Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics

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Preface xxiii

In Chap. 7 we connect the dressing method with the Hirota formalism.

We also explain how to construct in a general way B¨acklund transformations

proceeding from the explicit form of the DT. One more aspect of the dress-

ing theory appears within the Weiss–Tabor–Carnevale procedure of Painlev´e

analysis for partial diﬀerential equations. We derive DT formulas using the

singular manifold method. At the end of this chapter we comment on the

historical point connected with the appearance of the dressing method in the

ZS theory and suggest some revision of the technique.

The last three chapters deal with a realization of the dressing approach

in terms of complex analysis. In Chap. 8 we apply the local RH problem for

ﬁnding soliton (and some other) solutions of (1+1)-dimensional nonlinear in-

tegrable equations. The distinctive feature of the formalism used is the vector

parameterization of the discrete spectral data of the RH problem. Such a

parameterization arises naturally within the RH problem. Using an example

of the classical nonlinear Schr¨odinger equation, we demonstrate in detail the

dressing of the bare (trivial) solution which leads to the soliton. Each subse-

quent section in this chapter demonstrates a new peculiarity in the application

of the matrix RH problem. Besides, our formalism turns out to be eﬃcient

to obtain another class of solutions associated with the notion of homoclinic

orbits which arise in the case of periodic boundary conditions. The last section

contains the description of the well-known Korteweg–de Vries (KdV) equa-

tion. A purpose of this section is rather methodological: we discuss the KdV

equation in the manner most suitable for treating in the next chapter nonlin-

ear equations in terms of the nonlocal RH problem. We hope the content of

this chapter is useful to newcomers as a concise introduction to the modern

machinery of the theory of solitons.

Dressing by means of the nonlocal RH problem is the main topic of Chap. 9.

We consider three featured examples: the Benjamin–Ono (BO) equation, the

Kadomtsev–Petviashvili I (KP I) equation, and the Davey–Stewartson I (DS I)

equation. Despite the fact that all these equations are well known, most of

the results of Chap. 9 cannot be found in monographic literature. Namely,

for the BO equation we pose the reality condition from the very beginning

and account for important reductions in the space of spectral data. For the

KP I equation we describe a class of localized solutions which arise fr om the

eigenfunctions with multiple poles. The consideration of the DS I equation

is more traditional and aims to demonstrate peculiarities which occur when

using the matrix nonlocal RH problem.

Finally, Chap. 10 is devoted to the description of the

∂ method, as applied

to nonlinear integrable equations. First we develop in detail the technique,

which is based on a rather unusual symbolic calculation, and prove its eﬃ-

ciency. We apply this formalism for the analysis of nonlinear equations with

a self-consistent source (or with a nonanalytic dispersion relation) both in

1+1 and in 2+1 dimensions. The classic example of equations with a self-

consistent source is the Maxwell–Bloch equation. Following our approach, we

obtain the main results concerning the Lax pairs, the recursion operators,

xxiv Preface

gauge-equivalent counterparts, and so on. The KP II equation was histori-

cally the ﬁrst one to be successfully analyzed by means of the

∂ formalism.

We brieﬂy outline the main steps of such an analysis. The DS II equation

is considered in more detail. In particular, we describe a recently developed

metho d aimed at incorporating multiple-pole eigenfunctions for generating a

new class of localized solutions.

Some words about possible linkages of our book with those recently pub-

lished and devoted to similar subjects are in order. The part devoted to the

DT theory is complementary to the book of Matveev and Salle [324]. We

include mostly the results obtained after their book was published. We also

avoided discussing matters dealt with in the bo ok of Rogers and Schief [376]

and the quite new bo ok of Gu et al. [197] where the geometrical problems are

discussed from the scope of the Darboux appro ach. We almost do not touch

classical one-dimensional integrability discussed in the b ooks of Perelomov

[366, 367].

We are very grateful to our co lleagues P i lar Est´evez, Nadya Matsuka, Yury

Brezhnev, Marek Czachor, Vladimir Gerdjikov, Maciej Kuna, Franklin Lam-

bert, Vassilis Rothos, Mikhail Salle, Valery Shchesnovich, Johann Springael,

NikolaiUstinov,RafaelVlasov,JiankeYang,ArtemYurov,andAnatolyZa-

itsev for fruitful collaboration and exciting discussions. We a re also indebted

very much to Vladimir Matveev for valuable criticism and friendly recom-

mendations. Some ﬁgures were kindly provided by Robert Milson and Javier

Villarroel. E.V.D. is particularly thankful to the eJDS service of the Abdus

Salam International Centre for Theoretical Physics (Trieste) for information

support. Of 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

Mathematical preliminaries

In this chapter we sketch the basic mathematical notions used in this book,

starting from general relations and illustrating them by the simplest exam-

ples. We also brieﬂy review the ideas of the dressing from the viewpoint of

intertwining relations under the scope of Lie algebras [151]. There is a long

history of the applications of (semisimple) Lie algebras for determination of

operator spectra. One line dating back to Weyl [450] relates to the explicit

algebraic solution of an eigenvalue problem; an overview has been given by

Joseph and Coulson [223, 224, 225]. Perhaps the best known example of such

a construction is the quantum theory of angular momentum, including its

development for many-particle systems (from three particles to aggregates)

in terms of hyperspherical harmonics [154, 456]. The good old geometry of

surfaces and conjugate nets uses the Laplace equations and transformations

as a starting point [138]. The challenging problem of the Laplace operator

factorization, perhaps ﬁrst addressed by Laplace, created something like an

“undressing” procedure which, being cut at some step, leads to the complete

integrability. The direct attempt to extend the technique of the Lapla ce trans-

formations and invariants to higher-order operators was made in [264]. In [405]

this technique was generalized under the name of the Darboux integrability

including nonlinearity up to the ﬁrst derivatives. The search is still going on;

see the very recent paper of Tsarev [431]. It is not yet the Darboux transfor-

mation (DT) but it is precisely in this way that Moutard [340, 341] found its

transference.

Then we are concerned with the modern development of the determinant

theory related to non-Abelian rings. It appears under the name of quaside-

terminant [174]. Quasideterminants deﬁned for matrices over free skew-ﬁelds

are not an analog of the commutative determinants but rather of a ratio of

the determinant of n × n matrices to the determinants of (n − 1) × (n − 1)

submatrices. Such a deﬁnition is natural for the Darboux dressing. In the last

two sections we give basic notions of the Riemann–Hilbert (RH) problem and

∂ problem whi ch will be used in chapters devoted to solving soliton equations.

2 1 Mathematical preliminaries

1.1 Intertwining relation

We start from the notion of intertwining relation. Let us consider three

operators L, L

,andA,denotingD(L), D(L

), and D(A) their domains

of deﬁnition. Consider the equality

A = AL, (1.1)

named as an intertwining relation.

Proposition 1.1. Generally, if

Lψ =0,ψ∈ D(A), (1.2)

then

(Aψ)=0. (1.3)

In other words, the op erator A maps a solution of (1.2) onto a solution of

(1.3), if Aψ =0andAψ ∈ D(L

). The case of Aψ =0meansthatψ belongs

to the kernel of the op erator A.

Consider next an eigenvalue problem for the operator L whichactsina

Hilbert space H:

Lψ = λψ, ψ ∈ H. (1.4)

Then, owing to (1.1), L

(Aψ)=ALψ = λ(Aψ). This means that the map

A : ψ → ψ

, ψ

= Aψ links eigenspaces of operators L and L

,leaving

eigenvalues unchanged. If Aψ ∈ H for any λ and ψ, the operator A is referred

to as an isospectral transformation.

Remark 1.2. If for some ψ, Aψ = 0, then the eigenvalue λ of A does not belong

to the spectrum of A

Remark 1.3. If the operator L is factorizable, i.e., L = SA,thenA intertwines

L and

= 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 for L

= AA

Given an op erator algebra, we can derive comprehensive statements about

eigenvalues and eigenstates of operators. The important example of such a

construction (ladder operators) is given in the following section.

1.2 Ladder operators

Dressing by means of ladder operators is perhaps the most familiar example

of generating new solutions from the seed one. In this section we recall the

deﬁnition of ladder operators, discuss their Hermitian properties, and demon-

strate the diagonalization of the model Jaynes–Cummings (JC) Hamiltonian

by means o f a unitary dressing operator.

1.2 Ladder operators 3

1.2.1 Deﬁnitions and Lie algebra interpretation

The concept of ladder opera tors is widely used; they are discussed in [223,

224, 225], where their self-adjoint version is reviewed. Let us start from the

commutation relations

[M,A

]=A

, [M,A

−

]=−A

−

, (1.6)

where A

and A

−

are mutually adjoint operators. The link to the factor-

ization method (Chap. 2) is immediately seen. Rewriting, for example, the

ﬁrst relation in (1.6) as MA

= A

(M +1), one can easily check that

−

= A

−

M. So, the operators M and A

−

commute; hence, spec-

tral problems for both can be considered together and there exists a link

between the spectral parameters [80]. Such a property is often referred to as

supersymmetry [204].

The impo rtant link to the Lie algebra representation theory can be illus-

trated by the simplest example. The algebra su(1, 1) is generated by (1.6)

and

−

]=2M. (1.7)

The Casimir op erator C is constructed as the second-order Hermitian operator

C = M

−

+ A

−

), (1.8)

whose eigenvalues are equal to k(k − 1) for the unitary irreducible represen-

tations. This set deﬁnes the representation [positive discrete series D

(k)]

M|m, k >=(m + k)|m, k >, (1.9)

|m, k > =



(m +1)(m +2k)|m +1,k >,

−

|m, k > =



m(m +2k − 1)|m − 1,k >, (1.10)

where m =0, 1, 2,... . The operators A

act as lowering and raising ones for

Generally the ideas expressed by relations (1.7)–(1.10) are used in the

Cartan–Weyl representation theory of Lie algebras [205].

1.2.2 Hermitian ladder operators

The operators in (1.6), being mutually adjoint, cannot be Hermitian; how-

ever, some modiﬁcation of the theory is possible as mentioned in the previous

subsection in connection with [223, 224, 225].

4 1 Mathematical preliminaries

Considering the same example with the oper ators M , A

,andA

−

,letus

deﬁne two matrix op erators



M and





M =



M 0

0 −M





A =



0 A

−



. (1.11)

Both op erators are Hermitian by deﬁnition and their anticommutator is

[





≡ (





A +





M)=−



A, (1.12)

where the intertwining relation is also recognized:



A intertwines



and 11 −



With every eigenfunction ψ

of the operator M, a pair of eigenfunctions

(a)

and ψ

(b)

of the operator



M can be associated. The space of eigenfunctions

of this op erator is decomposed into a direct sum of two subspaces designated

by (a)and(b). The functions are spinorlike vectors,

(a)





,ψ

(b)





, (1.13)

that give solutions to the eigenvalue problems



Mψ

(a)

= mψ

(a)



Mψ

(b)

= −mψ

(b)

. (1.14)

Applying the operator



A to the eigenfunctions (1.13) and taking into account

the anticommutator (1.12) yields



Aψ

(a)

∼ ψ

(b)

m+1



Aψ

(b)

m+1

∼ ψ

(a)

Hence, we encounter again the ladder operators which move the eigenfunctions

from one subspace to the other , either increasing or decreasing the eigenvalue

m by 1.

Consider an example in which the Hermitian ladder operators app ear. Let

M be the operator of the projection of the angular momentum on the z-axis:

M = L

. (1.15)

Then we take

= L

± iL

. (1.16)

As a consequence, the operators



M and



A arewrittenintheform



M =



0 −L



= σ

(1.17)

1.2 Ladder operators 5

and



A =



0 L

− iL

+iL



= σ

+ σ

. (1.18)

Here σ

, σ

,andσ

are the Pauli matrices





,σ



0 −i



,σ



0 −1



. (1.19)

The operators A

are often rather complicated, while the Hermitian ones



and



M are simpler. The applications studied in [223, 224, 225] cover many

quantum problems, from the harmonic oscillator to the relativistic Kepler

problem; see also more recent pap ers [107, 301, 379, 457].

1.2.3 Jaynes–Cummings model

One more example of the dressing technique with the use of ladder operators

is concerned with the multimode JC model [204, 221, 300, 402].

Many phenomena of matter–radiation interaction can be described by a

model of (nearly) resonant interaction of a linearly coupled quantum radiation

ﬁeld and a two-level atomic system. The corresponding Hamiltonian is written

H = ω



[ǫ

−

+ ǫ

∗

], (1.20)

where ω and ǫ

are c-numbers. The creation and annihilation operators a

and a

obey the standard commutator algebra [a

]=δ

. The atomic spin

matrices S

, S

,andS

−

are expressed by the Pauli matrices (1.19) as

= σ

± iσ

This canonical approach has been previously applied to the original JC

model, and has proved itself to be much more eﬀective than the algebraic

approach in elucidating the physical origin of the dressing processes [82]. In

[299, 300] the dressing was applied for the two-photon-interaction Hamiltonian

H = ω



k,q

[ǫ

k,q

−

+ ǫ

∗

k,q

]. (1.21)

This Hamiltonian, like the original one-mode JC one (1.20), is diagonalizable

by a transformation in the op erator space. The eigenstates are known as states

of a dressed atom, a new physical object.

The ladder operators



k,q

−



k,q

∗

k,q

,M=



(2a

+1)

(1.22)

6 1 Mathematical preliminaries

with the commutators (1.6) and (1.7) generate the Lie algebra su(1, 1). The

corresponding group elements can be used to deﬁne generalized coherent

states [366]. If the coeﬃcients in the last term of the Hamiltonian (1.21) are

related to α

k,q

, ǫ

k,q

= ǫα

k,q

with real ǫ, it could be rewritten as

H = ω

+2ω(M −

)+V, (1.23)

where m is a boson mode number and

V =2ǫ(A

−

+ A

−

This Hamiltonian is reduced to the familiar two-mode two-quantum JC Hamil-

tonian [221] if α

k,q

=1− δ

k,q

2JC

= ω

+ ω(a

+ a

)+ǫ(a

−

+ a

). (1.24)

The Hamiltonian (1.21) is diagonalized by the dressing unitary operator

T =exp



2β

−

− A

−

)



, (1.25)

where the parameters θ and β are deﬁned by

tan θ = −

4ǫβ

2ω − ω

,β=



(N − 1+m/4)(N + m/4) − C,

where C is the Casimir op erator.

In the expression for the β, there is an eigenvalue N of the operator

ℵ =(M − m/4) + S

+1/2

that commutes with both H

and V and, owing to (1.22), has the sense of the

total numbers of excitations.

The other (group theoretical) aspect of this approach is connected with a

pioneering paper by Fock [153] reviewed in [150] and generalized in [139] to a

nonlinear case.

1.3 Results for diﬀerential operators

The greatest part of this b ook is directed to operator algebra a spects. We

consider mainly the so-called correspondences [165] of operators polynomial in

the diﬀerentiation and shift operators with matrix coeﬃcients. The particular

case of L = L

in (1.1) means simply that L and A commute. Such a “zero”

level of the study from the point of view of the intertwining relations was

initiated by Burchnal and Chaundy [80].

1.3 Results for diﬀerential operators 7

1.3.1 Commuting ordinary diﬀerential operators

Consider two commuting operators P

and Q

, polynomial in the diﬀeren-

tiation operator D and having ﬁnite-dimensional eigenspaces. There exists a

common solution η of these equations:

− λ)y =0, (Q

− μ)y =0. (1.26)

Eliminating m + n derivatives y, Dy,..., D

m+n−1

y in m + n linear homoge-

neous equations

− λ)y =0,r=0, 1,...,n− 1

and

− μ)y =0,s=0, 1,...,m−1

yields the connection (spectral curve) f (λ, μ) = 0. Analysis of terms with the

highest order in λ and μ and evaluation of the commutator P

− Q

that contains p owers of D from “0” to “m + n −1” lead to the conclusion that

the highest-order term of f(λ, μ)inλ is n and in μ is m.

The operator f (P

) maps to zero any linear combination of eigenfunc-

tions η

of diﬀerent eigenvalues of P ; hence, it is identically zero. Otherwise,

the relation f(P

) = 0 is fundamental. It could be shown that the inverse

statement is valid if m and n are interprime, i.e., when the highest (in D)

order of aQ

− bP

is equal to mn.

The algebraic construction of the “dressing” type was investigated in [80].

The existence of common solutions of (1.26) implies the existence of an op-

erator T such that the common solutions form the kernel of T ; hence, the

operators factorize, or P − h ≡ RT and Q − k ≡ ST. The commutativity of

P and Q yields RT ST = STRT and the op erator R, say, intertwines TS and

ST, i.e., RT S = STR,orS intertwines RT and TR. Such a phenomenon is

called a transference of the common factor b ecause the operators P

′

= TR

and Q

′

= TS commute. In Chap. 2 this phenomenon is used to introduce an

analog of the classical DT. By the transference of a new common factor (T

one gets P

− h

≡ T

, generating a sequence of operators.

Remark 1.4. The characteristic identity f (P, Q) = 0 is invariant with respect

to the transform P → P

′

,Q→ Q

′

The proof is based on the intertwining relat i on

Tf(P, Q)=f (P

′

)T,

which follows directly from the expansion f (P, Q)=



(P − h)

(Q − k)

after substitution of the factorized form of P and Q.

8 1 Mathematical preliminaries

Investigation of a general form of commuting operators begins with the

standard form that is obtained by a gauge transformation with a function θ:



P = θ

−1

Pθ,



Q = θ

−1

Qθ,

which leads to a unit leading co eﬃcient and to a constant second coeﬃcient.

Irreducibility of P and Q means that these operators are not represented as

functions of other commuting operators. The fact that Q−P

commutes with

P if n = mr excludes combinations of multiple orders of D; hence, the least

possible ord er is m =2,m>n, m = nr and the minimal noncommutative

duet comprises the oper ator P of the order 2 and the operator Q of the

order 3.

The case m = 2 is connected with the famous relation to stationary solu-

tions of the Korteweg–de Vries equation [353]. Here

P = D

+ u. (1.27)

The operator Q should be of odd order:

2n+1

= D

2n+1

+ β

2n−1

+ ... + θ

. (1.28)

The commutativity condition yields the recurrent formula for θ

n+1

= θ

′′′

/4+θ

′

+2θ

′

, (1.29)

which lead s to hyperelliptic functions. For example, for m =2and2n +1= 3

the diﬀerential condition which provides commutativity gives θ

′

n+1

=0.From

(1.29) we have

′′′

/4+3θ

′

=0.

Integration gives θ

′2

+4θ

= g

− g

. As a result, the explicit expressions

for P and Q are obtained in terms of the Weierstrass function ℘(x):

P = D

− 2℘(x),Q= D

− 3℘(x)D − 3℘

′

(x)/2.

1.3.2 Direct consequences of intertwining relations in the matrix

case and multidimensions

A link between the one-dimensional Schr¨odinger operators L

= −∂

/∂x

and

L = −∂

/∂x

+ u(x) in terms of the intertwining relations was established in

[137] for a potential with regular singularities (the order of poles is less than

3), vanishing at inﬁnity. We give here some details following [191].

Amonodromyψ → (∂ − σ)ψ, σ = σ(x) is called trivial if all the solutions

of the equation



−

∂

∂x

+ u(x)



ψ = λψ

are single-valued in the λ-plane. The monodromy is proved trivial iﬀ the op-

erator L is intertwined with L

by a ﬁnite product of the DTs. This means