Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
Подождите немного. Документ загружается.
9.9 Conserved quantities and conservation laws 235
as x →∞. By comparing coefficients we find that
a(k) = e
ikL
$
cos
L
√
H
2
+ k
2
− i
H
2
+ 2k
2
2k
√
H
2
+ k
2
sin
L
√
H
2
+ k
2
2
,
b(k) =
iH
2
2k
√
H
2
+ k
2
sin
L
√
H
2
+ k
2
.
Since the zeros of a(k) are purely imaginary, let k = iκ where κ ∈ R. Then
a(iκ) = e
−κL
$
cos
L
√
H
2
− κ
2
−
H
2
− 2κ
2
2κ
√
H
2
− κ
2
sin
L
√
H
2
− κ
2
2
.
Thus, the zeros of a(iκ) occur when
tan
L
√
H
2
− κ
2
=
2κ
√
H
2
− κ
2
H
2
− 2κ
2
.
Note that the right-hand side is a monotonically increasing function from
[0, ∞)on0 ≤ κ<H/
√
2 and from (−∞, 0] on H/
√
2 <κ≤ H, and that
tan
LH
1 − κ
2
/H
2
has LH/π periods for κ between 0 and H. Thus, the num-
ber of zeros of a(iκ) depends on the size of LH, such that there are n ∈ N zeros
of a(iκ)for0<κ<H when
(n − 1)π<LH ≤ nπ.
Additionally, using the same reasoning as above, there are up to n zeros for
−H <κ<0 when LH < nπ.For|κ| > H, we can write
tanh
L
√
κ
2
− H
2
=
2κ
√
κ
2
− H
2
H
2
− 2κ
2
,
and it is seen that there are no zeros for κ>H. As earlier, the time depen-
dence of the scattering data can be determined from a(k, t) = a(k, 0), b(k, t) =
b(k, 0) exp(8ik
3
t), etc.
9.9 Conserved quantities and conservation laws
As mentioned in Chapter 8, one of the major developments in the study of
integrable systems was the recognition that special equations, such as the KdV
equation, have an infinite number of conserved quantities/conservation laws.
These were derived by Gardner et al. (1967) by making use of the Miura trans-
formation. That derivation can be found in Section 8.3. In this section we use
IST to derive these results.
Recall from Lemma 9.1 that a(k) and M(x; k) = φ(x; k)exp(−ikx) are ana-
lytic in the upper half k-plane and that a(k) has a finite number of simple zeros
236 Inverse scattering transform for the KdV equation
on the imaginary axis: {k
m
= iK
m
}
N
m=1
, in the upper half k-plane and a(k) → 1
as k →∞. Moreover we have shown in Section 9.4 that a(k) is constant in
time. We will use this fact to obtain the conserved densities.
We can relate log a(k) to the potential u and its derivatives. From (9.5)
and (9.8)wehave
a(k) =
1
2ik
W(φ, ψ)
=
1
2ik
(φψ
x
− φ
x
ψ) = lim
x→+∞
1
2ik
φike
ikx
− φ
x
e
ikx
.
Letting
φ = e
p−ikx
, p → 0asx →−∞ (9.49)
yields
a(k) = lim
x→+∞
e
p
1 −
p
x
2ik
or
log a = lim
x→+∞
p + lim
x→+∞
log
1 −
p
x
2ik
. (9.50)
Substituting (9.49)into
φ
xx
+ (k
2
+ u)φ = 0
and calling μ = p
x
= ik +
φ
x
φ
gives
μ
x
+ μ
2
− 2ikμ + u = 0 (9.51)
which is a Ricatti equation for μ in terms of u. This equation has a series
solution, in terms of inverse powers of k, of the form
μ =
∞
n=1
μ
n
(2ik)
n
=
μ
1
2ik
+
μ
2
(2ik)
2
+
μ
3
(2ik)
3
+ ···, (9.52)
which yields a recursion relation for the μ
n
.From(9.51) the first three terms
are
μ
1
= u,μ
2
= u
x
,μ
3
= u
2
+ u
xx
(9.53)
and in general μ
n
satisfies
μ
n,x
+
n−1
q=1
μ
q
μ
n−q
− μ
n+1
= 0 (9.54)
9.9 Conserved quantities and conservation laws 237
which further yields μ
4
= (2u
2
+ u
xx
)
x
, μ
5
= μ
4,x
+ (u
2
)
xx
+ 2u
3
−u
2
x
, and so on.
From (9.53)–(9.54) we see that all of μ
n
are polynomial functions of u and its
derivatives, i.e., μ
n
= μ
n
(u, u
x
, u
xx
,...) and μ
n
→ 0asx →∞. Hence,
lim
x→+∞
log
1 −
p
x
2ik
= 0.
Then from (9.50) and (9.52),
log a = lim
x→+∞
x
−∞
p
x
dx =
∞
−∞
μ dx
=
∞
n=1
1
(2ik)
n
∞
−∞
μ
n
(u, u
x
, u
xx
,...) dx =
∞
n=1
C
n
(2ik)
2
, (9.55)
which implies we have an infinite number of conserved quantities
C
n
=
∞
−∞
μ
n
(u, u
x
, u
xx
,...) dx.
The first three non-trivial conserved quantities are, from (9.53)–(9.54)(C
2n
are
trivial)
C
1
=
∞
−∞
udx, C
3
=
∞
−∞
u
2
dx, C
5
=
∞
−∞
2u
3
− u
2
x
dx.
We can also use the large k expansion to retrieve the infinite number of
conservation laws
∂T
n
∂t
+
∂F
n
∂x
= 0, n = 1, 2, 3,...
associated with the KdV equation. As discussed earlier there is an associated
linear dependence with the Schr
¨
odinger scattering problem; that is,
φ
t
= (u
x
+ 4ik
3
)φ + (4k
2
− 2u)φ
x
. (9.56)
We note that the fixed boundary condition φ ∼ exp(−ikx)asx →−∞is con-
sistent with (9.56). Substituting φ = exp(p −ikx), μ = p
x
into (9.56) and taking
a derivative with respect to x yields after some manipulation
μ
t
=
∂
∂x
[2iku + u
x
+ (4k
2
− 2u)μ].
Then the expansion (9.52)gives
∂μ
n
∂t
+
∂
∂x
(μ
n+2
+ 2uμ
n
) = 0, n ≥ 1
238 Inverse scattering transform for the KdV equation
with the μ obtained from (9.53)–(9.54). Substitution of μ
n
yields the infinite
number of conservation laws. The first two non-trivial conservation laws are
∂
∂t
u +
∂
∂x
(u
xx
+ 3u
2
) = 0
∂
∂t
(u
2
+ u
xx
) +
∂
∂x
u
xxxx
+ 4(u
2
)
xx
+ 4u
3
− 3u
2
x
= 0.
Finally, we note that the conserved densities can also be computed in terms of
the scattering data. This requires some complex analysis.
Consider the function
α(k) = a(k)
N
#
B
m=1
k − k
∗
m
k − k
m
, (9.57)
where k
∗
m
is the complex conjugate of k
m
. This function is analytic in the upper
half k-plane with no zeros and α(k) → 1ask →∞. By the Schwarz reflection
principle (see Ablowitz and Fokas, 2003) the complex conjugate of (9.57)
α(k)
∗
= a(k)
∗
N
#
B
m=1
k − k
m
k − k
∗
m
(9.58)
is analytic in the lower half k-plane with no zeros. Then by Cauchy’s integral
theorem for Im{k} > 0
log α(k) =
1
2πi
∞
−∞
log α(z)
z − k
dz
0 =
1
2πi
∞
−∞
log α(z)
∗
z − k
dz.
Adding the above equations for Im{k} > 0 and using (9.57) and (9.58)gives
log a(k) = −
1
2πik
∞
−∞
log aa
∗
1 − z/k
dz +
N
#
m=1
log
1 − k
m
/k
1 − k
∗
m
/k
.
Then expanding for large k gives
log a(k) = −
1
2πi
∞
n=1
1
k
n
∞
−∞
z
n−1
log
(
aa
∗
(z)
)
dz
+
∞
n=1
1
k
n
N
#
m=1
33
k
∗
m
4
n
− k
n
m
4
n
=
∞
n=1
C
n
(2ik)
n
.
9.10 Outline of the IST for a general evolution system 239
The right-hand side of the above equation is also obtained from (9.55); thus C
n
are the conserved densities now written in terms of scattering data. Recall,
1 −
#
#
#
#
#
b
a
(k)
#
#
#
#
#
2
= 1 −|ρ(k)|
2
=
1
|a(k)|
2
.
Thus we can write the non-trivial conserved densities in terms of the data
obtained from log a(k):
C
2n+1
= (2i)
2n+1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2πi
∞
−∞
z
2n
log(1 −|ρ|
2
) dz +
N
#
m=1
3
k
∗
m
4
2n+1
− k
2n+1
m
2n + 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
for n ≥ 1, or noting that k
m
= iκ
m
, κ
m
> 0
C
2n+1
=
1
π
∞
−∞
(−1)
n
(2z)
2n
log(1 −|ρ(z)|
2
) dz +
2
2n + 1
N
#
m=1
(2κ
m
)
2n+1
,
which gives the conserved densities in terms of the scattering data
{ρ(z),κ
m
|
N
m=1
}.
9.10 Outline of the IST for a general evolution system –
including the nonlinear Schr
¨
odinger equation with
vanishing boundary conditions
In the previous sections of this chapter we concentrated on the IST for the
KdV equation with vanishing boundary values. In this section we will discuss
the main steps in the IST associated with the following compatible linear 2 ×2
system (Lax pair):
v
x
=
−ik q
rik
v, (9.59a)
and
v
t
=
AB
C −A
v, (9.59b)
where v is a two-component vector, v(x, t) =
v
(1)
(x, t), v
(2)
(x, t)
T
.
1
In Chap-
ter 8 we described how the compatibility of the above general linear system
yielded a class of nonlinear evolution equations. This class includes as special
cases the nonlinear Schr
¨
odinger (NLS), modified Korteweg–de Vries (mKdV)
1
Unless otherwise specified, superscripts
()
with = 1, 2 denote the th component of the
corresponding two-component vector and
T
denotes matrix transpose.
240 Inverse scattering transform for the KdV equation
and sine–, sinh–Gordon equations. The equality of the mixed derivatives of v,
i.e., v
xt
= v
tx
, is equivalent to the statement that q and r satisfy these evolu-
tion equations, if k, the scattering parameter or eigenvalue, is independent of x
and t. As before, we refer to the equation with the x derivative, (9.59a), as the
scattering problem: its solutions are termed eigenfunctions (with respect to the
eigenvalue k) and the equation with the t derivative, (9.59b), is called the time
equation.
For example, the particular case of the NLS equation results when we take
the time-dependent system to be
v
t
=
AB
C −A
v =
2ik
2
+ iqr 2kq + iq
x
2kr + ir
x
−2ik
2
− iqr
v, (9.60)
which by compatibility of (9.59a)–(9.60) yields
iq
t
= q
xx
− 2rq
2
(9.61a)
−ir
t
= r
xx
− 2qr
2
. (9.61b)
Then the reduction r = σq
∗
with σ = ∓1, gives, as a special case, the NLS
equation
iq
t
= q
xx
± 2|q|
2
q. (9.62)
We say the NLS equation is focusing or defocusing corresponding to σ = ∓1.
In what follows we will assume that the potentials, i.e., the solutions of the
nonlinear evolution equations q(x, t), r(x, t), decay sufficiently rapidly as x →
±∞;atleastq, r ∈ L
1
. The reader can also see the references mentioned earlier
in this chapter and in Chapter 8. Here we will closely follow the methods and
notation employed by Ablowitz et al. (2004b).
We can rewrite the scattering problem (9.59a)as
v
x
=
(
ikJ + Q
)
v
where
J =
−10
01
, Q =
0 q
r 0
.
and I is the 2 × 2 identity matrix.
9.10 Outline of the IST for a general evolution system 241
9.10.1 Direct scattering problem
Eigenfunctions and integral equations
When the potentials q, r → 0sufficiently rapidly as x →±∞the eigenfunc-
tions are asymptotic to the solutions of
v
x
=
−ik 0
0 ik
v
when
|
x
|
is large. We single out the solutions of (9.59a) (eigenfunctions)
defined by the following boundary conditions:
φ(x; k) ∼
1
0
e
−ikx
, φ(x; k) ∼
0
1
e
ikx
as x →−∞ (9.63a)
ψ(x; k) ∼
0
1
e
ikx
, ψ(x; k) ∼
1
0
e
−ikx
as x → +∞. (9.63b)
Note that here and in the following the bar does not denote complex con-
jugate, for which we will use the
∗
notation. It is convenient to introduce
eigenfunctions with constant boundary conditions by defining
M(x; k) = e
ikx
φ(x; k), M(x; k) = e
−ikx
φ(x; k)
N(x; k) = e
−ikx
ψ(x; k), N(x; k) = e
ikx
ψ(x; k).
In terms of the above notation, the eigenfunctions, also called Jost solutions,
M(x; k) and
N(x; k), are solutions of the differential equation
∂
x
χ(x; k) = ik
(
J + I
)
χ(x; k) +
(
Qχ
)
(x; k), (9.64)
while the eigenfunctions N(x; k) and
M(x; k) satisfy
∂
x
χ(x; k) = ik
(
J − I
)
χ(x; k) +
(
Qχ
)
(x; k), (9.65)
with the constant boundary conditions
M(x; k) →
1
0
,
M(x; k) →
0
1
as x →−∞ (9.66a)
N(x; k) →
0
1
,
N(x; k) →
1
0
as x → +∞. (9.66b)
242 Inverse scattering transform for the KdV equation
Solutions of the differential equations (9.64) and (9.65) can be represented by
means of the following integral equations
χ
(
x; k
)
= w +
+∞
−∞
G(x − x
; k)Q(x
)χ(x
; k) dx
χ
(
x; k
)
= w +
+∞
−∞
G(x − x
; k)Q(x
)χ(x
; k) dx
where w =
(
1, 0
)
T
, w =
(
0, 1
)
T
and the (matrix) Green’s functions G(x; k) and
G(x; k) satisfy the differential equations
[
I∂
x
− ik
(
J + I
)
]
G(x; k) = δ(x),
[
I∂
x
− ik
(
J − I
)
]
G(x; k) = δ(x) ,
where δ(x) is the Dirac delta (generalized) function. The Green’s functions are
not unique, and, as we will show below, the choice of the Green’s functions
and the choice of the inhomogeneous terms w,
w together uniquely determine
the eigenfunctions and their analytic properties in k.
By using the Fourier transform method, one can represent the Green’s
functions in the form
G(x; k) =
1
2πi
C
1/p 0
01/
(
p − 2k
)
e
ipx
dp
G(x; k) =
1
2πi
C
1/
(
p + 2k
)
0
01/p
e
ipx
dp
where C and
C will be chosen as appropriate contour deformations of the real
p-axis. It is natural to consider G
±
(x; k) and G
±
(x; k) defined by
G
±
(x; k) =
1
2πi
C
±
1/p 0
01/
(
p − 2k
)
e
ipx
dp
G
±
(x; k) =
1
2πi
C
±
1/
(
p + 2k
)
0
01/p
e
ipx
dp
where C
±
and C
±
are the contours from −∞ to +∞ that pass below (+ func-
tions) and above (− functions) both singularities at p = 0 and p = 2k,or
respectively, p = 0 and p = −2k. Contour integration then gives
G
±
(x; k) = ±θ(±x)
10
0 e
2ikx
,
G
±
(x; k) = ∓θ(∓x)
e
−2ikx
0
01
,
9.10 Outline of the IST for a general evolution system 243
where θ(x) is the Heaviside function (θ(x) = 1ifx > 0 and θ(x) = 0ifx < 0).
The “+” functions are analytic and bounded in the upper half-plane of k and
the “−” functions are analytic in the lower half-plane. By taking into account
the boundary conditions (9.66), we obtain the following integral equations for
the eigenfunctions:
M(x; k) =
1
0
+
+∞
−∞
G
+
(x − x
; k)Q(x
)M(x
; k)dx
(9.67a)
N(x; k) =
0
1
+
+∞
−∞
G
+
(x − x
; k)Q(x
)N(x
; k)dx
(9.67b)
M(x; k) =
0
1
+
+∞
−∞
G
−
(x − x
; k)Q(x
)M(x
; k)dx
(9.67c)
N(x; k) =
1
0
+
+∞
−∞
G
−
(x − x
; k)Q(x
)N(x
; k)dx
. (9.67d)
Equations (9.67) are Volterra integral equations, whose solutions can be sought
in the form of Neumann series iterates. In the following lemma we will show
that if q, r ∈ L
1
(R), the Neumann series associated to the integral equations for
M and N converge absolutely and uniformly (in x and k) in the upper k-plane,
while the Neumann series of the integral equations for
M and N converge
absolutely and uniformly (in x and k)inthelowerk-plane. This implies that
the eigenfunctions M(x; k) and N(x; k) are analytic functions of k for Im k > 0
and continuous for Im k ≥ 0, while
M(x; k) and N(x; k) are analytic functions
of k for Im k < 0 and continuous for Im k ≤ 0.
Lemma 9.2 If q, r ∈ L
1
(R), then M(x; k),N(x; k) defined by (9.67a)
and (9.67b) are analytic functions of k for Im k > 0 and continuous for
Im k ≥ 0, while
M(x; k) and N(x; k) defined by (9.67c) and (9.67d) are ana-
lytic functions of k for Im k < 0 and continuous for Im k ≤ 0. Moreover, the
solutions of the corresponding integral equations are unique in the space of
continuous functions.
The proof can be found in Ablowitz et al. (2004b).
Note that simply requiring q, r ∈ L
1
(R) does not yield analyticity of the
eigenfunctions on the real k-axis, for which more stringent conditions must be
imposed. Having q, r vanishing faster than any exponential as
|
x
|
→∞implies
that all four eigenfunctions are entire functions of k ∈ C.
From the integral equations (9.67) we can compute the asymptotic expan-
sion at large k (in the proper half-plane) of the eigenfunction. Integration by
parts yields
244 Inverse scattering transform for the KdV equation
M(x; k) =
1 −
1
2ik
x
−∞
q(x
)r(x
)dx
−
1
2ik
r(x)
+ O(1/k
2
) (9.68a)
N(x; k) =
⎛
⎜
⎜
⎜
⎜
⎝
1 +
1
2ik
+∞
x
q(x
)r(x
)dx
−
1
2ik
r(x)
⎞
⎟
⎟
⎟
⎟
⎠
+ O(1/k
2
) (9.68b)
N(x; k) =
⎛
⎜
⎜
⎜
⎜
⎝
1
2ik
q(x)
1 −
1
2ik
+∞
x
q(x
)r(x
)dx
⎞
⎟
⎟
⎟
⎟
⎠
+ O(1/k
2
) (9.68c)
M(x; k) =
1
2ik
q(x)
1 +
1
2ik
x
−∞
q(x
)r(x
)dx
+ O(1/k
2
). (9.68d)
9.10.2 Scattering data
The two eigenfunctions with fixed boundary conditions as x →−∞are lin-
early independent, and the same holds for the two eigenfunctions with fixed
boundary conditions as x → +∞. Indeed, let u(x, k) =
u
(1)
(x, k), u
(2)
(x, k)
T
and v(x, k) =
v
(1)
(x, k), v
(2)
(x, k)
T
be any two solutions of (9.59a) and let us
define the Wronskian of u and v, W(u, v), as
W(u, v) = u
(1)
v
(2)
− u
(2)
v
(1)
.
Using (9.59a) it can be verified that
d
dx
W(u, v) = 0
where W(u, v) is a constant. Therefore, from the prescribed asymptotic behav-
ior for the eigenfunctions, it follows that
W
φ,
φ
= lim
x→−∞
W
φ(x; k), φ(x; k)
= 1
W
ψ,
ψ
= lim
x→+∞
W
ψ(x; k), ψ(x; k)
= −1,
which shows that the solutions φ and
φ are linearly independent, as are ψ and
ψ. As a consequence, we can express φ(x; k) and φ(x; k) as linear combina-
tions of ψ(x; k) and
ψ(x; k), or vice versa, with the coefficients of these linear
combinations depending on k only. Hence, the relations
φ(x; k) = b(k)ψ(x; k) + a(k)
ψ(x; k) (9.69a)
φ(x; k) = a(k)ψ(x; k) + b(k)ψ(x; k) (9.69b)
hold for any k ∈ C such that the four eigenfunctions exist. In particular, (9.69)
hold for Im k = 0 and define the scattering coefficients a(k),
a(k), b(k), b(k)for