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

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2.9 A pair of diﬀerence operators 59

2.9 A pair of diﬀerence operators

Let us consider a pair of equations of the same type (2.63) for a function ψ:

(x, t)=



m=−M

ψ, (2.115)

(x, t)=

′



m=−M

′

ψ. (2.116)

The compatibility condition for them is the nonlinear equation

− V





s−k

) − U

s−k

)



(2.117)

for s = −M −M

′

, ..., N +N

′

,k∈{k

′

= − M

′

, ..., N

′

}∩{s−k = −M , ..., N }.

In the simplest case of the Zakharov–Shabat operators in both (2.115)

and (2.116) with the subclass of stationary in y solutions we obtain three

conditions:

= V

− U

= V

− U

+ V

T (U

) −U

T (V

) ,

and

T (U

)=U

T (U

) .

The connection with polynomials of a diﬀerential operator and hence with

the theory of classical Bell polynomials can be revealed if we change the

deﬁnition of potentials. It is clear that if the automorphism T is the shift

operator Tf(x)=f (x + δ), the coeﬃcients of the polynomials in T should be

arranged as follows:

(x, t)=



m=−M



r=0



m − r



(−1)

m−r

ψ. (2.118)

The recursion equation that deﬁnes classical diﬀerential Bell polynomials in

commutative variables y

, y

, ... [388],

m+1



r=0





m−r

r+1

together with the deﬁnition (2.65) of B

, conn ects these special functions.

Let us remark that the transformations for U

found in Sect. 2.6 give the

transforms for u

deﬁned by (2.118). The possibility of inverse transition

depends on the independence of functions (T −1)

f for a given T and the set

of functions ψ under consideration. The joint covariance of the system (2.115)

and (2.116) hence may be investigated along the guidelines of [260] and [270],

where the so-called binary Bell polynomials are used to form a convenient

basis.

60 2 Factorization and classical Darboux transformations

2.10 Non-Abelian Hirota system

Let us consider a pair of the Zakharov–Shabat type equations,

(x, y, t)= (V

+ V

T ) ψ (2.119)

and

(x, y, t)=



+ U

−1



ψ. (2.120)

It diﬀers from that used in the previous section by the change T → T

−1

the right-hand side of (2.120).

In a t-lattice version of equation (2.63) with j ∈ Z we go to

f (x, j +1)=



m=−M

f(x, j).

The case of a lattice in all variables is generated by the transition to

the discrete variables x, y, t → n, j, r ∈ Z, f(x, y, t) → f

(j, r), deﬁned as

in [321]. The operator T acts as the shift of n: Tf

(j, r)=f

n+1

(j, r). The

corresponding equations (2.119) and (2.120) are written as

(j − 1,r)=f

n+1

(j, r)+v(n, j, r)f

(j, r) (2.121)

and

(j, r − 1) = f

(j, r)+u(n, j, r)f

n−1

(j, r) (2.122)

with the potentials indicated. The compatibility condition of the linear equa-

tions (2.121) and (2.122) has the form

u(n, j − 1,r) − u(n +1,j,r)=v(n, j, r − 1) − v(n, j, r),

v(n, j, r − 1)u(n, j, r)=u(n, j − 1,r)v(n − 1,j,r). (2.123)

The second equation in (2.123) is automatically valid if

u(n, j, r)=τ

n+1

(j, r − 1)τ

−1

(j, r − 1)τ

n−1

(j, r)τ

−1

(j, r),

v(n, j, r)=τ

n+1

(j − 1,r)τ

−1

(j − 1,r)τ

(j, r)τ

−1

n+1

(j, r). (2.124)

It should be stressed that the order of the entries in these expressions is im-

portant. The substitution of (2.124) in the ﬁrst equation in (2.123) leads to

the generalized Hirota bilinear equation [210] (compare also with the gener-

alizations in [336]):

n+1

(j − 1,r− 1)τ

−1

(j − 1,r− 1)τ

n−1

(j − 1,r)τ

−1

(j − 1,r)

−τ

n+1

(j − 1,r− 1)τ

−1

(j − 1,r− 1)τ

n−1

(j, r − 1)τ

−1

(j, r − 1)

−τ

n+2

(j, r − 1)τ

−1

n+1

(j, r − 1)τ

(j, r)τ

−1

n+1

(j, r)

+τ

n+1

(j − 1,r)τ

−1

(j − 1,r)τ

(j, r)τ

−1

n+1

(j, r)=0. (2.125)

2.11 Nahm equations 61

In the scalar case the system reduces to the Hirota bilinear equation [321]

(j +1,r)τ

(j, r +1)− τ

(j, r)τ

(j +1,r +1)

+τ

n+1

(j +1,r)τ

n−1

(j, r +1)=0. (2.126)

Using (2.124) and the DT formalism, we could elaborate a non-Abelian

version of these equations that can be useful for applications in the theory of

quantum transfer matrices for fusion rules [255, 256] and of quantum corre-

lation functions [36, 37]. Note that the non-Abelian Hirota–Miwa equation is

discussed by Nimmo [351].

Let us return to the DT theory. Equations (2.119) and (2.120) are jointly

covariant; hence, solving equations (2.123) or (2.125) is based on the symmetry

that is generated by the joint covariance of (2.121) and (2.122) with respect

to the transformations of the type (2.111), namely,

−

(j, r)=ψ − σ

−

−1

f, σ

−

= ϕ



−1



−1

As can be easily seen, the form of both linear equations (2.121) and (2.122)

represents reductions of (2.119) and (2.120) with V

=1,V

= v, U

=1,and

−1

= u. We show further some details in the proof of the covariance theorem

because it demonstrates important features in the procedure of the derivation

of the chain equation. Let us start, say, from (2.122). The covariance conditions

are obtained from the coeﬃcients by ψ, T

−1

ψ,andT

−2

ψ. The ﬁrst o ne is valid

automatically,

−

= u − σ

−

(r − 1) + σ

−

(r) , (2.127)

−

−1

−

(r)=σ

−

(r − 1) u. (2.128)

2.11 Nahm equations

The Nahm equations [344] appear in conformal ﬁeld theory in connection with

the monopole problem. They are solved by the variational method in [129],

producing a parameterization of the Bogomolny equations. Their generaliza-

tions attract great attention in mathematical physics [101, 345].

In the following example, we change the DT formulas a bit, showing the

alternative version, similar to [381]. We stress, however, that the formulas

from Sect. 2.1 give an equivalent result. Some generalization will be needed

within the reduction constraints related to an additional (gauge) transforma-

tion denoted by g. This is expressed by the following:

Theorem 2.32. The equation

= uT ψ + vψ + wT

−1

ψ (2.129)

62 2 Factorization and classical Darboux transformations

is covariant with respect to the combined gauge–DT

ψ[1] = g(T −σ)ψ. (2.130)

Here σ =(Tφ)φ

−1

,whereφ is a solution of the same e quation (2.129) and g

is an invertible element of the ring. The tr a nsforms of the equation coeﬃcients

are

u[1] = gT(u)[T (g)]

−1

, (2.131)

v[1] = gT(v))g

−1

− gσug

−1

+ gT(u)T (σ)g

−1

+ g

−1

, (2.132)

w[1] = gσw[T

−1

(gσ)]

−1

. (2.133)

Proof. The substitution of (2.130) into the transformed equation (2.129) gives

four equations assuming T

ψ a re independent. Three of them yield trans-

formed potentials (2.131)–(2.133). The fourth equation after use of the trans-

forms takes the form

= σF − (TF)σ, (2.134)

where

F = uσ + v + w[T

−1

(σ)]

−1

One can check the condition (2.134) by direct substitution of the operator σ

and by use of the equation for φ.

Remark 2.33. Theorem 2.32 is evidently valid for the spectral problem

λψ = uT ψ + vψ + wT

−1

ψ (2.135)

with the only correction being that the last term for the transform v[1] is

absent. The equation goes to the “Riccati equation” analog for the function

σ:

μ = uσ + v + w[T

−1

(σ)]

−1

. (2.136)

Note that inserting the element σ =(Tφ)φ

−1

into (2.136) transforms it to the

spectral problem for φ (2.135) with the spectral parameter μ.

The Nahm equations can be written by means of the Lax representation

using the spectral equation (2.135) and the evolution equation

=(q + pT )ψ (2.137)

with potentials p and q. The covariance of this equation with respect t o the

DT (2.130) can be established similarly to Theorem 2.32 with account of the

y-evolution of σ(y):

= T (q)σ − σpσ + T (p)T (σ)σ − σq =0, (2.138)

which proves the following transformation formulas for the coeﬃcients in

(2.137):

p[1] = gT(p)[T (g)]

−1

2.11 Nahm equations 63

and

q[1] = g [T (q) − σp + T (p)T (σ)] g

−1

+ g

−1

The jo int covariance principle (Sect. 2.7 and [265]) deﬁnes the connectio n

between potentials p and q and u and v:

p = u + βI, q = v/2. (2.139)

Hence, the joint DT covariance means integrability of the compatibility con-

dition of equations (2.137) and (2.129), e.g., of the Nahm equations:

[uT (v) −vu]+β[T (v) − v],

= uT (w) − wT

−1

u + β[T (w) − w],

vw − wT − 1(v).

One mor e possible speciﬁcation is the use of periodic potentials in the

problem (2.137) with the evolution (2.129) with account for the connections

(2.139) that result in the appearance of commutators on the right-hand sides

of the equations. Some linear transformations and rescaling

u = α(−iϕ

/2 − ϕ

),v= ϕ

w = α

−1

(−iϕ

/2+ϕ

),q= ϕ

/2,

p = α(−iϕ

/2 − ϕ

)+βI

produce the Nahm equations for the periodic functions Tϕ

= ϕ

[periodicity

of ϕ

does not mean a periodicity of solutions ψ and φ of the Lax pair and

the corresponding σ =(Tφ)φ

−1

=iǫ

ikl

[ϕ

,ϕ

]. (2.140)

α and β are free parameters. This system is covariant with respect to the

combined DT–gauge transformations if the gauge transformation g =expG

is chosen as fo ll ows:

= α [(ϕ

+ ϕ

/2)T (σ) −σ(ϕ

+ ϕ

/2)] /2. (2.141)

Finally, the following theorem can be formulated:

Theorem 2.34. For Tϕ

= ϕ

the system (2.140) is invariant with resp ect to

the tr ansformations

[1] = g



(ϕ

/2 − iϕ

)T (g)

−1

+ σ(ϕ

/2+iϕ

)[T

−1

(gσ)]

−1



[1] = g [ϕ

+ α(iσϕ

/2 − iϕ

T (σ)/2+σϕ

− T (ϕ

σ))] g

−1

, (2.142)

[1] = g



(−iϕ

/2 − ϕ

)T (g)

−1

+ σ(−iϕ

/2+ϕ

)[T

−1

(gσ)]

−1



with the function g =expG,whereG is obtained by inte grating (2.141), if the

element σ is a solution of the system

μ = α(−iϕ

/2 − ϕ

)σ + ϕ

+ α

−1

(−iϕ

/2+ϕ

)[T

−1

(σ)]

−1

, (2.143)

=[ϕ

,σ]/2 −σ[α(−iϕ

/2 −ϕ

)+βI]σ +[α(−iϕ

/2 −ϕ

)+βI]T (σ)σ =0.

64 2 Factorization and classical Darboux transformations

The system (2.143) follows from (2.138) and (2.136).

Remark 2.35. A similar statement can be formulated for the discrete version

[342] of the Nahm system (2.140), as may be seen from the previous section.

2.12 Solutions of Nahm equations

Making use of the construction described in the previous section , we consider

a simple example. Let T be a shift operator Tψ(x, y)=ψ(x +1,y). As a seed

solution of the Nahm equations (2.140) take commuting constant matrices

= A

, i =1, 2, 3, which means constant u, v,andw. First of all we should

generate a solution of the Lax pair (2.135) and (2.137) that can b e found

in the form φ = ξ(t)Φ(x) (all elements are supposed to be invertible). The

equation for ξ is obtained as

=[v/2+(u + βI)T ]ξ = Zξ,

which is solved by

ξ =exp(Zt)ξ

Plugging Φ into (2.135) yields the spectral problem for the diﬀerence shift

operators:

μΦ(x)=ξ

−1

[uξΦ(x +1)+vξΦ + wξΦ(x − 1)].

Separating variables again, a class of particular solutions is built as

Φ = η exp(Σx);

hence, we arrive at the matrix spectral problem for η:

μη = ξ

−1

[uξη exp(Σ)+vξη + wξη exp(−Σ)] ,

with the operator on the right-hand side and, therefore, spectral parameter μ

parameterized by t. Finally, the matrix σ is composed as

σ = ξ(t)η exp(Σ)η

−1

(t).

An appropriate choice of commutator algebra for A

, Σ,andη allows us to

obtain an explicit form of σ and, hence, to construct and solve the following

equation for G:





ξ(t)η exp(Σ)η

−1

(t)

−ξ(t)η exp(Σ)η

−1

(t)





2.12 Solutions of Nahm equations 65

Its exponent (the matrix g) is necessary for the dressing formulas (2.142). We

would like to stress that the matrices σ and g do not depend on x; hence, the

dressed ϕ[i]alsodoesnot.

Starting from the known solution of (2.140), we arrive at the Euler system

for f

(y) tha t is solved in the Jacobi functions [129]. The solutions are dressed

by the transformations (2.142). A more general possibility is a direct series

solution of (2.138).

From elementary to twofold elementary

Darboux transformation

In this chapter we extend the results of Chap. 2 related to the classical Dar-

boux transformation (DT), by means of more detailed analysis of algebraic

aspects o f general t heory. Indeed, already in the pioneering paper by Matveev

[314] it was shown that the DT represents a universal algebraic operation. We

start from the intertwining relations (Sect. 1.1) and formulate in Sect. 3.1 a

general deﬁnition of the DT, as well as its connection with gauge transforma-

tions. We introd uce a concept of the elementary DT (eDT) [278] which will

play a similar role for constructing particular solutions of nonlinear equations

as the classical DT does (for a comprehensive study of the method see [433]).

In Sect. 3.2 we begin the development of a purely algebraic construction of

a matrix DT on the basis of two projectors [289]. The extension of the eDT

covariance based on the existence of idempotents and skew ﬁelds in an as-

sociative diﬀer ential ring is discussed in Sect. 3.3 using an example of three

basic projectors [267]. We stress that the twofold DT widely used as a dress-

ing too l represents a sequence of two eDTs deﬁned for mutually conjugated

Zakharov–Shabat (ZS) problems. After a detailed consideration of particular

cases in the preceding sections, we formulate in Sect. 3.4 the deﬁnitions of

the eDTs and twofold DTs for an arbitrary number of projectors [269]. Ex-

plicit formulas ar e given for both eDTs and twofold DTs. A special case of the

Schlesinger transformation is deﬁned in Sect. 3.5. The usefulness of this trans-

formation lies in the fact that it directly connects the seed and transformed

potentials. In Sect. 3.6 we demonstrate a generaliza tion of the known Bianchi–

Lie formula to the non-Abelian case. Section 3.7 is devoted to the non-Abelian

N-wave e q u ations with linear terms for illustrating the general case of an ar-

bitrary number of idempotents. We describe reduction constraints and soliton

solutions for N = 3 that account for damping and space asynchronism. In

Sect. 3.8 we show that a particular form of the twofold DTs determines a Lie

group. Hence, we can determine inﬁnitesimal transforms for the iterated DTs

which can be applied to study the stability of soliton solutions. In Sect. 3.9

we demonstrate an interesting example of using the Darboux-integration tech-

nique for solving a class of nonlinear von Neumann equations. In particular,

68 3 From elementary to twofold elementary Darboux transformation

we show that the existence of self-scattering solutions to these equations rep-

resents a generic consequence of the nonlinearities involved. In Sect. 3.10 we

introduce a notion of a compound eDT joining the structures of the classical

DTs and the eDTs. As an example, in the framework of this approach we pro-

duce explicit solutions to the integrable Korteweg–de Vries (KdV)–modiﬁed

KdV (MKdV) system.

3.1 Gauge transformations and general deﬁnition

of Darboux transformation

When dea li ng with general dressing procedures, two prolonged spaces are

usually used on which the problem can be posed: the ﬁrst one is spanned by

the derivatives ψ

(n)

, while the second space is determined by the successive

action of automorphism powers T

ψ. Both constructions were used in the

previous chapter. Here we restrict ourselves to the ﬁrst type of prolonged

space.

Let again A be a diﬀerential ring with a diﬀerentiation D. The generic

transform [466]

f = ǫDψ + Σψ , ǫ, Σ, ψ ∈ A (3.1)

gives the classical DT in the case of ǫ = 1, the gauge transformation if ǫ =0,

and the combination of the DT and gauge transformation if Σ = ǫσ. The ﬁrst

two cases have been well studied; the th ird one has been used for integration

of the Nahm system (Sect. 2.11). In this chapter we will consider a degenerate

operator ǫ which is prop ortional to a projector (idempotent). In Sects. 3.2–3.5

such a case is studied u nder the name of eDT. Next, a deﬁnite combination

of the eDTs produces a twofold DT, or more complicated transformations

(Sects. 3.6–3.9). In Sect. 3.10 the whole space is taken, so the form of the DT,

named a combined eDT, looks like a generic transform (3.1), with ǫ being a

projector.

Following [466], we assume a covariance of the evolution equation



ψ, (3.2)

with resp ect to the transformation (3.1). The transformed potentials for the

ZS problem (n =1)

[1] = ǫu

−1

[1] = ǫ[u

+ u

′

+[σu

] − u

−1

′

+ ǫ

]ǫ

−1

(3.3)

demonstrate new possibilities of the combined transformations. Additional

transformation of the independent variable leads to a possibility to widen the

class of covariant operators. The combinations of this sort will be used in

Chap. 4 for studying the shap e-invariant p otentials.