Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics
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9.1 Benjamin–Ono equation 279
Now we introduce the J ost solutions N
±
(x, k) of (9.4) by means of their
asymptotics:
N
+
(x →−∞,k)=e
ikx
,N
−
(x →∞,k)=e
ikx
,k>0. (9.5)
There exist the other pair M
±
(x, k) of the Jost solutions of the spectral equa-
tion (9.2), which are defined by the asymptotics
M
+
(x →−∞,k)=1,M
−
(x →∞,k) = 1 (9.6)
and are solutions of the inhomogeneous integrodifferential equation
iΨ
+
x
+ kΨ
+
+[uΨ]
+
− k =0. (9.7)
All the Jost functions are defined for positive values of k.Thesevaluesofk
comprise the continuous spectrum of the spectral problem (9.2).
The Jost functions introduced obey the following integral equations:
M
±
(x, k)=1+
∞
−∞
dx
′
G
±
(x − x
′
,k)u(x
′
)M
±
(x
′
,k), (9.8)
N
±
(x, k)=e
ikx
+
∞
−∞
dx
′
G
±
(x − x
′
,k)u(x
′
)N
±
(x
′
,k). (9.9)
Here
G
±
(x, k)=
1
2π
∞
0
dp
e
ipx
p − (k ± i0)
(9.10)
are limits of the Green function
G(x, k)=
1
2π
∞
0
dp
e
ipx
p − k
on the real k-axiswhichareanalyticeverywhereinthecomplexk-plane,
except for the positive part of the real k-axis. Equation (9.10) can be obtained
with account of the formula
1
x
′
− (x +iǫ)
=i
∞
0
dp exp[−ip (x
′
− x − iǫ)].
It should be stressed that the integral equations (9.8) and (9.9) are of
the Fredholm type, as distinct from the Volterra integral equations for the
(1+1)-dimensional equations of the preceding chapter. This fact is of crucial
importance because homogeneous versions of (9.8) and (9.9) can have non-
trivial solutions Φ
j
(x),
Φ
j
(x)=
∞
−∞
dx
′
G(x − x
′
,k
j
)u(x
′
)Φ
j
(x
′
),k
j
< 0, (9.11)
G(x, k
j
)=
1
2π
∞
0
dp
e
ipx
p − k
j
, (9.12)
280 9 Dressing via nonlocal Riemann–Hilbert problem
in some isolated points k
j
, j =1,...,N, lying on the negative part of the
axis Imk = 0. These solutions determine the discrete spectrum of the spectral
problem and are associated with solitons (or lumps) of the BO equation. Each
of these functions is analytic in the upper half z-plane.
Integrating (9.12) by parts, we obtain an asymptotic of G(x, k
j
):
G(x, k
j
)=
1
2πik
j
x
+ O (x
−2
).
Therefore,
Φ
j
(x) →
1
2πik
j
x
∞
−∞
dx
′
u(x
′
)Φ
j
(x
′
)atx →∞.
It is seen that it is natural to take the normalization of Φ
j
(x)as
xΦ
j
(x) → 1atx →∞. (9.13)
In this case, the eigenvalues k
j
are given by the functional
k
j
=
1
2πi
∞
−∞
dxu(x)Φ
j
(x),j=1,...,N. (9.14)
9.1.2 Scattering equation and symmetry relations
In the same way as for other spectral problems, the left and right Jost functions
are interrelated by scattering equations. To obtain the scattering equations,
we find at fir st a difference ∆(x, k)=M
+
(x, k)−M
−
(x, k)fork>0. It follows
from (9.8) and (9.9) that
∆(x, k)=
∞
−∞
dx
′
G
+
(x−x
′
,k)u(x
′
)M
+
(x
′
,k)−
∞
−∞
dx
′
G
−
(x−x
′
,k)u(x
′
)M
−
(x
′
,k).
As stated already, the Green function G(x, k) has a discontinuity across the
positive k-axis. We can calculate this jump:
G
+
(x, k) − G
−
(x, k)
=
1
2π
∞
0
dp e
ipx
1
p − k − i0
−
1
p − k +i0
=iθ(k)e
ikx
,
wherewehaveused
1
k ∓ i0
= ±πiδ(k)+p.v.
1
k
9.1 Benjamin–Ono equation 281
and θ(k) is the Heaviside step function. Hence,
∆(x, k)=
∞
−∞
dx
′
G
−
(x − x
′
,k)u(x
′
)∆(x
′
,k)+
∞
−∞
dx
′
(G
+
− G
−
)(x − x
′
,k)
= β(k)e
ikx
+
∞
−∞
dx
′
G
−
(x − x
′
,k)u(x
′
)∆(x
′
,k), (9.15)
where the reflection coefficient β(k) is defined as
β(k)=iθ(k)
∞
−∞
dxu(x)M
+
(x, k)e
−ikx
. (9.16)
On the other hand, it follows from (9.9) that
β(k)N
−
(x, k)=β(k)e
ikx
+
∞
−∞
dx
′
G
−
(x − x
′
,k)u(x
′
)β(k)N
−
(x
′
,k). (9.17)
Comparing (9.15) and (9.17), we find ∆(x, k)=β(k)N
−
(x, k). Thereby, we
obtain the important relation between the Jost functions:
M
+
(x, k)=M
−
(x, k)+β(k)N
−
(x, k). (9.18)
There also exists a connection between the functions N
+
and N
−
written
as [383]
N
+
(x, k)=Γ (k)N
−
(x, k) (9.19)
with some function Γ (k). Let us differentiate G
±
in k and i ntegrate the result
by parts. This gives
∂G
±
∂k
= −
1
2πk
± ixG
±
(x, k). (9.20)
On the other hand, writing N
−
(x, k) from (9.9) as
N
−
(x, k)e
−ikx
=1+
∞
−∞
dx
′
G
−
(x − x
′
,k)e
−ik(x−x
′
)
u(x
′
)N
−
(x
′
,k)e
−ikx
′
and taking the k-derivative, we find with account of (9.20)
∂
∂k
N
−
(x, k)e
−ikx
=
∞
−∞
dx
′
∂G
−
∂k
(x − x
′
,k)e
−ik(x−x
′
)
u(x
′
)N
−
(x
′
,k)e
−ikx
′
−i
∞
−∞
dx
′
G
−
(x − x
′
,k)e
−ik(x−x
′
)
(x − x
′
)u(x
′
)N
−
e
−ikx
′
+
∞
−∞
dx
′
G
−
(x − x
′
,k)e
−ik(x−x
′
)
u(x
′
)
∂
∂k
(N
−
e
−ikx
′
)
=e
−ikx
f
−
(k)+
∞
−∞
dx
′
G
−
(x − x
′
,k)e
−ik(x−x
′
)
u(x
′
)
∂
∂k
(N
−
e
−ikx
′
), (9.21)
282 9 Dressing via nonlocal Riemann–Hilbert problem
where
f
−
(k)=−
1
2πk
∞
−∞
dxu(x)N
−
(x, k),k>0. (9.22)
Multiplying M
−
(x, k) (9.8) by f
−
(k) and comparing the result with the right-
hand side of (9.21), we arrive at the differential connection between N
−
and
M
−
[158]:
∂
∂k
N
−
(x, k)e
−ikx
= f
−
(k)M
−
(x, k)e
−ikx
. (9.23)
Similarly,
∂
∂k
N
+
(x, k)e
−ikx
= f
+
(k)M
+
(x, k)e
−ikx
,
where
f
+
(k)=−
1
2πk
∞
−∞
dxu(x)N
+
(x, k),k>0. (9.24)
Equations (9.23) and (9.24) give the symmetry (or closure) relations for the
Jost solutions. Multiplying both sides of (9.19) by u(x) and integrating in x,
we fi nd a simple connection between f
+
(k)andf
−
(k):
f
+
(k)=Γ (k)f
−
(k). (9.25)
Eventually, (9.23), (9.24), (9.19), and (9.25) give
β(k)f
+
(k)=
∂
∂k
Γ (k). (9.26)
Now we should find out how the function M
−
behaves as k → k
j
. Define a
function M
(j)
−
(x, k) a s in [158] (we suppose here that M
−
allows simple poles
only; see Sect. 9.1 . 4)
M
(j)
−
(x, k)=M
−
(x, k) −
c
j
k − k
j
Φ
j
(x),
where c
j
is a normalization constant. Then we can write, taking into account
(9.8) and (9.11),
M
(j)
−
(x, k) −
∞
−∞
dx
′
G
−
(x − x
′
,k)u(x
′
)M
(j)
−
(x
′
,k) (9.27)
=1−
c
j
k − k
j
Φ
j
(x) −
∞
−∞
dx
′
G
−
(x − x
′
,k)u(x
′
)Φ
j
(x
′
)
.
When k → k
j
, the last term in (9.27) develops an uncertainty 0/0. The k-
derivative of the numerator in this term gives with regard to (9.20)
∞
−∞
dx
′
∂
∂k
G
−
(x − x
′
,k)
u(x
′
)Φ
j
(x
′
)
= −
1
2πk
∞
−∞
dxu(x)Φ
j
(x)+i
∞
−∞
dx
′
(x − x
′
)G(x −x
′
,k
j
)u(x
′
)Φ
j
(x
′
).
9.1 Benjamin–Ono equation 283
Hence,
M
(j)
−
(x, k) −
∞
−∞
dx
′
G(x − x
′
,k
j
)u(x
′
)M
−
(x
′
,k
j
) (9.28)
= a
j
+ic
j
∞
−∞
dx
′
(x − x
′
)G(x − x
′
,k
j
)u(x
′
)Φ
j
(x
′
),
where
a
j
=1−
c
j
2πk
j
∞
−∞
dxu(x)Φ
j
(x).
Putting c
j
= −igivesa
j
= 0. It is easy to see that a particular solution
to (9.28) has the form M
(j)
−
(x, k)=xΦ
j
(x). Then, applying the Fredholm
alternative [425], we obtain a connection b etween Φ
j
and M
−
:
lim
k→k
j
M
−
(x, k)+
i
k − k
j
Φ
j
(x)
=(x + γ
j
)Φ
j
(x). (9.29)
The normalization constant γ
j
will be time-dependent if we account for the
evolution of the potential u(x, t).
9.1.3 Adjoint spectral problem and asymptotics
Following the paper by Kaup and Matsuno [232], we define now the adjoint
spectral problem
iΦ
A
x
− kΦ
A
+ u
Φ
A
+
=0. (9.30)
Applying the projector P
−
to (9.30), we obtain the equation complex conju-
gate to (9.4) whose solution is N
∗
+
(x, k). The (+) projection of (9.30) gives
a linear inhomogeneous equation. As a result, a solution N
A
of the adjoint
problem is given by
N
A
(x, k)=N
∗
+
(x, k) − i
x
−∞
dx
′
uN
∗
+
+
(x
′
,k)e
−ik(x−x
′
)
. (9.31)
Similarly we can write a bound-state eigenfunction N
A
j
(x):
N
A
j
(x)=Φ
∗
j
(x) − i
x
−∞
dx
′
uΦ
∗
j
+
(x
′
)e
−ik
j
(x−x
′
)
.
In the following we will need asymptotic expressions for the eigenfunctions
at |x|→∞. Taking into accountthe definitions (9.5) and (9.6) and equations
284 9 Dressing via nonlocal Riemann–Hilbert problem
(9.18), (9.19), and (9.31), we have for x →−∞and k>0, in addition to (9.5)
and (9.6)
N
−
(x, k) → Γ (k)e
ikx
,M
−
(x, k) → 1+β(k)e
ikx
,N
A
(x, k) → e
ikx
(9.32)
and for x →∞and k>0
N
+
(x, k) → Γ (k)e
ikx
,M
+
(x, k) → 1+β(k)e
ikx
,N
A
(x, k) → Γ
∗
(k)e
−ikx
,
(9.33)
while for the bound states we have for x →−∞and k
j
< 0
Φ
j
(x) →
1
x
,N
A
j
(x) → 0,
and for x →∞and k
j
< 0
Φ
j
(x) →
1
x
,N
A
j
(x) →−ie
−ik
j
x
∞
−∞
dx
′
u(x
′
)Φ
∗
j
(x
′
)e
ik
j
x
′
.
Now we can concretize the form of the function Γ (k). Excluding the poten-
tial u(x) from the direct (9.4) and adjoint (9.30) spectral problems , we obtain
a relation (the Wronskian relation) between the left and right asymptotics:
Φ
A
(x, k
′
)Φ(x, k)|
+∞
x=−∞
=i(k − k
′
)
∞
−∞
dx
′
Φ
A
(x
′
,k
′
)Φ(x
′
,k). (9.34)
Taking k
′
= k, Φ
A
= N
A
,andΦ = N
−
in (9.34) and using the above asymp-
totics, we easily get
Γ
∗
(k)Γ (k)=1,
which allows us to introduce a real phase θ(k):
Γ (k)=e
−iθ(k)
.
To find the phase explicitly, we construct the Wronskian relation from the
spectral problems (9.7) and (9.30):
N
A
(x, k
′
)[M
−
(x, k) − 1]
+∞
x=−∞
(9.35)
=i(k − k
′
)
∞
−∞
dx
′
N
A
(x
′
,k
′
)[M
−
(x
′
,k) − 1] −i
∞
−∞
dx
′
u(x
′
)
N
A
(x
′
,k
′
)
−
.
Taking k
′
= k once again, we find
β(k)=−iΓ (k)
∞
−∞
dxu(x)
N
A
(x, k)
−
. (9.36)
Let us remember [see (9.31)] that
N
A
−
= N
∗
+
. Then, with account for the
connection (9.19) and the definition (9.22) of f
−
(k), we obtain from (9.36)
β(k)=−2πikf
∗
−
(k), i.e.,
f
−
(k)=
β
∗
(k)
2πik
. (9.37)
9.1 Benjamin–Ono equation 285
Because (∂/∂k)Γ = −iΓ (∂/∂k)θ, we find from (9.25), (9.26), and (9.37):
∂θ
∂k
=
|β(k)|
2
2πk
.
Hence, we represent explicitly the function Γ (k) in terms of the reflection
coefficient β(k):
Γ (k)=exp
1
2πi
k
0
dk
′
k
′
|β(k
′
)|
2
. (9.38)
It should be stressed that (9.37) and (9.38) hold for real potentials only. In
the case of complex potentials, the spectral functions β(k)andf
−
(k)are
independent [158].
We can make use of the Wronskian relation to derive integral expressions
for “squared” eigenfunctions. Taking Φ
A
= N
A
(x, k
′
)andΦ = N (x, k)in
(9.34) and using the asymptotics (9.32) and (9.33), we obtain
∞
−∞
dxN
A
(x, k
′
)N(x, k)=−iΓ
∗
(k
′
) lim
x→+∞
e
i(k−k
′
)x
k − k
′
+
i
Γ (k)
lim
x→−∞
e
i(k−k
′
)x
k − k
′
.
Invoking the formula
lim
x→+∞
p.v.
e
±ikx
k
= ±iπδ(k),
we find the following orthogonality relation for the continuous eigenfunctions:
∞
−∞
dxN
A
(x, k
′
)N
−
(x, k)=
2π
Γ (k)
δ(k − k
′
). (9.39)
It is evident (because k and k
j
belong to different parts of the real k-axis)
that the continuous and bounded eigenfunctions are mutually orthogonal:
∞
−∞
dxN
A
j
(x)N
−
(x, k)=
∞
−∞
dxN
A
(x, k)Φ
j
(x)=0, (9.40)
as well as
∞
−∞
dxN
A
j
(x)Φ
n
(x)=0, if j = n. (9.41)
Furthermore, it follows from (9.35) that
∞
−∞
dxN
A
j
(x)Φ
j
(x)=i
∞
−∞
dxu(x)
N
A
j
(x)
−
.
With account for N
A
j
= Φ
∗
j
and (9.14) we obtain
∞
−∞
dxN
A
j
(x)Φ
j
(x)=−2πk
j
. (9.42)
286 9 Dressing via nonlocal Riemann–Hilbert problem
Finally, the boundary condition (9.13) gives
∞
−∞
dxΦ
j
(x)=−πi. (9.43)
9.1.4 RH problem
Now we have all ingredients to formulate the RH problem for the BO equation.
Indeed, because the Green function G
+
is analytic in the upper half k-plane, it
follows from the theory of Fredholm integral equations [425] that the solution
M
+
(x, k) is analytic in the same region as well, except for possible poles
at isolated p oints k
j
, where nontrivial solutions of the homogeneous integral
equation exist. The same analytic properties but in the lower half plane are
inherent to the eigenfunction M
−
(x, k). As regards N
±
, they cannot in general
be continued off the real k-a xi s because of the exponent e
ikx
. On the other
hand, there is a differential connection (9.23) between the functions N
−
and
M
−
. This fact enables us, by means of (9.18), (9.23), and (9.37), to p ose the
RH problem of the form
M
+
(x, k)=M
−
(x, k)+
β(k)
2πi
k
0
dk
′
k
′
β
∗
(k
′
)e
−ik
′
x
M
−
(x, k
′
). (9.44)
The normalization of the RH problem (9.44) is given by
M
+
(x, k) → 1atk →∞. (9.45)
It should be noted that the RH problem (9.44) is essentially distinct from
the RH problems we dealt with in the preceding chapter. Namely, t he RH
problem (9.44) is nonlocal in that stems from the nonlocality of the BO equa-
tion. The spectral data of the RH problem (9.44) are determined by the set
{β(k),k > 0; k
j
, Reγ
j
,j =1,...,N}.
2
As usual, we can regularize the RH
problem and represent its solutions M
+
and M
−
as
M
+
(x, k)=1−i
N
j=1
(k −k
j
)
−1
Φ
j
(x)+ m
+
(x, k),m
+
(x, k →∞)=0, (9.46)
M
−
(x, k)=1−i
N
j=1
(k−k
j
)
−1
Φ
j
(x)+m
−
(x, k),m
−
(x, k →∞)=0, (9.47)
where the holomorphic functions m
±
(x, k)are(+)and(–)functionswith
respect to x, respectively. The fact that the spectral problem (9.2) allows
only simple poles was proved in [362].
2
It will be shown later that Imγ
j
canbeexpressedintermsofk
j
.
9.1 Benjamin–Ono equation 287
The solution M
−
,Imk ≤ 0 of the RH problem can be expressed in terms
of N
−
(x, k)forpositivek and of Φ
j
(x)atthepolesk
j
. Let us write (9.18) as
[see (9.46)]
1 − i
N
j=1
(k − k
j
)
−1
Φ
j
(x)+m
+
(x, k)=M
−
(x, k)+β(k)N
−
(x, k)
and apply the projector P
−
to it. This yields
M
−
(x, k)=1−i
N
j=1
(k −k
j
)
−1
Φ
j
(x)+
1
2πi
∞
0
dl
β(ℓ)
ℓ − (k − i0)
N
−
(x, ℓ). (9.48)
We see that the solution (9.48) clearly demonstrates the separation of con-
tributions of the discrete and continuous spectra. It follows from (9.29) and
(9.48) that in the limit k → k
j
(x + γ
j
)Φ
j
(x)+i
n=j
(k − k
j
)
−1
Φ
j
(x) −
1
2πi
∞
0
dℓ
β(ℓ)
ℓ − (k − i0)
N
−
(x, ℓ)=1.
(9.49)
Now we multiply (9.49) by Φ
∗
j
(x) and integrate in x with account for the
orthogonality conditions (9.39)–(9.43). This yields
∞
−∞
dx(x + γ
j
)|Φ
j
(x)|
2
= πi.
In accordance with (9.42) the last formula gives
γ
j
=
1
2πk
j
∞
−∞
dxx|Φ
j
(x)|
2
−
i
2k
j
; (9.50)
therefore,
Imγ
j
= −
1
2k
j
. (9.51)
Hence, only the Reγ
j
should be included in the set of spectral data of the RH
problem.
The reconstruction of the potential u(x) from the solution of the RH prob-
lem can be performed as follows. Integrating by parts the integral in (9.9), we
find
M
−
→ 1 −
[u]
+
k
,k→∞.
As a result, we obtain from (9.48) in the same limit k →∞
[u]
+
=
1
2πi
∞
0
dkβ(k)N
−
(x, k)+i
N
j=1
Φ
j
(x). (9.52)
Therefore, the real potential u(x)isgivenby
u(x)=[u(x)]
+
+[u(x)]
+∗
. (9.53)
288 9 Dressing via nonlocal Riemann–Hilbert problem
9.1.5 Evolution of spectral data
To determine the time evolution of the spectral data, we should consider the
Lax equation (9.3):
iM
+t
− 2ikM
+x
+ M
+xx
− 2i[u
x
]
+
M
+
+ νM
+
=0.
Since M
+
→ 1forx →−∞,wehaveν =0.Atx →∞this equation is
reduced to
iM
+t
− 2ikM
+x
+ M
+xx
=0. (9.54)
Let us substitute (9.18) into (9.54):
i(M
−t
+ β
t
N
−
+ βN
−t
) −2ik(M
−x
+ βN
−x
)+M
−xx
+ βN
−xx
=0. (9.55)
At x → +∞, M
−
→ 1andN
−
→ e
ikx
. As a result, (9.55) gives β
t
=ik
2
β,
i.e.,
β(k, t)=β(k, 0)e
ik
2
t
. (9.56)
Taking the time derivative of (9.29) yields
k
j
=const,γ
j
(t)=2k
j
t + γ
j,0
,γ
j,0
=const. (9.57)
Therefore, (9.56) and (9.57) give the time dependence of the spectral data.
9.1.6 Solitons of BO equation
In the case of the pure soliton potential u
s
(x, t) the reflection coefficient β(k)
vanishes; hence, the reconstruction of the potential is given by [see (9.52)]
[u
s
(x, t)]
+
=i
N
j=1
Φ
j
(x, t).
Bound states Φ
j
can be found from (9.49) as a solution of the algebraic system
(x + γ
j
)Φ
j
+i
n=j
(k
j
− k
n
)
−1
Φ
n
=1. (9.58)
For N = 1 we obtain from (9.58)
Φ
1
(x, t)=[x + γ
1
(t)]
−1
.
Putting k
1
= −(1/2)v, v>0 and accounting for the relation (9.51), we can
write γ
1,0
= −x
0
+i/v.Thenγ
1
(t)=−x
0
− vt +i/v and (9.52) and (9.53)
eventually give
u
s
(x, t)=
2v
1+v
2
(x − vt − x
0
)
2
. (9.59)