Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics
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xii Contents
8.4.2 Analyticsolutions .................................248
8.4.3 RHproblem ......................................250
8.4.4 Ablowitz–Ladiksoliton.............................252
8.5 Three-waveresonantinteractionequations..................254
8.5.1 Jost solutions.....................................255
8.5.2 Analyticsolutions .................................256
8.5.3 RHproblem ......................................257
8.5.4 Solitonsofthree-waveequations.....................258
8.6 Homoclinic orbits via dressingmethod .....................261
8.6.1 Homoclinic orbit for NLS equation ..................261
8.6.2 MNLS equation: Floquet spectrum
and Blochsolutions................................264
8.6.3 MNLS equation:dressingofplane wave ..............266
8.6.4 MNLS equation:homoclinicsolution.................267
8.7 KdVequation...........................................269
8.7.1 Jost solutions.....................................269
8.7.2 Scattering equationandRH problem.................271
8.7.3 Inverseproblem...................................272
8.7.4 EvolutionofRHdata..............................274
8.7.5 Solitonsolution ...................................274
9 Dressing via nonlocal Riemann–Hilbert problem ...........277
9.1 Benjamin–Onoequation..................................277
9.1.1 Jost solutions.....................................278
9.1.2 Scattering equation and symmetry relations . . . . . . . . . . 280
9.1.3 Adjointspectralproblemandasymptotics ............283
9.1.4 RHproblem .....................................286
9.1.5 Evolutionof spectraldata ..........................288
9.1.6 SolitonsofBOequation............................288
9.2 Kadomtsev–Petviashvili I equation—lump solutions . . . . . . . . . . 290
9.2.1 Laxrepresentation.................................291
9.2.2 Eigenfunctions andeigenvalues......................292
9.2.3 Scattering equation and closure relations . . . . . . . . . . . . . 296
9.2.4 RHproblem ......................................297
9.2.5 EvolutionofRHdata..............................298
9.2.6 Solitonsolution ...................................299
9.2.7 KPIequation—multiplepoles ......................300
9.3 Davey–StewartsonIequation .............................306
9.3.1 Spectral problem and a nalytic eigenfunctions . . . . . . . . . 308
9.3.2 Spectraldata and RH problem......................310
9.3.3 Time evolution of spectra l data and boundaries . . . . . . . 311
9.3.4 Reconstruction of p otential q(ξ, η,t) .................315
9.3.5 (1, 1)Dromionsolution ............................317
Conten ts xiii
10 Generating solutions via
¯
∂ problem ........................319
10.1 Nonlinear equations with singular dispersion relations: 1+1
dimensions .............................................319
10.1.1 Spectral transformandLaxpair.....................320
10.1.2 Recursionoperator ................................324
10.1.3 NLS–Maxwell–Blochsoliton ........................326
10.1.4 Gaugeequivalence.................................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
¯
∂ Problemandrecursionoperator ...................331
10.2.2 Gaugetransformation..............................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 Recursionoperator ................................340
10.4 Kadomtsev–PetviashviliIIequation........................342
10.4.1 Eigenfunctionsandscattering equation...............342
10.4.2 Inversespectralproblem ...........................344
10.5 Davey–StewartsonIIequation.............................345
10.5.1 Eigenfunctionsandscattering equation...............346
10.5.2 Discrete spectrum and inverse problem solution . . . . . . . 349
10.5.3 Lump solutions ...................................351
References .................................................355
Index ......................................................379
Preface
The emergence of a new par adigm 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 p ossible to find
exact solutions of a wide class of physically imp ortant 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 a s
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 R
3
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 incorp orates the
¯
∂ problem formalism. The
¯
∂ prob-
lem was put forward by Beals and Coifman [39, 40] as a generalization of
the RH problem a nd 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. Exp erimental asp ects 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].
xv
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 B¨acklund 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 Schr¨odinger 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 e ssentially two glob al 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 g o beyond the tradi-
tional scope o f the Darb oux–B¨acklund 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 unders tanding of dressing, we demonstrate efficient use of the
Darboux-like transformations for the discrete spectrum management in linear
quantum mechanics. The second aspect of the dressing concept is largely an-
alytical and is based on the RH and ∂ formalisms following most closely the
Zakharov and Shabat ideas.
The DTs, as the representative of the direct methods in soliton theory,
provide a powerful tool to analyze and solve nonlinear equations [324] a nd
allow far-reaching generalizations. On the other hand, direct methods are not
very suitable for solving the initial-value (Cauchy) problems or to describe in-
teraction of radiation with localized objects. Therefore, the second main topic
of our book is devoted to solving the Ca uchy problem and finding localized
Preface xvii
solution of various nonlinear integrable equations in both 1+1 and 2+1 di-
mensions by means of the RH and
¯
∂ problems.
Let us briefly comment on a modern state of the art of the subjects our
book is devoted to. If ψ(x, λ)andϕ(x, μ) ∈ C are linearly independent solu-
tions of the linear equation
−ψ
xx
+ u(x)ψ = λψ
associated with the parameters λ and μ,then
ψ[1] = ψ
x
+ σψ , σ = −ϕ
x
/ϕ
is the solution of the equation
−ψ[1]
xx
+ u[1]ψ[1] = λψ[1] ,
with
u[1](x)=u +2σ
x
.
They are the analytic expressions of ψ[1] and u[1] in terms of ψ, ϕ,andu that
determine the DT.
Already the pioneering papers of Matveev [313, 314, 315] have shown that
the DT represents in fact a universal algebraic operation up to the most
advanced one [321] for associative rings. The Matveev theorem provides a
natural generalization of the DTs in the spirit of the classical approach of
Darboux [102] with a great variety of applications. Let us start with the class
of functional-differential equations for some function f(x, t) and coefficients
u
m
(x, t) belonging to the ring,
f
t
(x, t)=
N
m=−M
u
m
(x, t)T
m
(f) ,t∈ R ,
where T is an automorphism. This equation is covariant with respect to the
DT:
D
±
f = f − σ
±
T
±1
f,
with σ
±
= ϕ[T
±1
(ϕ)]
−1
. It is possible to reformulate the result for differential-
difference or difference-difference equations and give the explicit expres-
sions for the transformed coefficients [321]. From this result, the lattice and
q-deformation DTs for matrix-valued functions follow in a straightforward
way:
T (f)(x, t)=f (x + δ, t) ,x,δ∈ R
or
T (f)(x, t)=f (qx, t) ,x,q∈ R ,q=0.
xviii Preface
It is sufficient to take the limit
σf − f
x
= lim
δ→0
1
δ
D
±
f = lim
q→1
(x − xq)
−1
D
±
f
to repro duce the formalism in the case of classical differential operators [321].
The general form of the DT p ermits us to incorporate the Combesqure
and Levy transforms of conjugate nets in classical differential geometry [138],
as well as the vectorial DTs for quadrilateral lattices [128, 307].
Being the covariance transformation, the DT can be iterated and this thus
constitutes an important feature of the dressing procedure. The result of the
iterations is expressed through determinants of the Wronskian type [94]. The
universal way to generate the iterated transforms for different versions of the
DT including those containing integral operators is given in [324]; e.g., the
Abelian lattice DT results in the Casorati determinants [314, 322].
The DT theory is strictly connected with the problem of the factorization
of differential and difference T operators [271] and hence with the technique
of symbolic manipulations [298, 429, 431]. Namely, let Q
±
= ±D + σ and
H
(0)
= −D
2
+ u = Q
−
Q
+
,H
(1)
= Q
+
Q
−
= −D
2
+ u[1] .
The operators H
(i)
play an important role in quantum mechanics as the one-
dimensional energy operators. The spectral parameter λ stands for the energy
and the relation Q
+
Q
−
(Q
+
ψ
λ
)=λ(Q
+
ψ
λ
) shows the property of DTs Q
±
to
be the ladder operators. The majority of explicitly solvable models of quantum
mechanics are connected with those properties that allow us to generate new
potentials together with eigenfunctions [190, 214, 324]. The op erator of the
DT deletes the energy level that corresponds to the solution ϕ.Conversely,the
invers e transformation adds a level. So, there is a possibility to manage the
spectrum by a sequence of DTs. The intertwining relation H
(1)
Q
+
= Q
+
H
(0)
gives rise to supersymmetry algebra that is an example of infinite-dimensional
graded Lie algebras or, more generally, the Kac–Mo ody algebras. The Moutard
transformation is a map of the DT type: it connects solutions and potentials
of the equation
ψ
xy
+ u(x, y)ψ =0,
so that if ϕ and ψ are different solutions, then the solution of the twin equation
with ψ → ψ[1] and u(x, y) → u[1](x, y) can be constructed solving the system
(ψ[1]ϕ)
x
= −ϕ
2
(ψϕ
−1
)
x
, (ψ[1]ϕ)
y
= ϕ
2
(ψϕ
−1
)
y
.
The transformed potential is given by
u[1] = u − 2(log ϕ)
xy
= −u + ϕ
x
ϕ
y
/ϕ
2
together with the transformation of the wave function
ψ[1] = ψ − ϕΩ(ϕ, ψ)/Ω(ϕ, ϕ) ,
Preface xix
where Ω is the integral of the exact differential form
dΩ = ϕψ
x
dx + ψϕ
y
dy.
The Moutard equation, by a complexification of independent variables, is
transformed to the two-dimensional Schr¨odinger equation and studied in con-
nection with problems of classical differential geometry [242]. In the soliton
theory it enters the Lax pairs for some (2+1)-dimensional nonlinear equations
[3, 58]. Another generalization of the Moutard transformations leads after it-
erations to multidimensional Toda-like lattice models [435]. Note that there
is a possibility of local approximation of solutions by a sequence of Moutard
and Ribacour transformations [170]. Other applications of the DT theory in
multidimensions can be found in [26, 228, 278, 281, 287, 277]. A useful chrono-
logical survey of DTs, intertwining relations, and the factorization method is
given by Rosu [377].
A wide class of geometrical ideas and particular results of soliton surfaces
[417] in real semisimple Lie a lgebras is connected with the concept of the
Darboux matrix that seems to b e the most “Darboux-like” approach in the
whole of DT theory. Note also in this con nection the application of the DTs in
vortex and relativistic string problems initiated by the pap er of Nahm [344].
In searching for alternative formulations of the method containing the prin-
cipal ideas of the Darb oux approach, the so-called elementary DT [279] on a
differential ring was introduced [467] . Its particular case that does not depend
on solutions (only on p o tentials) is referred to as the Schlesinger transforma-
tion [389, 467]. The elementary DT in combination with a conjugate to it
generates a new transformation. This construction was named the binary DT
in [267, 270, 281]. Such a name intersects with the notion introduced in [317];
for details, see [324]. Therefore, we use the new term of twofold elementary
DT throughout this book. This transformation strictly realizes the dressing
procedure for solutions of integrable nonlinear equations. Namely, the twofold
elementary DT solves the matrix RH problem with zeros.
One of the main purposes in introducing the concept of the twofold DT
directly concerns the problem of reductions [331]. The properties of the
Zakharov–Shabat (ZS) spectral problem and its conjugate give the possi-
bility to establish a clas s of reductions by solving the simple conditions for
parameters of the elementary DTs which comprise the twofold combination
[279, 280, 434]. The symmetric form of the resulting expressions for p otentials
and wave functions make almost obvious the heredity of reduction restric-
tions [281] and underlying authomorphisms [181, 331, 361] 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]) un-
der 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 the time-dependent Schr¨odinger equation and a special one
for a conjugate problem. Combinations of twofold elementary DTs were used
xx Preface
to obtain multisolitons and other solutions of the three-level Maxwell–Bloch
equation [279]. A natural generalization of this construction consists in replac-
ing matrix elements by appropriate matrices. The most promising ap p li cations
of the technique are related to operator rings. Such an example was considered
in [267].
As regards the RH problem, its application to the study of spectral equa-
tions goes back to the 1975 pap er by Shabat [394], though Zakharov and
Shabat [473] in their classic paper used in fact a formalism closely related to
that of the RH problem. A status of the “keystone” of the soliton theory was
acquired by the RH problem as a result of the 1979 paper by Zakharov and
Shabat [475]. The next imp ortant step is associated with Manakov [305], who
put forward a concept of the nonlocal RH problem. This idea turned out to be
very profitable for integration of (2+1)-dimensional nonlinear equations (and
some integro-differential equations in 1+1 dimensions as well). In addition to
the results described in the aforementioned monographs, mention should be
made of more recent papers devoted to the application of the RH problem
to the soliton theory. This includes integration of equations associated with
more complica ted spectral problems than the ZS one (e.g., the modified Man-
akov equation [125] and the Ablowitz–Ladik equation [122, 185]). Results of
principal importance were obtained by Shchesnovich and Yang [400, 401], who
derived a novel class of solitons in 1+1 dimensions that corresponds to higher-
order zeros of the RH problem data. The soliton solutions associated with
multiple-pole eigenfunctions of the spectral problems for (2+1)-dimensional
nonlinear equations were obtained by Ablowitz and Villarroel [14, 439, 440].
The RH problem has been proved to be efficient for analysis of nearly inte-
grablesystemsaswellaswhensolitonsare subjected to smallperturbations.
The soliton perturbation theory has been elaborated on the basis of the RH
formalism in a number of papers [122, 123, 237, 398, 397, 399]. A connection
between the RH problem and the approximation theory and random matrix
ensembles is demonstrated in [113], where the steepest descent analysis for
the matrix RH problem was performed, and in [160], where the matrix RH
problem was associated with the problem of reconstructing orthogonal poly-
nomials. A closely related area of problems focuses on finding th e semiclassical
limit of the N-soliton solution for large N [302, 333].
As is known, solving the RH problem amounts to reconstructing a section-
ally meromorphic function from a given jump condition at some contour (or
contours) of the domains of meromorphy and discrete data given at the pre-
scribed singularities. Studying some nonlinear equations in 2+1 dimensions
reveals a situation when we cannot formulate the RH problem because of the
absence of domains of meromorphy. In other words, functions we work with are
nowhere meromorphic. Beals and Coifman [41] and Ablowitz et al. [1] invoked
a new tool for studying nonlinear equations, the
¯
∂ problem, which amounts
to overcoming the difficulty with meromorphy. The
¯
∂-dressing method consti-
tutes now a true foundation of the soliton theory. As the latest development
of the
¯
∂-dressing formalism, a derivation of the quasiclassical limit of the
Preface xxi
scalar nonlo cal
¯
∂-dressing problem should be mentioned [245]. Besides, the
¯
∂
problem with conjugation has been analyzed within the dressing approach by
Bogdanov and Zakharov [57].
The book is organized as follows. We begin in Chap. 1 with the introduc-
tion of some mathematical notions used throughout the book. This chapter
reviews concisely the operator technique that can be considered as one o f the
sources of the dressing ideas. We discuss its origin in Lie algebra theor y and
applications in quantum mechanics (creation–annihilation operators, angular
momentum, and spin theory), as well as in classical mechanics in the Poisson
representation. We also give the main definitions and results co ncerning the
RH boundary-value problem, both scalar and matrix, and the
¯
∂ problem.
The other important idea of the dressing technology goes back to fac-
torization of differential and difference operators discussed in Chap. 2. The
story of the factorization of operators of linear equations starts perhaps from
the classic papers by Euler [147] and Jacobi [218] (see the historical essay
in [52]). We present here a rather general construction of the factorization
[467], necessary from the point of view of the dressing theory. Of course, the
result of a right/left division of the differential operators strongly depends
on the ring/field used in the construction, but the link between factors and
the eigenstates is universal. To explain the thesis, note that the factorization
of the second-order differential operator produces the DT by the operator
L
σ
=(D − σ) [324]. The factorization of L =(−D − σ)(D − σ)=L
+
σ
L
σ
yields a new op erator L[1] = L
σ
L
+
σ
that is intertwined with L:
L[1]L
σ
= L
σ
L. (0.1)
This rela t ion is the basis of the algebraic dressing procedure, when applied
to some eigenstate of L. The theory was developed in [102] in connection
with applications in geometry [103]; it has been attracting more and more
attention from researchers since its introduction (for many developments, see
[197, 376]).
We elaborate a compact form of the solution of the factorization problem
by introducing special (Bell) polynomials for a general non-Abelian case. It
gives a direct link to the DT derivation, a cova riance theorem formulation, and
proof. Some examples complementary to those used in the books mentioned
are demonstrated. A natural connection with supersymmetry is shown.
In Chap. 3 we introduce a general non-Abelian version of the elementary
and twofold elementary DT constructed by means of an arbitrary number
of orthogonal projectors p
i
. The order of the elements in determining the
equations is therefore essential. The resulting expressions for transformations
may be represented both in general operator form and by means of “matrix
elements” x
ik
= p
i
xp
k
of the ring element x (x stands for either a potential
or a solution of the linear problem).
A comparison with the relations originating from the matrix RH problem
with zeros demonstrates the possibility to generate the projectors that con-
nect solutions of the RH problem in a simple algebraic way. More detailed
xxii Preface
exposition of this subject is given in Chap. 8. Moreover, for the same reason,
the limiting procedures may be explicitly performed without any reference to
analytic prop erties of the entries. Note that there are lots of other (advanced
in comparison with twofold) possibilities to combine elementary DTs as well
as to use them directly. It is shown how the non-Abelian geometry is induced
by the DT on a differential ring.
In the last part of Chap. 3 we study a generalization of the theory of small
deformations of iterated transforms with respect to intermediate parameters
that appear within the iteration procedure of twofold elementary DTs. The
perturbation formulas allow us to define and investigate generators of the cor-
responding group, being a symmetry group of a given hierarchy associated
with the ZS problem. Then we give examples that generali ze the N-wave sys-
tem as a zero-curvature condition of an appropriate pair of the ZS problems.
This case is chosen to show the imp ortance of this approach in both geome-
try and applied mathematics, with a perspective to apply the DT theory to
computations of eigenfunctions and eigenval u es.
The nontrivial development of methods aimed at solving spectral problems
and nonlinear equations is associated with dressing chain equations produced
by iterated DTs (Chap. 4). It is first of all a link of the DT theory to the
finite-gap p o tentials (also as solutions of integrable equations) and to the
investigation of asymptotic behavior. The role of the complete set of the DT-
covariance conditions (the so-called Miura maps) is studied. As the new object,
t-chains are constructed and superposed with the x-chains in 1+1 dimensions.
In Chap. 5 we show in detail recent results on integrable n onlinear
equations in two space and one time variables that could be solved by
the Moutard-like and the Goursat-like transformations. We use examples of
(2+1)-dimensional Boiti–Leon–Manna–Pempinelli and Boiti–Leon–Pempinelli
equations. The asymptotic formulas for the multikink solutions are analyzed.
Chapter 6 is devoted to applications of the dressing method to linear prob-
lems of quantum and classical mechanics, exemplifying thereby the “inverse”
influence of the nonlinear theory on the linear one. We br iefly review ex-
actly solvable quantum-mechanical problems on a line with potentials from
the review paper by Infeld and Hull [214] subjected to algebraic deformations.
Next we report results concerned with the radial Schr¨odinger equation and
treat via the dressing procedure the popular model of zero-range potentials.
In particula r, we dress the zero-range potentials and consider the dressing of
scattering data. Considering the DT that preserves a potential, we can con-
clude about the spectrum and eigenfunctions of the spectral problem. Going
to the problem of dressing of differential equations with matrix coefficients, we
show links to relativistic quantum equations. Some classical wave and heat-
conduction equations can be solved by the Green function constructed via
the dressing procedure. For the classical n-point system, we can associate the
Poisson bracket with a differentiation, which leads to the possibility to treat
the dressing of classical evolution as a generalized DT.