator .3 Nonlinear equations with singular dispersion relation: 2+1 dimensions .4 Kadomtsev–Petviashvili II equation .1 Eigenfunctions and scattering equation .2 Inverse spectral problem .5 Davey–Stewartson II equation .1 Eigenfunctions and scattering equation .2 Discrete spectrum and inverse problem solution .com Preface The emergence of a new paradigm in science offers vast perspectives for future investigations, as well as providing fresh insight into existing areas of knowl- edge, discovering hitherto unknown relations between them. We can observe this kind of process in connection with the appearance of the concept of soli- tons [465]. Understanding the fact that nonlinear modes are as fundamental as linear ones, with the advent of a rigorous formalism making it possible to find exact solutions of a wide class of physically important nonlinear equations, gave rise to “a revolution that has quietly transformed the realm of science over the past quarter century” [392].
The inverse spectral (or scattering) transform (IST) method serves as the mathematical background for the soliton theory. The development of the IST formalism affects many fields of mathematics, revealing on frequent oc- casions unexpected links between them. For example, the theory of surfaces in R3 can be considered as a chapter of the theory of solitons [468]. The modern version of IST 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 then using the Riemann–Hilbert (RH) problem [475].
The most powerful version of the dressing method incorporates the ∂¯ problem formalism. The ∂¯ prob- lem was put forward by Beals and Coifman [39, 40] as a generalization of the RH problem and was applied to the study of first-order one-dimensional spectral problems. The full-scale opportunities provided by this formalism came to be clear after the paper by Ablowitz et al. [1] devoted to solving the Kadomtsev–Petviashvili II equation.
The main achievements within this sub- ject have been summarized in the excellent books by Novikov et al. [354], Fad- deev and Takhtajan [148], Ablowitz and Clarkson [3], and Belokolos et al. [45], published more than a decade ago. Experimental aspects of the soliton physics are presented in the book by Remoissenet [373].
The elegant group-theoretical approach to integrable systems was presented in a recent book by Reyman and Semenov-tyan-Shansky [374].com xvi Preface Generally, the term “dressing” implies a 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, are realized in terms of the Bäcklund transformations which act in the space of solutions of the nonlinear equation, or the Darboux transformations (DTs) acting in the space of solutions of the associated linear problem. At the same time, it should be stressed that the term “dressed” has ap- peared for the first time perhaps in quantum field theory that operates with the states of bare and dressed particles or quasiparticles. These states are in- terconnected by operators whose properties have much in common, no matter whether we speak about electrons or phonons.
The study of these operators, which goes back to Heisenberg and Fock, was in due course one of the stimuli for active promotion of the methods of the Lie groups and algebras in physics. In mathematical physics, the operators of this sort occur under different names, like creation–annihilation, raising–lowering, or ladder operators. The factorization method [214] widely applicable in quantum mechanics consists in fact in dressing of the vacuum state by the creation operators which are obtained as a result of the factorization of the Schrödinger operator. The property of intertwining of the dressing operators is ultimately connected with the algebraic construction known as supersymmetry.
Hence, the concept of dressing is in fact considerably wider than if we were to take into account its application in soliton theory alone. Evidently, an attempt to span all the diversity of dressing applications treated in the aforementioned extended sense under the cover of a single book seems too ambitious. With regard to the authors’ scientific interests, we restrict our consideration to essentially two global aspects of the dressing method. The first one is mostly algebraical and relates to an extension of the possibili- ties of the DTs and Moutard transformations invoking new constructions and enhancing classes of objects used.
In essence, we aim to go beyond the tradi- tional scope of the Darboux–Bäcklund transformations towards the modern development like dressing chains, operator factorization on associative rings, a nonlinear von Neumann equation for the density matrix, and so on. Following our extended understanding of dressing, we demonstrate efficient use of the Darboux-like transformations for the discrete spectrum management in linear quantum mechanics.