Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons

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9.1 Direct scattering problem 215

S (λ, t = 0) via (9.1). The evolution of the scattering data S (λ, t) is determined

from (9.2). Then u(x, t) is recovered from inverse scattering.

9.1 Direct scattering problem for the time-independent

Schr

odinger equation

Suppose λ = −k

; then the time-independent Schr

odinger equation (9.1)

becomes

;

u(x) + k

v = 0, (9.4)

where, for convenience, we have suppressed the time dependence in u. We shall

further assume that u(x) decays suﬃciently rapidly, which for our purposes

means that u lies in the space of functions



∞

−∞

(1 + |x|

)|u(x)|dx < ∞,

with n = 2(Deift and Trubowitz, 1979). We also remark that the space L

was

the original space used by Faddeev (1963); this space was subsequently used

by Marchenko (1986) and Melin (1985). Associated with (9.4) are two com-

plete sets of eigenfunctions for real k that are bounded for all values of x, and

that have appropriate analytic extensions. These eigenfunctions are deﬁned by

the equation and boundary conditions; that is, we identify four eigenfunctions

deﬁned by the following asymptotic boundary conditions:

φ(x; k) ∼ e

−ikx

, φ(x; k) ∼ e

ikx

as x →−∞,

ψ(x; k) ∼ e

ikx

, ψ(x; k) ∼ e

−ikx

as x →∞.

(9.5)

Therefore φ(x; k), for example, is that solution of (9.4) that tends to e

−ikx

x →−∞. Note that here the bar does not denote complex conjugate, for which

we will use the ∗ notation. From (9.4) and the boundary conditions (9.5), we

see that

φ(x; k) =

φ(x; −k) = φ

∗

(x, −k), (9.6a)

ψ(x; k) =

ψ(x; −k) = ψ

∗

(x, −k). (9.6b)

Since (9.4) is a linear second-order ordinary diﬀerential equation, by the linear

independence of its solutions we obtain the following relationships between

the eigenfunctions

216 Inverse scattering transform for the KdV equation

φ(x; k) = a(k)

ψ(x; k) + b(k)ψ(x; k), (9.7a)

φ(x; k) = −¯a(k)ψ(x; k) +

b(k)

ψ(x; k) (9.7b)

with k-dependent functions: a(k), ¯a(k), b(k),

b(k). These functions form the

basis of the scattering data we will need.

The Wronskian of two functions ψ, φ is deﬁned as

W(φ, ψ) = φψ

− φ

and for (9.4), from Abel’s theorem, the Wronskian is constant. Hence

from ±∞:

W(ψ,

ψ) = −2ik = −W(φ, φ).

The Wronskian also has the following properties

W(φ(x; k),ψ(x; k)) = −W(ψ(x; k),φ(x; k)),

W(c

φ(x; k), c

ψ(x; k)) = c

W(φ(x; k),ψ(x; k)),

with c

, c

arbitrary constants.

From (9.7) it follows that

a(k) =

W(φ(x; k),ψ(x; k))

2ik

, b(k) = −

W(φ(x; k),

ψ(x; k))

2ik

(9.8)

and

a(k) =

φ(x; k), ψ(x; k))

2ik

b(k) =

φ(x; k),ψ(x; k))

2ik

The formula for a(k) follows directly from W(φ(x; k),ψ(x; k)) and using the

ﬁrst of equations (9.7); the others are similar.

Rather than work with the eigenfunctions φ(x; k),

φ(x; k), ψ(x; k) and

ψ(x; k),

it is more convenient to work with the (modiﬁed) eigenfunctions M(x; k),

M(x; k), N(x; k) and

N(x; k), deﬁned by

M(x; k) = φ(x; k) e

ikx

M(x; k):=

φ(x; k) e

ikx

, (9.9a)

N(x; k) = ψ(x; k) e

ikx

N(x; k) =

ψ(x; k) e

ikx

; (9.9b)

then

M(x; k) ∼ 1,

M(x; k) ∼ e

2ikx

,asx →−∞, (9.10a)

N(x; k) ∼ e

2ikx

N(x; k) ∼ 1, as x →∞. (9.10b)

9.1 Direct scattering problem 217

Equations (9.7a) and (9.9) imply that

M(x; k)

a(k)

N(x; k) + ρ(k)N(x; k),

M(x; k)

¯a(k)

= −N(x; k) + ¯ρ(k)

N(x; k),

(9.11)

where

ρ(k) =

b(k)

a(k)

, (9.12a)

¯ρ(k) =

b(k)

¯a(k)

; (9.12b)

τ(k) = 1/a(k) and ρ(k) are called the transmission and reﬂection coeﬃcients

respectively. Equations (9.6b), (9.9b) imply that

N(x; k) =

N(x; −k) e

2ikx

. (9.13)

Due to this symmetry relation we will only need two eigenfunctions. Namely,

from (9.11) and (9.13) we obtain

M(x; k)

a(k)

N(x; k) + ρ(k) e

2ikx

N(x; −k), (9.14)

which will be the fundamental equation. Later we will show that (9.14)is

equivalent to what we call a generalized Riemann–Hilbert boundary value

problem (RHBVP); this RHBVP is central to the inverse problem and is a

consequence of the analyticity properties of M(x; k),

N(x; k) and a(k) that are

established in the following lemma.

Lemma 9.1 (i) M(x; k) can be analytically extended to the upper half

k-plane and tends to unity as |k|→∞for Im k > 0;

(ii)

N(x; k) can be analytically extended to the lower half k-plane and tends to

unity as |k|→∞for Im k < 0.

These analyticity properties are established by studying the linear integral

equations that govern M(x; k) and

N(x; k). We begin by transforming (9.4)via

v(x; k) = m(x; k)e

−ikx

. Then m(x; k) satisﬁes

(x; k) −2ikm

(x; k) = −u(x)m(x; k).

Then assuming that m → 1, either as x →∞,orasx →−∞, we ﬁnd

m(x; k) = 1 +



∞

−∞

G(x −ξ; k) u(ξ) m(ξ; k) dξ,

218 Inverse scattering transform for the KdV equation

where G(x; k) is the Green’s function that solves

− 2ikG

= −δ(x).

Using the Fourier transform method, i.e., G(x; k) =



G(p, k)e

ipx

dp with

δ(x) =



ipx

dp, we ﬁnd

G(x; k) =

2π



ipx

p(p − 2k)

dp,

where C is an appropriate contour. We consider G

(x; k) deﬁned by

(x; k) =

2π



ipx

p(p − 2k)

dp,

where C

and C

−

are the contours from −∞ to ∞ that pass below and above,

respectively, both of the singularities at p = 0 and p = 2k. Hence by contour

integration

(x; k) =

2ik

(1 − e

2ikx

)H(x) =

⎧

⎪

⎨

⎪

⎩

2ik

(1 − e

2ikx

), if x > 0,

0, if x < 0,

−

(x; k) = −

2ik

(1 − e

2ikx

)H(−x) =

⎧

⎪

⎨

⎪

⎩

0, if x > 0,

−

2ik

(1 − e

2ikx

), if x < 0,

where H(x) is the Heaviside function given by

H(x) =

⎧

⎪

⎨

⎪

⎩

1, if x > 0,

0, if x < 0.

We require that M(x; k) and

N(x; k) satisfy the boundary conditions (9.10).

Since from (9.10a), M(x; k) → 1asx →−∞, we associate M(x; k) with the

Green’s function G

(x; k), so the integral is from −∞ to x, and analogously

since

N(x; k) → 1asx →∞, we associate

N(x; k) with the Green’s function

−

(x; k). Therefore it follows that M(x; k) and

N(x; k) are the solutions of the

following integral equations

M(x; k) = 1 +



∞

−∞

(x − ξ; k) u(ξ) M(ξ; k) dξ

= 1 +

2ik



−∞



1 − e

2ik(x−ξ)



u(ξ) M(ξ; k) dξ, (9.15)

N(x; k) = 1 +



∞

−∞

−

(x − ξ; k) u(ξ)

N(ξ; k) dξ

= 1 −

2ik



∞



1 − e

2ik(x−ξ)



u(ξ)

N(ξ; k) dξ. (9.16)

9.2 Scattering data 219

These are Volterra integral equations that have unique solutions; their

Neumann series converge uniformly with u in the function space L

, for each

k: M(x; k)Imk ≥ 0, and for

N(x; k)Imk ≤ 0(Deift and Trubowitz, 1979).

One can show that their Neumann series converge uniformly for u in this func-

tion space. We note: G

(x; k) is analytic for Im k

0 and vanishes as |k|→∞;

M(x; k) and

N(x; k) are analytic for Im k > 0 and Im k < 0 respectively, and

tend to unity as |k|→∞. In the same way it can be shown that N(x; k)e

−2ikx

ψ(x; k)e

ikx

is analytic for Im k > 0 and M(x; k)e

2ikx

= ψ(x; k)e

−ikx

is analytic

for Im k < 0.

9.2 Scattering data

A corollary to the above lemma is that a(k) can be analytically extended to the

upper half k-plane and tends to unity as |k|→∞for Im k > 0. We note that

Deift and Trubowitz (1979) also show that a(k) is continuous for Im k ≥ 0. The

analytic properties of a(k) may be established either from the Wronskian rela-

tionship (9.8), noting that W(φ(x; k),ψ(x; k)) = W(M(x; k), N(x; k)) or by using

a suitable integral representation for a(k). To derive this integral representation

we deﬁne

Δ(x; k) = M(x; k) − a(k)

N(x; k),

and then from (9.15) and (9.16) it follows that

Δ(x; k) = 1 − a(k) +

2ik



∞

−∞



1 − e

2ik(x−ξ)



u(ξ) M(ξ; k) dξ

−

2ik



∞



1 − e

2ik(x−ξ)



u(ξ) Δ(ξ; k) dξ. (9.17)

Also, from (9.14)wehave

Δ(x; k) = b(k) e

2ikx

N(x; −k);

therefore from (9.16)

Δ(x; k) = b(k) e

2ikx

−

2ik



∞



1 − e

2ik(x−ξ)



u(ξ) Δ(ξ; k) dξ. (9.18)

Hence by comparing (9.17) and (9.18), and equating coeﬃcients of 1 and e

2ikx

we obtain the following integral representations for a(k) and b(k):

a(k) = 1 +

2ik



∞

−∞

u(ξ) M(ξ; k) dξ, (9.19)

b(k) = −

2ik



∞

−∞

u(ξ) M(ξ; k) e

−2ikξ

dξ. (9.20)

220 Inverse scattering transform for the KdV equation

From these integral representations it follows that a(k) is analytic for Im k > 0,

and tends to unity as |k|→∞, while b(k) cannot, in general, be continued

analytically oﬀ the real k-axis. Furthermore, since M(x; k) → 1asκ →∞,

then it follows from (9.19) that a(k) → 1asκ (for Im k > 0). Similarly it can

be shown that

a(k) can be analytically extended to the lower half k-plane and

tends to unity as |k|→∞for Im k < 0.

The functions a(k), ¯a(k), b(k),

b(k) satisfy the condition

a(k)¯a(k) + b(k)

b(k) = −1. (9.21)

This follows from the Wronskian and its relations:

W(φ(x; k),

φ(x; k)) = W(a(k)

ψ(x; k) + b(k)ψ(x; k),

− ¯a(k)ψ(x; k) +

b(k)

ψ(x; k))



a(k)¯a(k) + b(k)

b(k)



W(ψ(x; k),

ψ(x; k)),

with W(φ(x; k),

φ(x; k)) = 2ik = −W(ψ(x; k),

ψ(x; k)). From the deﬁnitions

the reﬂection and transmission coeﬃcients, we see that τ(k) and ρ(k)

satisfy

|ρ(k)|

+ |τ(k)|

= 1.

Using (9.4)–(9.7) and their complex conjugates we have the symmetry condi-

tions

¯a(k) = −a(−k) = −a

∗

b(k) = b(−k) = b

∗

(9.22)

where a

∗

), b

∗

) are the complex conjugates of a(k), b(k), respectively.

The zeros of a(k) play an important role in the inverse problem; they are

poles in the generalized Riemann–Hilbert problem obtained from (9.14). We

note the following: the function a(k) can have a ﬁnite number of zeros at

, k

,...,k

, where k

= iκ

, κ

∈ R, j = 1, 2,...,N

(i.e., they all lie on

the imaginary axis), in the upper half k-plane.

First we see that from equations (9.21), (9.22) that

|a(k)|

−|b(k)|

= 1,

for k ∈ R; hence a(k)  0fork ∈ R. Recall that φ(x; k), ψ(x; k) and

ψ(x; k)are

solutions of the Schr

odinger equation (9.4) satisfying the boundary conditions

φ(x; k) ∼ e

−ikx

, as x →−∞,

ψ(x; k) ∼ e

ikx

, as x →∞,

ψ(x; k) ∼ e

−ikx

, as x →∞,

(9.23)

9.2 Scattering data 221

together with the relationship

φ(x; k) = a(k)

ψ(x; k) + b(k)ψ(x; k). (9.24)

From (9.8) it follows that

W(φ(x; k),ψ(x; k)) = φ(x; k)ψ

(x; k) −φ

(x; k)ψ(x; k) = 2ika(k).

Thus at any zero k

of a(k), we have that φ(x; k),ψ(x; k) are linearly indepen-

dent. In order to have a rapidly decaying state; i.e., a bound state where φ(x; k

)

is in L

(R), we must have that Im k

> 0.

The eigenfunction φ(x; k) and its complex conjugate φ

∗

(x; k

∗

) satisfy the

equations



u(x) + k



φ = 0,

∗



u(x) + (k

∗

)



∗

= 0,

(9.25)

respectively. Hence

∂

∂x

W(φ, φ

∗

) +



∗

)

− k



φφ

∗

= 0,

which implies



∗

)

− k





∞

−∞

|φ(x; k)|

dx = 0 (9.26)

since φ and its derivatives vanish as x →±∞. Further, calling k

= ξ

+ η

,we

see by taking the real part of (9.26):



∗



− k

= −2ξ

= 0, any zero of a

must be purely imaginary; i.e., Re k

= ξ

= 0.

To show that a(k) has only a ﬁnite number of zeros, we note that a → 1as

|k|→∞; furthermore, it is shown by Deift and Trubowitz (1979) that a(k)is

continuous for Im k ≥ 0. Hence there can be no cluster points of zeros of a(k)

either in the upper half k-plane or on the real k-axis and so necessarily a(k) can

possess only a ﬁnite number of zeros.

We can also show that the zeros of a(k) are simple by studying the derivative

da/dk with respect to k: but we will not do this here (see Deift and Trubowitz,

1979).

Hence, as long as u ∈ L

, M(x; k)/a(k) is a meromorphic function in the

upper half k-plane with a ﬁnite number of simple poles at k = iκ

, iκ

,...,iκ

Then set

M(x; k)

a(k)

= μ

(x; k) +



j=1

(x)

k − iκ

, (9.27)

222 Inverse scattering transform for the KdV equation

where μ

(x; k) is analytic in the upper half k-plane. Integrating (9.27) around

iκ

and using (9.14) and the fact that at a discrete eigenvalue M, N are

proportional, M(x; iκ

) = b

N(x; iκ

), shows that

(x) = C

N(x; −iκ

)exp(−2κ

x),

where C

= b



. Hence from (9.14) we obtain

(x; k) =

N(x; k) −



j=1

k − iκ

exp(−2κ

N(x; −iκ

)

+ ρ(k)exp(2ikx)

N(x; −k). (9.28)

Furthermore, μ

(x; k) → 1asκ → 1, for Im k > 0.

9.3 The inverse problem

Equation (9.28) deﬁnes a Riemann–Hilbert problem (cf. Ablowitz and Fokas,

2003) in terms of the scattering data S (λ) =



(κ

, C

)

j=1

,ρ(k), a(k)



. We will

use the following.

Consider the P

projection operator deﬁned by

f )(k) =

2πi



∞

−∞

f (ζ)

ζ − (k ± i0)

dζ = lim

↓0

2πi



∞

−∞

f (ζ)

ζ − (k ± i)

dζ

(9.29)

Suppose that f

(k) is analytic in the upper/lower half k-plane and f

(k) → 0as

|k|→∞(for Im k

0). Then

∓

)(k) = 0,

)(k) = ±f

(k).

These results follow from contour integration.

To explain the ideas most easily, we ﬁrst assume that there are no poles;

that is, a(k)  0. Then (9.28) reduces to (9.14), because M(x; k)/a(k) = μ

In (9.14) unity from both

N and M/a and operating on (9.14) with P

−

and

using the analytic properties of M,

N and a yields

−



M(x; k)

a(k)

− 1



= P

−



(

N(x; k) −1) + ρ(k)e

2ikx

N(x; −k)



This implies

0 =



N(x; k) −1



2πi



∞

−∞

ρ(ζ) e

2iζx

N(x; −ζ)

ζ − (k − i0)

dζ.

9.3 The inverse problem 223

Then employing the symmetry: N(x; k) = e

2ikx

N(x; −k) and N(x; −k) =

−2ikx

N(x; k), it follows that

N(x; k) = 1 +

2πi



∞

−∞

ρ(ζ) N(x; ζ)

ζ − (k − i0)

dζ,

N(x; k) = e

2ikx

1 +

2πi



∞

−∞

ρ(ζ) N(x; ζ)

ζ + k + i0

dζ

(9.30)

As k →∞, equation (9.30a) shows that

N(x; k) ∼ 1 −

2πi



∞

−∞

ρ(ζ) N(x; ζ) dζ, (9.31)

From the integral equation for

N(x; k), equation (9.16), using the Riemann–

Lebesgue lemma, we also see that as κ →∞

N(x; k) ∼ 1 −

2ik



∞

u(ξ) dξ. (9.32)

Therefore, by comparing (9.31) and (9.32), it follows that the potential can be

reconstructed from

u(x) = −

∂

∂x



∞

−∞

ρ(ζ) N(x; ζ) dζ

. (9.33)

For the case when a(k) has zeros, one can extend the above result. Suppose

that

a(iκ

) = 0,κ

∈ R, j = 1, 2,...,N

;

then deﬁne

(x) = M(x; iκ

(x) = exp(−2κ

N(x; −iκ

for j = 1, 2,...,N

.In(9.28) subtracting unity from both μ

(x, k) and

N(x; k) and applying the P

−

projection operator (9.29)to(9.28), recalling

that N(x; k) = e

2ikx

N(x; −k) and carrying out the contour integration of simple

poles, we ﬁnd the following integral equations

N(x; k) = e

2ikx

⎧

⎪

⎨

⎪

⎩

1 −



j=1

(x)

k + iκ

2πi



∞

−∞

ρ(ζ) N(x; ζ)

ζ + k + i0

dζ

⎫

⎪

⎬

⎪

⎭

, (9.34)

(x) = exp(−2κ

⎧

⎪

⎨

⎪

⎩

1 + i



j=1

(x)

+ κ

2πi



∞

−∞

ρ(ζ) N(x; ζ)

ζ + iκ

dζ

⎫

⎪

⎬

⎪

⎭

(9.35)

224 Inverse scattering transform for the KdV equation

for p = 1, 2,...N

, with the potential reconstructed from [following the same

method that led to (9.33)]

u(x) =

∂

∂x

⎧

⎪

⎨

⎪

⎩



j=1

(x) −



∞

−∞

ρ(k) N(x; k) dk

⎫

⎪

⎬

⎪

⎭

. (9.36)

Existence and uniqueness of solutions to equations such as (9.34)–(9.36)is

given by Beals et al. (1988). We also note that the above integral equa-

tions (9.34) and (9.35) can be transformed to Gel’fand–Levitan–Marchenko

integral equations, see Section 9.5 below, from which one can also deduce the

existence and uniqueness of solutions (cf. Marchenko, 1986).

9.4 The time dependence of the scattering data

In this section we will ﬁnd out how the scattering data evolves. The time evo-

lution of the scattering data may be obtained by analyzing the asymptotic

behavior of the associated time evolution operator, which as we have seen

earlier for the KdV equation is

= (u

+ γ)v + (4k

− 2u)v

with γ a constant. If we let v = φ(x; k) and make the transformation

φ(x, t; k) = M(x, t; k) e

−ikx

M then satisﬁes

= (γ − 4ik

+ u

+ 2iku)M + (4k

− 2u)M

. (9.37)

Recall that from (9.11)

M(x, t; k) = a(k, t)

N(x, t; k) + b(k, t)N(x, t; −k),

where ρ(k, t) = b(k, t)/a(k, t). From (9.10) the asymptotic behavior of M(x, t; k)

is given by

M(x, t; k) → 1, as x →−∞,

M(x, t; k) → a(k, t) + b(k, t) e

2ikx

,asx →∞.

By using using the fact that u → 0 rapidly as x →±∞and the above equation

it follows from (9.37) that

γ − 4ik

= 0, x →−∞

+ b

2ikx

= 8ik

2ikx

, x → +∞,