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

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3.2 Zakharov–Shabat equations for two projectors 69

Further, if ǫ depends on a solution of (3.2), e.g., ǫ = AφBφ

−1

D,whereA,

B,andD are constants [466], the transform of the potentials u

, k =0, 1for

arbitrary n looks like a recurrence

k−1

[1] = ǫ(u

+ u

′

k−1

)ǫ

−1

⎛

⎝

σu

−



p=k

[1](C

(p−k)

−C

k−1

(p−k−1)

)

⎞

⎠

−1

[1] = ǫ(u

+ u

′

)ǫ

−1



σu

−



p=1

[1](C

(p−1)

− C

(p)

)



−1

+ ǫ

−1

.(3.4)

This recurrence r educes to the known Matveev formulas from [314] when

A = B = D =

11. Let us also mention the search for a general dressing scheme

performed in [478] along the lines of the ZS metho d.

3.2 Zakharov–Shabat equations for two projectors.

Elementary Darboux transformation

In the algebra A we ﬁx the element

J = a

p + a

q, a

= a

where a

and a

belong to the ﬁeld K and p and q are projectors (p

= p,

= q, pq = qp =0,p + q = e, e is the identity). It is easy to verify that

pJ = J

= a

p, qJ= J

= a

q. (3.5)

Moreover, for every element x ∈ A the following equality holds:

[J, x]=a(x

− x

) ∈ A

⊕ A

,a= a

− a

. (3.6)

Here A

= pAq and A

= qAp in accordance with the decomposition

A = pAp + pAq + qAp + qAq.

Deﬁnition 3.1. The ZS operator in a module M is a K-linear operator

: ψ → Dψ − (λJ + u)ψ,

where λ ∈ K, u ∈ A,andψ ∈ M .

In the same way, the ZS operators are deﬁned in modules A

and A

where A

= A

⊕ A

and A

= A

⊕ A

. Taking into account standard

applications (spectral problems for linear operators and nonlinear evolution

equations of mathematical physics), we shall refer to λ as a sp ectral parameter

and u as a p otential.

70 3 From elementary to twofold elementary Darboux transformation

Let us s uppose that the potential u of the ZS operator satisﬁes the

restriction

= pup =0,u

= quq =0, (3.7)

i.e., u ∈ A

⊕ A

It is not necessary to pose this restriction at the very beginning; instead it

can be realized by means of an appropriate “gauge” transformation (see the

previous section).

Deﬁnition 3.2. For the potential u let there exist a new p otential ˜u, satisfying

the condition (3.2), and the elements σ ∈ A such that for all λ ∈ K the

following intertwining r elation holds:

ℓL

= L

˜u

ℓ, (3.8)

where the action of a K-linear operator ℓ in module M is determined by the

equality

ℓψ =(λp − σ)ψ, ψ∈ M.

Then the tr ansformation

u → ˜u, ψ→

ψ = ℓψ (3.9)

is referred to as the elementary DT (eDT) connected with the projector p.

The simp lest consequence of the intertwining relatio n (3.8) represents the

well-knownfactthatifψ is a solution (partial or general) of the ZS equation

ψ =0,then

ψ (3.9) is a solution of the transformed equation L

˜u

ψ =0.

Evidently, the signiﬁcance of the DT is not exhausted by this property.

Taking into account (3.5), we can easily verify the identity

(ℓL

−L

˜u

ℓ)ψ = λ(˜up −pu −[J, σ]) + (∂σ − ˜uσ + σu))ψ, ψ∈ M.(3.10)

From this and the condition of nondegeneracy of the module M we have the

following:

Lemma 3.3. The intertwining relation (3.8) identically holds for all λ ∈ K

and ψ ∈ M if and only if

˜up = pu +[J, σ] ,∂σ=˜uσ − σu .

For the proof of this assertion it is suﬃcient to derive the identity (3.10)

or to calculate

ℓL

=(λ − σ)(D − λJ − u)=λpD − σD − λ

pJ − λpu + σu ,

accounting for the deﬁnition of the diﬀerentiation D and the equality (3.5).

The equalities from the statement of Lemma 3.3 nullify the right-hand side

of (3.10) identically; therefore, (3.8) holds. The left-hand side of (3.10) is

obviously reconstructed from the equalities of Lemma 3.3.

3.2 Zakharov–Shabat equations for two projectors 71

The problem of constructing the eDT and intertwining operators is

therefore reduced to th e solution of the equations of Lemma 3.3 with respect to

˜u and ˜σ. Splitting them with account of (3.6), we obtain after simpliﬁcations

˜u

= −aσ

,σ

= −a

−1

, (3.11)

∂σ

=˜u

+ a

−1

, ˜u

= σ

− a

−1

∂u

∂σ

= −aσ

− σ

,∂σ

=0. (3.12)

Two cases should be discriminated:

1. σ

=0.Designateσ

= c and suppose that c ∈ ker D \{0}. Then (3.11)

and (3.12) are transformed to the following equations:

˜u

=(∂σ

+ σ

−1

, (3.13)

∂σ

=(σ

− a

−1

∂u

−1

+ a

−1

∂σ

= −aσ

− cu

. (3.14)

Now we consider the auxiliary ZS problem in a module A

∂ϕ =(λ

J + u)ϕ, λ

∈ K, ϕ∈ A

. (3.15)

Lemma 3.4. Assume that the ring A contains the division ring B and

denote C = B

ker ∂, i.e., C  0 is the set of invertible constants of the

ring A.Lettheelementϕ ∈ A

satisfy (3.15) and ϕ

= pϕ ∈ B

\{0},

i.e., ϕ

has the inverse element ϕ

−1

in algebr a A

. Then the element

ξ = ϕ

−1

, ϕ

= qϕ satisﬁes the Riccati equation

∂ξ = −ξu

ξ − aλ

ξ + u

. (3.16)

Proof. Using once again the deﬁnition of D and equalities that directly

follow from the ZS equation (3.15) in module A

∂ϕ

=(qλ

J + qu)ϕ

and similarly with p,wecalculate

∂ξ =(qλ

Jϕ)ϕ

−1

+(quϕ)ϕ

−1

− ϕ

−1

(pλ

Jϕ)ϕ

−1

− ϕ

−1

(puϕ)ϕ

−1

As [q, λ

]=0andqJ = a

q [see (3.6)], it follows that

∂ξ = λ

ξ +(qu(q + p)ϕ)ϕ

−1

− ξλ

− ξpu(q + p)ϕϕ

−1

where p + q = e is inserted. Taking into account the deﬁnition of the

potential u

= puq and gauge conditions (3.2), we go to the statement of

Lemma 3.4.

72 3 From elementary to twofold elementary Darboux transformation

Theorem 3.5. L et the condition of Lemma 3.4 be valid. Then the

formulas

= λ

+ a

−1

ξandσ

= −cξ (3.17)

give a solution of equations (3.14).

The proof of Theorem 3.5 can be performed by a direct substitution of

(3.17) in (3.14) with account for (3.16).

Substituting (3.17) in (3.13) and (3.11), we obtain the transformed

potential ˜u in the form:

˜u = acξ + a

−1

(aλ

+ u

ξu

− ∂u

−1

. (3.18)

Taking together (3.11) and (3.16), we get

σ = λ

+ a

−1

ξ − a

−1

− cξ + c. (3.19)

Thus, transformations (3.9), where ˜u and σ are deﬁned by (3.18) and

(3.19), give a (one possible) DT that corresponds to the projector p.

Interchanging p and q, we obtain the DT corresponding to the projec-

tor q.

2. σ

= 0. Now equations (3.12) take the following form:

= a

−1

∂u

,∂σ

= −aσ

, (3.20)

and

˜u

= ∂σ

− a

−1

. (3.21)

If there exists the recipro cal element u

−1

, then the solution of (3.20) is

given by the formulas

= a

−1

(∂u

−1

,σ

= c

−1

∈ C

\{0} .

Substituting them in (3.21) and taking into account (3.11), we go to

Theorem 3.6:

Theorem 3.6. L et the condition σ

=0for a deﬁnition of l hold. Then

the tr a nsformed potential of the ZS operator L

is given by

˜u

= a

−1

(∂

− u

−1

∂u

−1

− u

−1

˜u

= −ac

−1

. (3.22)

The proof of the theorem is a chain of equalities b efore its formulation.

Hence, in case 2 we have built once again the DT corresponding to the projec-

tor p. Its peculiarity is that in this case it is not necessa ry to use a solution of

the auxiliary ZS equation (3.15) and the transformed potential ˜u is expressed

explicitly via the seed potential u (unlike the DT in case 1), as the formu-

las (3.22) show. The relations (3.22) generalize the well-known Schlesinger

transformation [389].

3.3 Zakharov–Shabat equation with three projectors 73

Both cases may be eﬀectively used when we g o down from the abstract

level to speciﬁc examples. Such examples can be explicitly constructed for the

diﬀerential r ings of matrices. If matrix elements ar e functions of parameters,

diﬀerentiation may be deﬁned as a derivative with respect to a parameter.

Such a matrix realization of the eDT was introduced and applied in [278] for

an arbitrary matrix dimension. Similar realizations were used in [281] and the

combinations of the eDTs leading to a binary DT were constructed and used

to obtain multisoliton formulas.

3.3 Elementary and twofold Darboux transformations

for ZS equation with three projectors

We continue to develop a rather abstract extension o f the eDT covari ance

based on the existence of idempotents and division rings (skew ﬁelds) in an

associative diﬀerential ring A over the ﬁeld K [267]. Let D be the diﬀer-

entiation map on A and let us ﬁx orthogonal (pq = qp =0)idempotents

(projectors) p and q, such that p, q ∈ ker D. Then the element s = e − p − q

is the third orthogonal projector. We choose here the case of three basic pro-

jectors for it covers features of a general formulation but nevertheless has a

clear exp l icit form.

For every x ∈ A we denote x

αβ

= αxβ,whereα, β ∈ p, q, s, so we split

theringintothedirectsum

A = ⊕

α,β

αβ

We ﬁx the element J = a

p + a

q + a

s, a

∈ K, a

= a

The degenerate case o f equal a

can be considered in a similar manner [278].

Deﬁnition 3.7. The ZS operator L

is the linear operator in A,

: ψ → Dψ +(λJ − u)ψ,

where λ ∈ K, u, ψ ∈ A.

Suppose the potential u of the ZS op erator satisﬁes the gauge restrictions

= pup =0, u

= quq =0, u

= sus =0.

Deﬁnition 3.8. Let for the potential u there exist a new potential u

and the

element σ ∈ A such that for all λ ∈ K the following intertwining identity

holds

= L

E, (3.23)

where the action of a K-linear operator E is determined by the equality

Eψ =(λp + σ)ψ. (3.24)

74 3 From elementary to twofold elementary Darboux transformation

Then the tr ansformation

u → u

,ψ→ ψ

= Eψ

is referred to as the eDT connected with the projector p.

It follows from (3.23) that

Dσ = u

σ − σu (3.25)

and

p = pu +[J, σ] . (3.26)

Let a solution of the ZS equation for λ = μ be ϕ and ϕ

= pϕp ≡ ϕ

has

the inverse element ϕ

−1

in A

. Then the elements ξ = ϕ

−1

and η = ϕ

−1

with ϕ

= qϕ and ϕ

= sϕ satisfy the system of the Riccati-type equations

[240] and the following theorems may be proved as in [324].

Theorem 3.9. Let σ

= c =0, σ

= d =0, c, d ∈ ker D. Then for the

transforme d potential u

the following formulas are valid:

=(σ

− b

−1

− a

−1

= −aσ

=(σ

− a

−1

− b

−1

= −bσ

=[(1− a/b)σ

+ cu

−1

=[(1− b/a)σ

+ du

−1

, (3.27)

where a = a

− a

and b = a

− a

, and the relations

= −μp + a

−1

ξ + b

−1

η,

= −a

−1

,σ

= −cξ , (3.28)

= −b

−1

,σ

= −dη , σ

= σ

deﬁne the eDT. Each additional pr ojector intro duces an eDT (obviously a

diﬀerent one) in a similar way.

We mention here additional possibilities of the Schlesinger transformations

[389] in the case of the constraints σ

=0orσ

=0.FortheSchlesinger

transformations it is not necessary to use the auxiliary solutions of the ZS

equation: the transformed potential u

is directly expressed via the s eed po-

tential u.Thechoiceσ

=0andσ

= 0 introduces constraints for the

potential u [240].

For the conjugate problem

∗

φ + φ(κJ − u) = 0 (3.29)

3.3 Zakharov–Shabat equation with three projectors 75

the equations for σ

and u

determine the transform (the upper index c label s

the conjugate transform)

= φE

= κp − σ

. (3.30)

The equations for σ

and u

diﬀer from (3.25) and (3.26) only in the order of

the op erators and the form of D

∗

The solution of these equations is determined again by solutions with pro-

jectors multiplied from the right, φ

= φp and so on, with the spectral pa-

rameter μ.Namely,

Theorem 3.10. Let σ

= c

∗

=0,σ

= d

∗

=0,andφ

−1

=(pφp)

−1

∈ A

exists. Then the transform (3.30) is determined by the elements

= −μ + a

−1

+ b

−1

= −φ

−1

∗

,σ

= −a

−1

, (3.31)

= −φ

−1

∗

,σ

= −b

−1

,σ

= σ

=0.

The set of formulas (3.31) in turn deﬁnes the eDT of eigenfunctions (3.30) for

the conjugate problem (3.29) via constants c

∗

∈ ker D entering equations

(3.31).

The p otential is transformed similarly to (3.27):

= −aσ

= −bσ

= c

−1

∗

− b

−1

− a

−1

) ,

= d

−1

∗

− a

−1

− b

−1

) ,

= c

−1

∗

(1 − b/a) − u

−1

∗

] ,

= d

−1

∗

(1 − a/b) − u

−1

∗

] . (3.32)

We consider th e case D

∗

= −D; hence, ζE

−1

,whereζ stands for another

solution of the ZS spectral problem, is the solution of the conjugate equation

(3.29) and the transformed potential is identical to u

. Namely, this point

allows us to carry out the twofold transformations. Elements of E

−1

asso ciated

with the corresponding projectors ca n be subsequently calculated. After that

the twofold DT is constructed and the respective theorem may be proved. We

introduce the twofold transformation as a sequence of two elementary ones

in b oth spaces. In the case of the conjugate space, the ﬁrst transformation

is made by the inverse operator to E with the spectral parameter μ and the

respective solution ϕ of the direct problem. The second map is generated

by the resulting function χ

expressed in the same way but from a linearly

independ ent seed solution χ of the equation (3.29) with the parameter ν .For

the direct case, the order is the opposite. The ﬁnal formulation is given by

Theorem 3.11:

76 3 From elementary to twofold elementary Darboux transformation

Theorem 3.11. L et the inverse of the element pχϕp exist. The twofold DTs

for solution of the direct ZS problem and for the conjugate one (3.29) with

∗

= −D ar e given by the following equalities:

=(p + c

−1

∗

c + d

−1

∗



ψ +

ν − μ

λ − ν

(

χ, ϕ)

−1

χψ



(3.33)

and

= ζ

(κp + σ



ζ −

ν − μ

κ − μ

ζϕ(χ, ϕ)

−1



(p + c

−1

∗

+ d

−1

∗

) .

(3.34)

As regards the potentials, we have for example

= u

+ aϕ

(χ, ϕ)

−1

(ν − μ)

and

= c

−1



+ aϕ

(χ, ϕ)

−1



(ν − μ) , (3.35)

where c

= c

−1

∗

(for other elements it is suﬃcient to change the relevant

indices).

The proof of (3.34) may be provided by the substitution of σ

from for-

mulas for the conjugate problem (3.31) and (3.32) with u → u

and φ → χ

The derivation of (3.33) is performed ﬁrst by (3.27) and (3.28) and then by

the inverse of the transformation (3.31) based on the results of the previous

step. For (3.35) the Riccati-like system for ξ and η is useful. Here the scalar

product analog is introduced as

(ζ,ϕ)

= pζϕp ∈ A

and c

1,2

appear owing to an arbitrary choice of the constant diagonal elements

of σ for both elementary transformations (Theorems 3.9 and 3.10).

It is easy to check that the transform of the potential may be rewritten in

terms of idempotents

P = φ(χ, ϕ)

−1

χ (3.36)

up to the choice c

= 1. Hence,

u[1] = u + δ[J, P ] . (3.37)

The twofold-like DT has been used to create multisolitons and other solu-

tions o f nonlinear evolution equations [35, 240, 314, 354].

3.4 Elementary and tw ofol d Darboux transformations. General case 77

3.4 Elementary and twofold Darboux transformations.

General case

Let D be the diﬀerentiation map on an associative diﬀerential ring A over the

ﬁeld K and assume there are idempotents (projectors) p

∈ ker D such that



= e and p

= p

= 0 . For every x ∈ A we denote x

= p

,so

we can split the ring into the direct sum of A

, x

∈ A

Let us ﬁx the element J =



, a

∈ K, a

= a

. The degenerate

case of equ al a

is considered in a similar manner (see [449] where the matrix

example is studied).

Deﬁnition 3.12. The ZS operator is the linear operator in A acting in ac-

cordance with

: ψ → Dψ +(λJ − u)ψ,

where λ ∈ K, ψ, u ∈ A.

Suppose the potential u of the ZS operator satisﬁes the restriction u

=0.

Deﬁnition 3.13. Let for the potential u there exist a new potential u

and

the element σ ∈ A such that for all λ ∈ K the following intertwining relation

holds:

= L

E, (3.38)

where the action of a K-linear operator E is determined by the equality

Eψ =(λp + σ)ψ.

Then the tr ansformation

u → u

,ψ→ ψ

= Eψ ,

where ψ is a solution of the equation

Dψ +(λJ − u)ψ =0, (3.39)

is referred to as the eDT connected with the projector p.

It follows from (3.38) that

Dσ = u

σ − σu (3.40)

and

p = pu +[J, σ] . (3.41)

Theorem 3.14. Let ϕ be a solution of the auxiliary ZS e quation for λ = μ

and ϕ

= pϕp := ϕ

has the inverse element ϕ

−1

in A

. Assume also that

= c

∈ A

and c

−1

exists (this me ans that c

−1

= c

−1

= p). Then the

operator of the eDT is deﬁned by

σ = −μp +



i=2

&

− a

)

−1

− c



− (a

− a

)

−1

, (3.42)

78 3 From elementary to twofold elementary Darboux transformation

where ξ

= ϕ

−1

. For the elements of the transformed potential u

the

following formulas are valid:



−



k=2

− a

)

−1

− (a

− a

)

−1



−1

=(a

− a

= c



− a



−1

. (3.43)

The expressions (3.43) deﬁne the eDT, i, k =2,...,n.Of course, any other

projector introduces the eDT in a similar way. First we note that c

∈ ker D

and the expressions for u

can be found in the spirit of the original Darboux

approach. Then the elements ξ

= ϕ

−1

with ϕ

= p

ϕ, i =2,...,n satisfy

the system of the Riccati-type equations and the relevant theorems may be

proved along the lines of [314].

For the conjugate problem

∗

φ + φ(κJ − u) = 0 (3.44)

the analog of Theorem 3.14 can be formulated and equations for σ

and u

diﬀer from (3.40) and (3.41) only in the order of the op erators. Hence, similarly

to (3.30) we deﬁne

= φE

= κp − σ

, (3.45)

where

= −μp +



i=2



− a

)

−1

(ξ

∗i

− 1)u

− ξ

∗i



, (3.46)

and

= c

−1

∗i



−



k=2

− a

)

−1

+(a

− a

)

−1

∗



=(a

− a

)ξ

∗i

= c

−1

∗i

− a

∗i

∗k

, (3.47)

with the same a

∗i

= σ

, but ξ

∗i

= φ

−1

,φ

= φp

; the existence of c

−1

∗i

and φ

−1

in A

is implied.

For the constraints σ

=0, for all or some i, the transformed potential u

is expressed via the seed potential u only. We denote this additional possibility

as the Schlesinger transformation. The case n = 2 wa s described by Matveev

[314].

It is important to note that ζE

−1

is the solution of the conjugate equation

(3.44) when D

∗

= −D and the eDT-transformed potential is identical to u

The inverse operator of E can be written in terms of projectors p

−1

= E

−1

so that for the ﬁrst idempotent p = p

we have

−1

=(λ − μ)

−1