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

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Dressing chain equations

One very promising approach to solving integrable systems is based on the

notion of a dressing chain [395, 438, 448]. This method covers soliton, ﬁnite-

gap, rational, and other important solutions [423, 438] within the universal

scheme, reducing the starting problem to a soluti on of clo sed sets of nonlinear

ordinary equations with the bi-Hamiltonian structure [438]. Here we derive

the dressing chain equations and study the above classes of soluti ons.

A scheme to reconstruct the potential entering an a ssociated linear prob-

lem depends on a class to which solutions of the inverse problem belong [378].

The c oeﬃcients (potentials) of the linear equation are elements of some alge-

braic structure. Such a structure is generated by transformations that preserve

the functional form of the equation. Hence, a general set of potentials splits

into subsets invariant with respect to the action of the transformations. These

transformations, for example, the Darb oux (Schlesinger, Moutard) transfor-

mations, are generated by transformations of eigenfunctions φ of a given

diﬀerential operator. As a general remark, note that it is possible to approx-

imate locally solutions of the linear problem by a sequence of the Moutard

and Ribacour transformations [170]. The Darboux transformation (DT) pro-

duces naturally an intermediate object σ =(Dφ)φ

−1

,whereD is the diﬀer-

ential operator, which is related to the potential by the generalized Miura

transformation and satisﬁes the Riccati equation (compare with [164]). The

main ideas of Darboux permit us, in principle, to determine the form of the

transformation [102] by means of a factorization of the operator under con-

sideration. The structure of the transformation depends on the ring to which

the op erator coeﬃcients belong, as well as on the realization of an abstract

diﬀerentiation operation [265, 321]. The DT and Miura transformations for

non-Abelian entries (diﬀerential ring) are studied in [467]. This method allows

us to operate eﬀectively with the spectra l data. Here we restrict ourselves to

one-dimensional problems, but steps towards multidimensions have already

been made [378].

If we substitute the potentials expressed in terms of σ in any iteration

of the DT, we get the chain equations in the form of diﬀerential-diﬀerence

109

110 4 Dressing chain equations

equations. They can be equally well treated as the transformed form of the

spectral problem we start with. This fact was established for the simplest

one-dimensional problem in [ 448] and extended in [395]. Section 4.1 contains

simple but instructive examples of the appearance of dressing chains and their

usefulness in applications. The dressing chain formalism opens new possibil-

ities to produce explicit solutions as well as to study diﬃcult questions of

a uniform approximation of the potentials [355]. The technique of dressing

chains is directly connected to the quantum inverse problem [84] and integra-

tion of soliton equations. In Sect. 4.2 we consider a general spectral problem,

polynomial in diﬀerentiation. We start from an appropriate evolution equation

and reduce the consideration to the stationary case that generates a spectr al

problem.

The solution of the chain equation can be analyzed from the point of

view of the bi-Hamiltonian structure [53, 438]. Some important structures

connected to the chain equations were studied in [165]. The symmetry of the

system is naturally related to the DT a nd generates a ﬁnite group that we use

to simplify the problem. In Sect. 4.3 we introduce projection operators for the

irreducible subspaces of the symmetry group and the corresponding variables.

Section 4.4 is devoted to symmetry (in particular, permutation symmetry)

of the dressing chain equations. In Sect. 4.5 we concretize the results for the

speciﬁc number of iterations in the dressing chain system.

In Sects. 4.6–4.8 we discuss a class of periodic or quasiperio dic p otentials

and associate with them a notion of the spectral curve, Dubrovin equations

and general ﬁnite-gap potentials. We consider a transition to new variables

in which solutions of the chain equations are expressed in quadratures. Note

also that an important application of the dressing theory is concerned with

the possibility to combine the ﬁnite-gap [45] and localized (solitonic) solutions

(see the discussion in [324]). This idea, following the Shabat scheme [393],

was implemented ﬁrst by Kuznetsov and Mikhailov [258] using an example of

dressing the cnoidal wave (stationary two-zone solution of the KdV equation)

with N solitons. By means of the ﬁnite-gap integration theory, solutions of this

type were also obtained in [24, 216, 251]. In Sect. 4.9 we formulate the DT

for the non-Abelian Zakharov–Shabat (ZS) problem. Section 4.10 contains

a derivation of the dressing chain equations produced by DT of operators

polynomial in an automorphism of a ring. Taking this result, we build in Sect.

4.11 a dressing chain equation for the non-Abelian Hirota mo del. Section 4.12

contains some comments.

4.1 Instructive examples

As shown in Chap. 2, the operator of the classical DT has the universal form

= D − σ

and intertwines, for example, the operators of the equation

−ψ

+ u

ψ = λψ. (4.1)

4.1 Instructive examples 111

The p otential u

is linked to u

i+1

= u

− 2σ

′

, (4.2)

with σ

satisfying

′

+ σ

+ μ

= u

, (4.3)

where μ

is an eigenvalue. Equation (4.3) is solved by σ

= φ

′

−1

,whereφ

is a solution of the problem (4.1) with the eigenvalue μ

. Substitution of u

from (4.3) into (4.2) yields the so-called dr essing chain:

(σ

+ σ

i+1

)

′

= σ

− σ

i+1

+ μ

− μ

i+1

. (4.4)

The important problems of quantum mechanics can be solved by the

dressing chain formalism. Let us look for solutions of (4.4) in the form

[395]

= ξ

a(x)+η

,j =0, ±1,... . (4.5)

Plugging (4.5) into (4.4) produces the Riccati equation for a(x):

(ξ

j+1

+ξ

′

=(ξ

−ξ

j+1

−2(ξ

j+1

−ξ

)a−η

j+1

+η

+μ

−μ

j+1

. (4.6)

The solutions a(x) via the relation (4.3) (known as the Miura transformation,

see Chap. 2) produce the potentials

= ξ

′

(x)+(ξ

a(x)+η

)

+ μ

The case a = x means ξ

j+1

= ±ξ

, η

j+1

= ±η

, j =0, 1,...,andtherelation

(4.6) reads as the recurrence for the eigenvalues,

j+1

= μ

+2ξ

, (4.7)

and gives the equidistant spectrum for the choice ξ

j+1

= ξ

= ξ, η

j+1

= η

η. The harmonic oscillator potential for x ∈ (−∞, ∞) is directly obtained in

such a way:

= −ξ + ξ

+2ξηx + η

+ μ

The ground state φ

is annihilated by the DT:

(D − σ)φ

=0; φ

′

−1

= ξx + η.

This fundamental property of the DT deﬁnes the explicit form of the func-

tion φ

and hence of all the dressed ones. Another choice ξ

j+1

= − ξ

−(−1)

ξ, η

j+1

= −η

= −(−1)

η does not produce essentially new po ten-

tials.

The radial Schr¨odinger equation in atomic units,



−

l(l +1)

+ u

− E



(r) = 0 (4.8)

(for the scattering problem see Sect. 6.2.3), is transformed to (4.1) by ψ

ψ/r, r → x and hence the corresponding chain equation is equivalent to

112 4 Dressing chain equations

(4.4). Equation (4.6) for the choice a =1/r yields two constraints for the

constants:

(ξ

j+1

+ ξ

)(ξ

j+1

− ξ

− 1) = 0,ξ

j+1

− ξ

=0.

The next ch oice

j+1

= ξ

+1,ξ

yields for j =1, 2,...

j+1

which results in

The s pectrum is determined by

j+1

− μ

= η

− η

j+1

with the obvious solution

= −η

= −

This means that (4.7) and

(r)=

exp





σ(x)dx



solve the Coulomb quantum problem for l = j − 1 and arbitrary principal

number j. More details and two more potentials linked to the solutions of

(4.6), namely, exp x and tan x are considered in Sect. 6.2.3.

4.2 Miura maps and dressing chain equations

for diﬀerential operators

4.2.1 Linear problems

Reproducing for convenience the general conjectures from Chap. 2, let us take

the diﬀerential operator

L =



n=0

(4.9)

on a diﬀerential ring A with coeﬃcients (potentials) a

∈ A, and the evolution

equation

= Lψ, ψ ∈ A. (4.10)

4.2 Miura maps and dressing chain equations for diﬀerential operators 113

Here the operator D is a diﬀerentiation with respect to some variable (maybe,

the abstract one) and ψ

is the derivative with respect to another variable (see

[271, 321] for details and generalizations). Denote Dψ = ψ

′

. The transforma-

tion of the solutions is taken in the standard Darboux form

ψ[1] = Dψ − σψ,

where

σ = φ

′

−1

(4.11)

with a linearly independent invertible solution φ of (4.10).

The transformation of the coeﬃcients of the r esulting operator

L[1] =



n=0

[1]D

is determined by

[1] = a

(4.12)

and for all the other n =0,...,N − 1by

[1] = a



k=n+1

k,k−n

+(a

′

− σa

k−1,k−1−n

] (4.13)

that yields the covariance principle. This means that the function ψ[1] is a

solution of the equation

[1] = L[1]ψ[1].

This result is a compact reformulation of the Matveev theorem [321].

The functions B

m,n

were introduced in [467] and represent the gen eralized

Bell polynomials deﬁned in Sect. 2.2.

Proposition 4.1. If the function σ satisﬁes the equation

σ ≡ σ

= Dr +[r, σ], (4.14)

where r =



n=0

(σ), the oper ator L

= D −σ intertwines the operators

− L and D

− L[1].

For the derivation of the dressing chain equations we consider the station-

ary solution of the evolution equation (4.10):

φ = φμ.

This gives D

σ =0. For example, in the matrix case μ =diag{μ

,...,μ

Note that (4.14) for σ taken from (4.11) becomes the identity. The param-

eters μ

have the sense of eigenvalues of the operator L.

114 4 Dressing chain equations

Hence, the consequence of (4.14),

Dr +[r, σ]=0, (4.15)

is the analog of the Riccati equation that we call the generalized Miura map.

It connects the potentials [coeﬃcients of the operator L (4.9)] and σ at every

step i of the DT iterations. Further we will supply the functions with the

upper ind ex i to show the number of iterations made. In the scalar case the

commutator is zero and (4.15) reads



n=1

(σ

)=c =const. (4.16)

If we have the single potential a

with the other ones being invariant, a

i+1

, n = 0, then the expression for the iterated potential is given by

= −



n=1

(σ

)+c

. (4.17)

In this case the derivation of the chain equation is made by the substitution

of (4.17) into (4.13) for n = 0 and with the indices i and i +1. Equation

(4.17) for N = 2 gives the link between σ

and the p otential u

that enters

the second-order operator

L = −D

+ u

. (4.18)

Supplying (4.16) for N = 2 with indices, we obtain

i,x

+ σ

+ c

= u

. (4.19)

The next ch oice N = 3 leads to the generalized Miura transformation

i,xx

(σ

)

+ σ

+ u

+ w

= μ

, (4.20)

where w

= a

and u

= a

. It connects the coeﬃcients u

and w

with σ

.For

both cases the DT has a similar form

i+1

= u

− 2σ

i,x

(4.21)

(for N =3wehavea

= 1; hence −2 in the DT should be changed to +3).

Finally, if one starts from the second-order Sturm–Liouville equation (4.18),

the associated dressing chain equation is written as

i+1,x

+ σ

i+1

+ μ

i+1

= −σ

i,x

+ σ

+ μ

. (4.22)

This dressing chain equation was studied in Sect. 4.1.

4.2 Miura maps and dressing chain equations for diﬀerential operators 115

Further generalization is concerned with a nonisospectral DT. Such a DT

has the same form as (4.21) but it links diﬀerent λ-subspaces:

i+1

(λ + α

)=(D − σ

)ψ

(λ). (4.23)

It speciﬁes the factorization condition as

i+1

= u

− 2σ

i,x

+ α

(4.24)

and shifts the eigenvalue by α

. Substitution of (4.19) into (4.24) yields

i+1,x

+ σ

i,x

= σ

− σ

i+1

+ α

+ c

− c

i+1

. (4.25)

Note that c

are arbitrary integration constants and could b e ﬁxed as, e.g.,

in the instructive example of Sect. 4.1, where α

=0andc

= μ

,that

corresponds to the choice of σ

= φ

′

/φ (check by direct substitution!). In

Sects. 4.6–4.8, where the dressing chain equation is studied and used to build

ﬁnite-gap solutions [422, 438], c

= 0 is chosen to simplify the dressing chain

equation Lax pair.

For the third-order operator, if w

= 0, the dr essing chain equation is

written as

(σ

i+1,xx

−μ

i+1

)/σ

i+1

+3σ

i+1,x

+ σ

i+1

=(σ

i,xx

−μ

)/σ

+6σ

i,x

+ σ

. (4.26)

Thecaseofzerow

is obviously a reduction for the space of solutions of the

linear problem and some modiﬁcation of the DT formula for the eigenfunctions

is necessary [281]. Important connections with the Hamiltonian structures of

some third-order problems (related to the Sawada–Kotera/Gibbon and Kaup–

Kupershmidt equations) were studied in [164].

4.2.2 Lax pairs of diﬀerential operators

As regards a Lax pair of a nonlinear system, one should consider an additional

operator, say,

A =



n=0

, (4.27)

leading to the evolution equation

= Aψ. (4.28)

The coeﬃcients of both equations (4.9) and (4.27) depend on a set of p otentials

,...,u

, eventual solutions of nonlinear equations. In accordance with the

joint covariance property [270, 271], the DTs of the coeﬃcients induce the

DTs of the potentials (Sect. 2.7). We can formulate a proposition similar to

Proposition 4.1:

116 4 Dressing chain equations

Proposition 4.2. If a function σ satisﬁes (4.14) and the equation

σ ≡ σ

= Dq +[q, σ], (4.29)

where q =



n=0

(σ), then the operator L

= D − σ intertwines the

pairs of operators (D

− L, D

− A) and (D

− L[1],D

− A[1]).Thismeans

the integrability of the c ompatibility condition of (4.1) and (4.28) in the

sense of the symmetry existence with respect to the DT u

→ u

[1] of the

potentials.

The compatibility condition of (4.14) and (4.29) yields the extra equation

+[q

,σ]+[q, Dr +[r, σ]] = Dr

+[r

,σ]+[r, Dq +[q, σ]], (4.30)

which links the potentials and the element σ. In the case of the single potential

u, it is possible to express it as a function of σ [273]. Considering the iterated

potentials

= f

(σ

)

(now the index is again a number of iterations) allows us to pro duce the

dressing chain equation substituting the function into the DT formula u

[1] =

i+1

The scalar case is much simpler. From (4.30) it follows that D(q

−r

)=0,

−r



n=0

(σ)+b

′

(σ)Dq]−



n=0

(σ)+a

′

(σ)Dr]=const.

(4.31)

The good example of this case is the Kadomtsev–Petv iashvili equatio n and

its dressing chain [273], whence the potential is extracted from (4.31). In the

theory of solitons (4.26) generates the Sawada–Kotera equation, while (4.20)

corresponds to the famous KdV equation (see also Sect. 8.7). From the point of

view of the derivation of the dressing chain equation, other reductions are more

complicated: it is necessary to express the potential from (4.20). For the chain

equation asso ciated with the reduction of the Boussinesq equation see [281].

4.3 Periodic closure and time evolution

The periodic closure of the dressing chain equation (4.22) for the KdV equa-

tion produces a ﬁnite system of equations that possesses the bi-Hamiltonian

structure [438]. As Veselov and Shabat [438] wrote about the case N =3

(one-gap potentials), “it is a useful exercise to derive explicit formulas for

directly from equations of the chain.” Below we brieﬂy show how to do

that and give the formula. There is an important question that occurs from

this direct way: How does one extract the dependence of the potential on the

additional parameter t from the Lax pair?

4.3 Periodic closure and time evolution 117

We propose to specify the “time” evolution via the t-dependence of x-

conserved quantities [274]. In this section we begin to study this problem in

terms of the same chain variab les supplied with a time d ependence. Let us

start from the system for three functions σ

for the simpl est nontrivial closure

(d/dx)σ

(x)=σ

(x) − σ

(x)+μ

− μ

(d/dx)σ

(x)=σ

(x) − σ

(x)+μ

− μ

(d/dx)σ

(x)=σ

(x) − σ

(x)+μ

− μ

(4.32)

This direct way to produce the bi-Hamiltonian formalism was initiated in

[438] by introducing a generating function and extracting conserved quantities

(integrals). In this simplest example we take the integrals to solve the problem

completely. If we express the third va riable σ

as the linear combination of the

other ones by means of the ﬁrst integral (Casimir function) c = σ

+σ

and

substitute it into the other equations of (4.32), then, after use of the second

integral A, we arrive at the diﬀerential equation of the ﬁrst order for elliptic

functions. It is convenient to show this fact i n terms of other variables [438]:

(x)=σ

(x)+σ

(x),g

(x)=σ

(x)+σ

(x),g

(x)=σ

(x)+σ

(x).

Now we exclude g

(x)bytherelationg

(x)=2c −g

(x) −g

(x). Further, if

we omit the argument x, the inverse transformations become

= −g

+ c, σ

= −c + g

+ g

,σ

= c − g

Inserting the transforms into the system (4.26), we obtain two diﬀerential

equations for new variables:

= μ

−μ

+2cg

−g

−2g

= −2cg

+2g

+ g

+ μ

−μ

The second integral of motion in terms of g

is more compact than in σ

A = g

+ μ

. (4.33)

It allows us to express g

as a function of g

. Therefore,

(x)=−μ

+ μ

+2cg

− g

− 2



−g

+ μ

− μ

+2g

+(g

− 4g

c +2g

+2c

− 2μ

+ μ

(4.34)

+4A − μ

c − μ

+ μ

− 2μ

)

1/2



The next problem is to solve (4.34) in elliptic functions. The Weierstrass

or Legendre canonical form of the integral yields a solution of the problem

after the Abel transformation [208] and use of the algebraic formulas that give

2,3

. Finally, we have the explicit dependence of σ

on x and parameters c and

A [275]; see also Sect. 4.5.

118 4 Dressing chain equations

Let us turn to the problem of a time evolution arising from the Lax repre-

sentation for some nonlinear equation. The main instruction from the above

example is to search for the time dependence of the x-independent entries c

and A. Let us consid er a DT-covariant “time” evolution. In the KdV case the

second Lax operator has the form of (4.9). The account of a connection (4.19)

between the potential and σ produ ces the modiﬁed KdV (MKdV) equatio n

for the function σ. The substitution into the equation of the x-derivatives

from the system (4.32) yields

∂

∂t

(x, t)=

(μ

+ μ

− 5μ

− F )

∂

∂x

(x, t),

F =(σ

+ σ

)

= c

. (4.35)

The equations for σ

1,2

are w ritten similarly if we use the cyclic symmetry in

the indices. Such equation s are general; we will call them the t-chain equations

(for t-chains see also [313]). The form of the equations is typical for the so-

called hydrodynamic-type equations. The system is diagonal and Hamiltonian,

it can be integrated by the hodograph method [428]. The ﬁnal step of this

direct construction is the use of (4.35) and similar equations for σ

1,2

.Firstwe

can check that

∂

∂t

c = −3σ

+3σ

− 3σ

+3σ

− 3σ

+3σ

=3c

which is zero if the x-dep end ence of σ

is governed by (4.32). If one plugs the

t-derivatives of σ

from formulas like (4.35) into the derivative A

,thisgives

= g

(3μ

− 3μ

)+g

(3μ

− 3μ

)+g

(3μ

− 3μ

)

(6μ

− 6μ

)+g

(6μ

− 6μ

)+g

(6μ

− 6μ

)

+3 (μ

− μ

)(μ

− μ

)(μ

− μ

) .

Further analy sis leads to Proposition 4.3:

Proposition 4.3. If a common period X exists for all g

,thex-independent

polynomial c does not depend on t, and A is a line ar function of t.

For the proof it is enough to notice that the second-order combination

(6μ

−6μ

)+g

(6μ

−6μ

)+g

(6μ

−6μ

) is a linear

combination of the x-derivative and constant because, for example,

− g

(x)+

(x)+μ

− μ

After integration of this equation in x over the period X and use of the

conservation laws for the KdV and MKdV equations in combination with

(4.26), we arrive at the linear dep endence of AX on t. The coeﬃcient may be

recognized in the expression for A

: it is a combination of the eigenvalues μ

For example, the



,i =1, 2, 3 and other more complicated conserved

quantities [443] sh ould be accounted for, and t h e integrals over X of the

combinations are zero. The resulting formula for A should be substituted into

the solution of (4.34).