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

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8.7 KdV equation 269

8.7 KdV equation

At ﬁrst sight it might appear as rather strange to devote the very last section in

this chapter to one of the most popular and most studied nonlinear integrable

equations. Our intention here is to pe rform an analysis of the KdV equation in

the form most suitable for a generalization to (2+1)-dimensional equations.

It is precisely this approach that will be realized in the next chapter, ﬁrst

using an example of the (1+1)-dimensional integrodiﬀerential Benjamin–Ono

equation and then for proper (2+1)-dimensional equations. We will not dwell

on a detailed exposition of various results concerning the KdV equation and

its solutions. There exists a vast amount of literature devoted to this topic (for

example, [13, 81, 117, 348]). The traditional treatment of the KdV equation

was mentioned in Sect. 7.5.1. Instead, our main goal in this section is to

illustrate the basic steps of the approach, which goes back to Novikov et al.

[354]. We recommend the reader to consult the content of this section when

studying the next chapter in order to see strict parallels in the strategy of

solving (1+1)- and (2+1)-dimensional equations.

8.7.1 Jost solutions

The KdV equation

+6uu

+ u

xxx

for a smooth real function u(x, t) is integrated by means of the sp ectral prob-

lem having the form of the time-independent Schr¨odinger equation [173]

+(u + k

)φ =0. (8.109)

To have linear dependence on the spectral parameter k, we perform a trans-

formation m(x, k)=φ(x, k)exp(ikx). Hence, th e s pectral problem for KdV is

written as

− 2ikm

+ um =0. (8.110)

The evolution part of the Lax pair has the form

=(α − 4ik

+ u

+2iku)m +(4k

− 2u)m

,α=const. (8.111)

Analysis of the spectral problem will be performed by means of the Green

function method. The Green function G(x, k) solves the equation

− 2ikG

= −δ(x)

and has the form

G(x, k)=

2π



∞

−∞

ipx

p (p − 2k)

We see that the Green function has a discontinuity across the real axis of the

complex k-plane. Accordingly, we can determine two Green functions G

(x, k)

which are analytic in the half planes Imk ≷ 0:

(x, k)=±

2ik



1 − e

2ikx



θ(±x), (8.112)

270 8 Dressing via local Riemann–Hilbert problem

where θ(x) is the Heaviside step function, θ(x)=1forx>0andθ(x)=0

for x<0. Hence, the fundamental solutions (the Jost functions) M

(x, k)

of the spectral equation (8.110) can be deﬁned by means of the integral

equations

(x, k)=1+



∞

−∞

′

(x − x

′

,k)u(x

′

,k), (8.113)

with the property M

(x, k) → 1atk →∞. M

can be analytically extended

in the upper and lower halves of the k-plane, respectively. Besides, these func-

tions obey the boundary conditions M

(x, k) → 1atx →∓∞. Indeed, though

the integral equations (8.113) have been written in the Fredholm form, they

are in fact of the Volterra type because of the step function θ(±x).

The free term 1 in (8.113) represents a solution of the spectral equation

with zero potential. This corresponds to dressing of the trivial solution. How-

ever, the second-order spectral equation with zero potential allows one more

linearly independent solution e

2ikx

. Accordingly, we can deﬁne two other fun-

damental solutions

(x, k)=e

2ikx



∞

−∞

′

(x − x

′

,k)u(x

′

,k), (8.114)

with the limits N

(x, k) → e

2ikx

at x →∓∞.AsdistinctfromM

,the

eigenfunctions N

, due to the exponential free term, cannot in general be

analytically continued o ﬀ the real axis Imk =0.

It follows from (8.112) that

(x, k) → a

(k) ± b

(k)e

2ikx

(8.115)

for x →±∞,where

(k)= 1±

2ik



∞

−∞

dxu(x)M

(x, k),b

(k)=−

2ik



∞

−∞

dxu(x)M

(x, k)e

−2ikx

(8.116)

are the scattering coeﬃcients. For real potentials u(x) there are involution

relations for scattering coeﬃcients:

∗

(k)=a

(−k),b

∗

(k)=b

(−k),a

∗

(k)=a

−

(k),b

∗

(k)=b

−

(k),

(k)|

=1+|b

(k)|

It is important that a

(k) are analytic functions in Imk ≷ 0, as is seen from

(8.116). Therefore, M

are meromorphic functions in Imk ≷ 0witha

ﬁnite number of poles in zeros of a

(k).

8.7 KdV equation 271

8.7.2 Scattering equation and RH problem

In accordance with (8.115), for x → +∞

(x, k)

(k)

→ 1+ρ

(k)e

2ikx

,ρ

At the same time M

−

(x, k) → 1atx → +∞. Let us now calculate a jump

∆(x, k)=M

− M

−

of these functions across the real axis Imk =0.

Calculation gives

∆(x, k)=(1/a

) − 1



∞

−∞

′

−G

−

)(x −x

′

,k)u(x

′

)

′

,k)

(k)



∞

−∞

′

−

(x −x

′

,k)∆(x

′

,k).

Since (G

−G

−

)(x, k)=(1/2ik)



1 − e

2ikx



, in virtue of the deﬁnitions (8.116)

we obtain

∆(x, k)=ρ

(k)e

2ikx



∞

−∞

′

−

(x − x

′

,k)u(x

′

)∆(x

′

,k).

On the other hand [see (8.114)],

(k)N

−

(x, k)=ρ

(k)e

2ikx



∞

−∞

′

−

(x − x

′

,k)u(x

′

)ρ

(k)N

−

′

,k).

Comparing the last two e qu ations, we conclude owing to the uniqueness of

the solution of the above integral equations that ∆(x, k)=ρ

(k)N

−

(x, k).

Therefore, we arr ive at the scattering equation

(x, k)

(k)

= M

−

(x, k)+ρ

(k)N

−

(x, k). (8.117)

In order to have reasons to treat (8.117) as the RH problem, we should ex-

press N

−

in terms of M

−

.TodothatwenoteﬁrstofallthatG

(x, k)=

(x, −k)e

2ikx

. Hence,

−

(x, −k)=1+



∞

−∞

′

−

(x − x

′

,k)u(x

′

−

′

, −k)

=1+



∞

−∞

′

−2ik(x−x

′

)

−

(x − x

′

,k)u(x

′

−

′

, −k)

−2ikx



2ikx



∞

−∞

′

−

(x − x

′

,k)u(x

′

−

′

, −k)e

2ikx

′



272 8 Dressing via local Riemann–Hilbert problem

Comparing this with the integral equation for N

−

(8.114), we get the symme-

try relation:

−

(x, k)=M

−

(x, −k)e

2ikx

. (8.118)

Therefore, we can write the scattering equation (8.117) in the form of the RH

problem withashift:

(x, k)

(k)

= M

−

(x, k)+ρ

(k)e

2ikx

−

(x, −k), Imk =0. (8.119)

The shift is referred to the change of sign in k inthelastterm.Thenormal-

ization of the RH problem (8.119) is canonical, M

→ 1ask →∞.

The form (8.119) of the RH problem corresponds to the jump problem for

a piecewise analytic function Ψ (x, k),

Ψ =



, Imk>0,

−

, Imk<0,

(8.120)

with discontinuity along the contour Imk = 0. In previous sections we dealt

with the factorization form of the RH problem. Evidently, both versions are

equivalent. Indeed, the RH problem (8.17) for the NLS equation with the

matrix G = I + g takes the form Φ

= Φ

−

+ gΦ

−

with the same structure as

(8.119).

8.7.3 Inverse problem

To solve the inverse problem of determining the potential u(x), we reconstruct

the function Ψ(x, k) (8.120). This function is analytic in the entire complex

k-plane, except for a ﬁnite number N of simple poles at the points k

=iκ

> 0,j =1,...,N, and a discontinuity across the real axis. Accordingly,

using the Cauchy formula, we obtain

Ψ(x, k)=1+



l=1

(x)

k − k

2πi



∞

−∞

dℓ

ℓ − k

ρ(ℓ)e

2iℓx

−

(x, −ℓ). (8.121)

Here Φ

(x) is a residue of Ψ in k

.AsetofallΦ

(x), j =1,...,N comprises the

set of eigenfunctions corresponding to the discrete spectrum (k

,j =1,...,N)

of the problem (8.110). For Imk>0 (8.121) and (8.118) give

(x, k)

(k)

=1+



j=1

(x)

k − k

2πi



∞

−∞

dℓ

ℓ − k

ρ(ℓ)N

−

(x, ℓ).

To determine the behavior of M

for k → k

, we deﬁne a regular part of

(j)

(x, k)=

(x, k)

(k)

−

(x)

k − k

8.7 KdV equation 273

With account for the integral equation (8.113) for M

,wecanwrite

(j)

(x, k) −



∞

−∞

′

(x − x

′

,k)u(x

′

(j)

′

,k)

(k)

−

k − k



−



∞

−∞

′

(x − x

′

,k)u(x

′

)Φ

′

)



The left-hand side of this equation is free of singularities for k = k

; hence,

the diverging terms on the right-hand side have to be canceled. This gives the

integral equation for discrete eigenfunctions:

(x)=iγ

−1



∞

−∞

′

(x − x

′

)u(x

′

)Φ

′

),γ

=i(da

/dk)

(8.122)

Comparing (8.122) with the integral equation for M

, we ﬁnd

lim

k→k

(x, k)=−iγ

(x). (8.123)

In turn, (8.123) and asymptotics of M

at x →±∞give

→



iγ

−1

,x→−∞

−2κ

,x→ +∞.

(8.124)

Here c

is a normalization constant. Finally, it follows from (8.122) and (8.124)

that it is possible to express eigenvalues as functionals of discrete eigenfunc-

tions:

= −



∞

−∞

dxu(x)Φ

(x).

To reconstruct the potential, we at ﬁrst derive a system of linear equa tions

for eigenfunctions. Φ

canbeexpressedintermsofM

−

(x)=

(x, k

)=ic

−

(x, k

)=ic

−2κ

−

(x, −k

On the other hand, we obtain from (8.121)

−

(x, −k)=1−



j=1

(x)

k + k

2πi



∞

−∞

dℓ

ℓ + k +i0

ρ(ℓ)N

−

(x, ℓ) (8.125)

and for k = k

−

(x, −k

)=1+i



j=1

(x)

+ κ

2πi



∞

−∞

dℓ

ℓ +iκ

ρ(ℓ)N

−

(x, ℓ).

Therefore, Φ

(x) takes the form

(x)=ic

−2κ

⎛

⎝

1+i



j=1

(x)

+ κ

2πi



∞

−∞

dℓ

ℓ +iκ

ρ(ℓ)N

−

(x, ℓ)

⎞

⎠

(8.126)

274 8 Dressing via local Riemann–Hilbert problem

Further, in virtue of (8.118) and (8.125),

−

(x, k)=e

2ikx

⎛

⎝

1 −



j=1

(x)

k +iκ

2πi



∞

−∞

dℓ

ℓ + k +i0

ρ(ℓ)N

−

(x, ℓ)

⎞

⎠

(8.127)

As a result, we have a closed system of linear equations (8.126) and (8.127)

to ﬁnd Φ

(x)andN

−

(x, k). This system is determined by the RH data

{ρ

(k),κ

,j =1,...,N}. Then we expand the integral equation (8.114)

for N

−

in the asymptotic series in k

−1

and compare it with (8.127) taken for

k →∞. As a result, we get the reconstruction formula

u(x)=∂

⎛

⎝



j=1

(x) −



∞

−∞

dkρ(k)N

−

(x, k)

⎞

⎠

. (8.128)

The ﬁrst term on the right-hand side of (8.128) contributes from the discrete

spectrum, while the second term is responsible for the continuous spectrum.

It should be stressed that it is the nonanalytic eigenfunction N

−

that

enters the complete s et of eigenfunctions {N

−

,Φ

,j =1,...,N} [363]. It

is not accidental because the spectral problem (8.110) is non-self-adjoint in

contrast to the standard sp ectral problem (8.109).

8.7.4 Evolution of RH data

Substituting the asymptotics of M

at x →±∞into the evolution equation

(8.111), we obtain

α =4ik

(k, t)=a

(k, 0),ρ

(k, t)=ρ

(k, 0)e

8ik

Because a

,t) = 0, we get κ

= const. Besides, owing to M

(x, k

) →

−2κ

at x → + ∞, we ﬁnd

(t)=c

(0)e

8κ

Hence, the time dependence of the RH data is extremely simple.

8.7.5 Soliton solution

Solitons correspond to reﬂectionless potentials, i.e., ρ(k)=0.Thesimplest

one-soliton solution is given by u

(x, t)=2iΦ

(x, t). It follows from (8.126)

that

(x, t)=2iκ

−2κ

(x−4κ

t−x

)

1+e

−2κ

(x−4κ

t−x

)

2κ

(0)

2κ

Hence, the one-soliton solution to the KdV equation has the form

(x, t)=2κ

sech2κ

(x − 4κ

t − x

8.7 KdV equation 275

which goes back to Korteweg and de Vries [249]. An N-soliton solution is also

well known [354].

Let us remember that the aim of this chapter is purely methodological.

We will see in the next chapter that the main ideas developed above will be

naturally developed for (2+1)-dimensional equations.

Dressing via nonlocal

Riemann–Hilbert problem

In the previous chapter we illustrat ed the eﬃciency of the dressing approach

using the local Riemann–Hilbert (RH) problem for solution of the Cauchy

problem for a number of (1+1)-dimensional nonlinear integrable equations.

The essential progress in the development of the inverse spectral transform

(IST) formalism has been achieved owing to the p erception that the nonlocal

RH problem can serve as a natural frame for solving nonlinear equations

in 2+1 dimensions. Manakov [305] was the ﬁrst to apply the nonlocal RH

problem to treat the Kadomtsev–Petviashvili (KP) equation by means of the

IST metho d. Besides, there exists an imp ortant class of (1+1)-dimensional

nonlinear integrodiﬀerential equations which cannot be solved by the methods

discussed in Chap. 8.

This chapter contains an exposition of basic points related to the appli-

cation of the nonlo cal RH problem. We consider thr ee featured examples. In

Sect. 9.1 we consider the (1+1)-dimensional integrodiﬀerential Benjamin–Ono

(BO) equation . At the very beginning we work with real function u(x, t)tak-

ing into account important constraints imposed on the spectral data by the

reality condition, due to Kaup, Lakoba, and Matsuno [231]. Basic steps in ap-

plication of the nonloc al RH problem developed for the BO equation are then

used in Sect. 9.2 for the KP I equation. Along with the classical lumps, we

discuss locali zed solutions associated with multiple-pole eigenfunctions found

by Ablowitz and Villarroel [439]. Finally, Sect. 9.3 is devoted to solution of

the initial boundary value problem for the Davey–Stewartson I (DS I) equa-

tion. It is this equation that allows true exponentially localized solitons in

2+1 dimensions discovered by Boiti, Leon, Martina, and Pempinelli [62]. In

our consideration we follow the formalism adopted by Fokas and Santini [162].

9.1 Benjamin–Ono equation

The BO equation for a scalar function u(x, t),

+2uu

+ Hu

=0,Hu(x)=

p.v.



∞

−∞

′

u(x

′

)

′

− x

, (9.1)

277

278 9 Dressing via nonlocal Riemann–Hilbert problem

where Hu(x) is the Hilbert transform of u(x) and p.v. means the principal

value of the integral, describes the propagation o f long internal waves in a

stratiﬁed ﬂuid [46, 106, 195, 357]. We require the function u(x, t) to be real, in

accordance with its physical meaning. In the general case of complex u(x, t)

Foka s and Ablowitz [158] develop ed an IST approach to solve the Cauchy

problem for the BO equation within the class of initial conditions which have

suﬃcient decay at inﬁnity, i.e., u(x, 0) ≡ u

(x) → 0at|x|→∞. The consid-

eration of real potentials u(x)

imposes nontrivial restrictions o n the spectral

data of the corresponding RH problem and, in addition, makes it possible

to derive a body of practically important results, such as the prediction of

a number of possible bound states (solitons) that can be produced by the

initial data [232, 231]. Furthermore, we will restrict ourselves to the so-called

nongeneric potentials which include, in particular, all N -solito n solutions and

zero background [232, 362]. The nongeneric case has distinctly diﬀerent fea-

tures, compared with the general situation. The N-soliton solution of the BO

equation was obtained for the ﬁrst time in [83, 86] in the framework of the

decomposition of u(x, t) in a ﬁnite number of simple poles, a s well as in [310]

by the Hirota method . The IST method permits us to carry out the complete

study of the Cauchy problem for the BO equation.

9.1.1 Jost solutions

The Lax pair for the BO equation (9.1) has the form [54, 346]

iΦ

+ k(Φ

− Φ

−

)+uΦ

=0, (9.2)

iΦ

− 2ikΦ

+ Φ

− 2i[u]

+ νΦ

=0. (9.3)

Here Φ

(k, x) are limit values of the analytic function Φ at y →±0inthe

upper and lower halves of the complex z-plane, where z = x +iy is a com-

plexiﬁcation of the physical variable x, k is a spectral parameter, an d ν is an

arbitrary constant. The functions [u]

(x) are deﬁned as [u]

= P

u,where

are projectors





(x)=±

2πi



∞

−∞

′

u(x

′

)

′

− (x ± i0)

In other words, [u]

means that one takes the part of u(x) that is analytic in

the upper half z-plane. It is interesting that the spectral equation (9.2) can

be considered as a diﬀerential RH problem.

Taking the (+) part of the spectral equation (i.e., acting on it by the

projector P

), we arrive at the integrodiﬀerential equation

iΦ

+ kΦ

+[uΦ]

=0. (9.4)

On o ccasion, we will suppress for convenience time dependence in the potential

and other quantities.