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

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7.5 Zakharov–Shabat dressing method via operator factorization 219

normalization constants a

= lim

x→∞

exp(κ

)ψ

for eigenfunctions ψ

nor-

malized as



∞

−∞

|ψ

dx = 1, and the reﬂection coeﬃcient v(k). The last one is

extracted from the asymptotic behavior of the continuum spectrum solutions

ψ(x, k) ≃



exp(−ikx)+v(k)exp(ikx),x→∞,

w(k)exp(−ikx),x→−∞.

(7.146)

Solving the scattering problem, we arrive at the function F (x) [354]:

F (x)=



exp(−κ

x)+

2π



∞

−∞

v(k)exp(ikx)dk, (7.147)

which determines the kernel of the Gel’fand–Levitan–Marchenko (GLM) in-

tegral equation

K(x, y)+F (x + y)+



∞

K(x, s)F (s + y)ds =0,x≤ y. (7.148)

Then the potential u(x) is retrieved from th e solution K(x, y) of (7.148) as

u(x)=2

K(x, x). (7.149)

Equation (7.148) links K and F ; it maps the scattering data to the poten-

tial and is referred to a s the inverse scattering transformation. The Gardner–

Green–Kruskal–Miura theory, using the second op erator of the Lax pair (see

Chap. 3), gives explicit dependence of the scattering data on time, a

(t)and

v(k, t), via the initial values of a

(0) and v(k, 0).

The GLM equation (7.148) is solved explicitly in some of the simplest

cases [354]. The multisoliton solutions correspond to zero v (reﬂectionless

potentials). The kernel of the integral operator factorizes in this case and has

a ﬁnite number of terms, as is seen from (7.147).

7.5.2 Dressible operators

The idea of the dressing method in its original IST version [474] (we follow

the modiﬁcation given in [466]) uses the fact that each function F generates

the function K and hence a potential. Let us write (7.147) symbolically as

K + F + K

∗

F =0, (7.150)

where the asterisk denotes the action of the integral operator and the function

F (x, y)goestoF (x + y) for the standard GLM equation. Consider a pair of

operators M and



M which obey the equation



MK + MF +(



MK)

∗

F + K

∗

(MF)=0. (7.151)

220 7 Important links

The operator M is named the “bare” operator, and



M is the “dressed” oper-

ator. Suppose the function F obeys the equation

MF =0. (7.152)

Then we have



MK =0, (7.153)

if the operator



M exists. The set of pairs (M ,



M) forms a vector space.

As an example, consider the operator

M = ∂

+ ∂

. (7.154)

In this case (7.151) takes the form

MK(x, y)+MF(x, y)+∂



∞

K(x, s)F (s, y)ds+



∞

K(x, s)∂

F (s, y)ds =0.

(7.155)

Evidently,

∂



∞

K(x, s)F (s, y)ds = −K(x, x)F (x, y)+



∞

[(∂

K(x, s)] F (s, y)ds.

(7.156)

Integration by parts gives

I =



∞

[∂

K(x, s)] F (s, y)ds = −



∞

K(x, s)∂

F (s, y)ds − K(x, x)F (x, y).

(7.157)

In (7.155) take into account (7.156) and introduce I as

MK(x, y)+MF(x, y) −K(x, x)F (x, y)+



∞

[∂

K(x, s)] F (s, y)ds



∞

K(x, s)∂

F (s, y)ds + I − I =0. (7.158)

For +I substitute the mid-positioned term in (7.157), and for −I the right-

hand side of (7.157) with the opposite sign:

MK(x, y)+MF(x, y) −K(x, x)F (x, y)+



∞

[∂

K(x, s)] F (s, y)ds



∞

K(x, s)∂

F (s, y)ds +



∞

[∂

K(x, s)] F (s, y)ds



∞

K(x, s)∂

F (s, y)ds + K(x, x)F (x, y)=0. (7.159)

7.5 Zakharov–Shabat dressing method via operator factorization 221

Ordering the ter ms, we get

(∂

+ ∂

)K(x, y)+MF(x, y) (7.160)



∞

[(∂

+ ∂

) K(x, s)] F (s, y)ds +



∞

K(x, s)(∂

+ ∂

) F (s, y)ds =0.

Hence, the following operator arises:



M = M = ∂

+ ∂

. (7.161)

The operator M is called dressible. A set of dressible operators forms linear

space.

A connection between scattering data and a potential U = U (x, t)with

the additional (time) parameter is used in integrable equations via the Lax

representation [335] and, directly, in quantum evolution problems. The prob-

lems in which potentials are functions of time can be studied by the present

metho d because the operator of the time derivative ∂

is dressible. As before,

from the equation

MF(x, y, t)=0

we obtain



MK(x, y, t)=0.

Let a function ψ be a solution of two equations

(∂

− L[U])ψ =0, (7.162)

(∂

− A[U])ψ =0. (7.163)

If derivatives with r espect to t and y commute, then the Lax representation

− L

=[A, L]. (7.164)

Proposition 7.4. If two operators M and



M are such that there exists a

solution of (7.151) (op erator M is dressible) and if the operator M forms the

Lax pair with N, then the operators



M and



N also form a Lax pair. If a pair

of operators M and N produces a nonlinear system, then the pair



M and



produces the same system.

The next example is

M = α

∂

∂t

+ ∂

− ∂

. (7.165)

We want to dress the operator M, applying it to the GLM equation (7.151).

Integrating by parts yields the operator



M = α

∂

∂t

+ ∂

− ∂

+ U(x), (7.166)

where

U(x)=−2

K(x, x). (7.167)

222 7 Important links

Note that a function U has appeared in the dressed operator, while for the

ﬁrst-order operator (7.154) the dressed operator is the same as the bare one

(7.161).

In general, we put

= l

(x,t,...)∂

. (7.168)

Consider the operator D of the following structure:

DF = α∂

F + L

F − FL

, (7.169)

where L

is the Hermitian conjugate to L and acts to the left.

Proposition 7.5. The operator (7.169) is dr essible. The dressed operator



DK = α∂

K + LK − KL

, (7.170)

where

L = L



L (7.171)

and



L =

∂

n−1

+ ...,

∼ (∂

− ∂

)



y=x

. (7.172)

7.5.3 Example

Let us take

= ∂

⇒ L = ∂

+ U.

Solving the equation MF =0yields



MK = 0; hence, some class of solvable

equations appears, with some linear space.

Let us consider operators D

and D

F = α

∂

F + L

(1)

− FL

(1)+

F = α

∂

F + L

(2)

− FL

(2)+

This class o f operators contains the Lax representation

∂

(2)

− α

∂

(1)



(1)

(2)

=0.

For relevant forms of the operators L

(1)

and L

(2)

and for α

= α, α

= −1,

= y,andt

= t we obtain the KP equation

∂

+6uu

+ u

xxx

)+α

=0.

In the case of α

= −1 we have the KP I equation; otherwise, if α

=1we

have the KP II equation. The KP equation is the two-dimensional equation

that contains the KdV equation as a y-independent reduction:

+6uu

+ u

xxx

=0.

7.5 Zakharov–Shabat dressing method via operator factorization 223

It was demonstrated in [324] that the triangular (Volterra) factorization

of the operator

F =(1+K

)

−1

(1 + K

−

)

proved by Zakharov and Shabat [474] links the Zakharov–Shabat dressing

scheme to the DT dressing.

Dressing via local Riemann–Hilbert problem

Beginning with this chapter, we proceed to a description of the second (mostly

analytic) aspect of the dressing method. In this chapter we will show how to

dress a seed solution of a (1+1)-dimensional nonlinear equation making use

of the local Riemann–Hilbert (RH) problem. First we formulate in Sect. 8.1 a

general approach to the RH problem based dressing method [354] in terms of

the Lax representation asso ciated with a given nonlinear equation. Then in

the subsequent sections we will illustrate with examples of speciﬁc nonlinear

equations the power of the RH problem method. Throughout this chapter we

stress two basic facts concerning the applicability of the RH problem to solve

nonlinear equations: (1) the RH problem naturally arises in the co ntext of non-

linear equations and (2) this approach is substantially universal. In Sect. 8.2

we concretize the main ideas by means of the classic example of the nonlinear

Schr¨odinger (NLS) equation. Sections 8.3 and 8.4 are devoted to mathemat-

ically more complicated equations: the modiﬁed NLS (MNLS) equation and

the Ablowitz–Ladik (AL) equation. These two examples are particularly in-

teresting from the point of view of the RH problem. Indeed, the reader will

see that the formalisms we apply for solving the MNLS equation and the AL

equation are practically the same though these equations are completely dif-

ferent: one of them is a partial diﬀerential equation (MNLS), while the other is

a diﬀerential-diﬀerence equation (AL). Section 8.5 demonstrates some novel

features of the RH problem formulation which arise in the case of higher -

order matrix spectral problems. As an example, we consider in this section

the three- wave resonant interaction equations. These equations are of inter-

est in themselves because they represent the so-called disp e rsionless nonlinear

equations. The non-Abelian version of this system was discussed in Sect. 3.7.

In Sect. 8.6 we give one more argument in favor of the universality of the dress-

ing method formalism developed. Namely, we will obtain the homoclinic orbits

for the NLS and MNLS equations. Strictly speaking, this problem is not per-

tinent to the RH problem in the context of this chapter because we will dress

the plane wave solution under periodic boundary conditions. Nevertheless, our

formalism exhibits its eﬀectiveness for solving nonsolitonic problems as well.

225

226 8 Dressing via local Riemann–Hilbert problem

Finally, in the last section we brieﬂy consider the well known Korteweg–de

Vries (KdV) equation. This consideration is based on the method which allows

a straightforward generalization to (2+1)-dimensional nonlinear equations and

servesasabridgetogointhisdirection.

8.1 RH problem and generation of new solutions

As indicated in previous chapters, the Lax representation [263] (or the zero-

curvature representation) is of primary importance for the integration of non-

linear equations. In the framework of the Lax representation, a system of two

linear matrix equa tions (sometimes these equation s are scalar ones)

= Uψ, ψ

= Vψ (8.1)

is associated with a given nonlinear equation. Here the matrices U(x, t, k)and

V (x, t, k) depend on a solution of the nonlinear equation and on a complex

spectral parameter k independent of the coor din ates (x, t). These matrices are

chosen in such a way that the compatibility condition [7]

− V

+[U, V ] = 0 (8.2)

resulting from the equality of mixed derivatives ψ

= ψ

and providing the

existence of a common solution for the system (8.1) would produce exactly the

nonlinear equation we are considering. The matrices U and V have noncoin-

ciding sets of poles (divisors) in some p oints of the extended complex k-plane

C = C ∪{∞}. In fact, they are the divisors that determine all the essential

features of the nonlinear equation with a given Lax representation.

Suppose we know some seed solution u

of the nonlinear equation (8.2).

On frequent occasions, trivial solutions like zero can serve as the seed solution.

We therefore know explicitly the matrices U

and V

which corresp ond to this

solution. As a result, we c an solve a system of linear equations

= U

E, E

= V

E (8.3)

for the matrix function E(x, t, k). Now we will demonstrate, following Za-

kharov and Shabat [475] (see also [148]), that there exists a possibility to

build a class of new solutions o f the nonlinear equation (8.2), this class being

parameterized by a closed oriented contour L on the extended plane

C and

by a nondegenerate bounded matrix function G

(k) deﬁned on the contour.

For this purpose we introduce ﬁrst a matrix function G(x, t, k),

G(x, t, k)=E(x, t, k)G

(k)E

−1

(x, t, k),k∈ L, (8.4)

where E(x, t, k) solves the system (8.3). Then we pose the RH problem [167]

for the matrix G(x, t, k)onthecontourL as

−1

−

(x, t, k)Φ

(x, t, k)=G(x, t, k),k∈ L. (8.5)

8.1 RH problem and generation of new solutions 227

In other words, we want to factorize the matrix G(x, t, k)intoaproductof

the matrix functions Φ

and Φ

−

in such a way that Φ

(Φ

−

) is analytic inside

(outside) the contour L and b oth of them satisfy equation (8.5) on the contour.

Such an analytic factorization problem r epresents one of the formulations of

the RH problem. If the contour L coincides with the real axis of the k-plane,

then the matrix Φ

is analytically continuable to the upper half plane, while

−

is analytically continuable to the lower half plane.

The solution of the RH problem in the form formulated above is not unique.

Evidently, matrices Φ

′

= M (x, t)Φ

with an arbitrary nondegenerate matrix

M solve the same factorization problem. To provide the uniqueness of the

solution, we should pose the normalization condition. This means that we

should set a deﬁnite value of one of the matrices Φ

in some predetermined

point of the k-plane. Usually the inﬁnite point k = ∞ is taken as the reference

point. If Φ

−

(∞)=11, then this normalization is called canonical.

Let us diﬀerentiate (8.5) with respect to x,takingintoaccountthatG

,G]. Then we obtain

= UΦ

− Φ

whereweintroduceamatrixfunctionU(x, t, k),

U = Φ

−

−1

−

+ Φ

−x

−1

−

= Φ

−1

+ Φ

−1

It is clear thatthematrixU has the same set of poles as U

,ifthepoles

do not lie on the contour. Moreover, if the contour contains the pole k

with

multiplicity n

, we should demand an additional property of the matrix G

(k),

namely,

(k)=11+O (|k − k

) ,

near the p oint k

; 11 is the identity matrix.

Similarly, the diﬀerentiation of (8.5) in t gives a matrix V ,

V = Φ

−

−1

−

+ Φ

−t

−1

−

= Φ

−1

+ Φ

−1

entering the equation

= VΦ

− Φ

The last step is to introduce functions ψ

= Φ

E which satisfy the compatible

linear equations

±x

= Uψ

,ψ

±t

= Vψ

Hence, we constructed new matrices U and V which obey the compati-

bility condition (8.2), provided that we are able to solve the RH prob-

lem (8.5). It is important to stress that the matrices U and V have the

same structure as U

and V

.Inotherwords,U and V depend on a new

solution u(x, t)inthesamewayasU

and V

depend on the seed so-

lution u

(x, t). Therefore, we can restore purely algebraically the solution

u(x, t).

228 8 Dressing via local Riemann–Hilbert problem

A distinction will be made between two types of the RH problem. The

ﬁrst one is the so-called regular RH problem when both Φ

and Φ

−1

−

have no

zeros in their domains of analyticity. In other words, det Φ

=0insidethe

contour L and det Φ

−1

−

= 0 outside the contour. Otherwise we will deal with

the RH problem with zeros. It is the RH problem with zeros that leads to

soliton solutions o f nonlinear equations.

The scheme described above g ives general principles o f dressing the seed

solution. In the examples we give, we demonstrate that the RH problem arises

naturally within the inverse spectral transform (IST) approach. M oreover, it

will be clear how to adopt involutions that impose some restrictions on the

general Lax representation (8.2) and reduce the matrices U and V to those

belonging to some complex Lie algebras or symmetric spaces [331].

8.2 Nonlinear Schr¨odinger equation

Here we demonstrate the main stages of the application of the RH problem

to obtain a soliton solution of the NLS equation

+ u

+2|u|

u =0. (8.6)

A vast amount of literature exists about the NLS equation, the most important

books are by Novikov et al. [354], Lamb [259], Do dd et al. [117], Ablowitz

and Segur [13], Calogero and Degasp eris [81], Newell [348], and Faddeev and

Takhtajan [148]. Rememb er that solutions of the NLS equation in terms of

the elliptic functions were given in Sects. 3.5 and 4.9.

8.2.1 Jost solutions

As is well known [473], the NLS equation (8.6) can be represented as the

compatibility condition of the system (8.1) of two linear matrix equations

with the 2 × 2 matrices U and V of the form

U = −ikσ

+ Q, Q =



0 u

−¯u 0



, (8.7)

V = −2ik

+2kQ +iσ

− iQ

this compatibility condition being fulﬁlled for arbitrary constant spectral pa-

rameter k.ThematrixQ stands for the potential in the spectral equation

= Uψ. It will be more convenient for us to write the spectral equation

in terms of the matrix J = ψE

−1

,whereE =exp(−ikxσ

) is a solution of

the sp ectral equation for zero potential. Hence, the spectral equation we shall

deal with is written as

= −ik[σ

,J]+QJ. (8.8)

We consider the zero solution of the NLS equation as the seed solutio n to be

dressed and are interested in deriving localized solutions.