Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 9 (2013), 006, 13 pages
On the N -Solitons Solutions
in the Novikov–Veselov Equation
Jen-Hsu CHANG
Department of Computer Science and Information Engineering,
National Defense University, Tauyuan, Taiwan
E-mail: jhchang@ndu.edu.tw
Received October 01, 2012, in final form January 12, 2013; Published online January 20, 2013
http://dx.doi.org/10.3842/SIGMA.2013.006
Abstract. We construct the N -solitons solution in the Novikov–Veselov equation from the
extended Moutard transformation and the Pfaffian structure. Also, the corresponding wave
functions are obtained explicitly. As a result, the property characterizing the N -solitons
wave function is proved using the Pfaffian expansion. This property corresponding to the
discrete scattering data for N -solitons solution is obtained in [arXiv:0912.2155] from the
∂-dressing method.
Key words: Novikov–Veselov equation; N -solitons solutions; Pfaffian expansion; wave functions
2010 Mathematics Subject Classification: 35C08; 35A22
1
Introduction
The Novikov–Veselov equation [3, 9, 34, 41] is defined by
Ut = ∂z3 U + ∂¯z U + 3∂z (V U ) + 3∂¯z (V ∗ U ),
∂¯z V = ∂z U,
∂z V ∗ = ∂¯z U.
(1)
When z = z̄ = x, we get the famous KdV equation (U = Ū = V = V̄ )
Ut = 2Uxxx + 12U Ux .
The equation (1) can be represented as the form of Manakov’s triad [24]
Ht = [A, H] + BH,
where H is the two-dimension Schrödinger operator
H = ∂z ∂¯z + U
and
A = ∂z3 + V ∂z + ∂¯z3 + V̄ ∂¯z ,
B = Vz + V̄z̄ .
It is equivalent to the linear representation
Hψ = 0,
∂t ψ = Aψ.
(2)
We see that the Novikov–Veselov equation (1) preserves a class of the purely potential selfadjoint operators H. Here the pure potential means H has no external electric and magnetic
fields. The periodic inverse spectral problem for the two-dimensional Schrödinger operator H
2
J.H. Chang
was investigated in terms of the Riemann surfaces with some group of involutions and the
corresponding Prym Θ-functions [5, 10, 22, 27, 28, 33, 37]. On the other hand, it is known
that the Novikov–Veselov hierarchy is a special reduction of the two-component BKP hierarchy
[23, 36, 40] (and references therein). In [23], the authors showed that the Drinfeld–Sokolov
hierarchy of D-type is a reduction of the two-component BKP hierarchy using two different
types of pseudo-differential operators, which is different from Shiota’s point of view [37]. Also,
in [26], it is shown that the Tzitzeica equation is a stationary symmetry of the Novikov–Veselov
equation. Finally, it is worthwhile to notice that the Novikov–Veselov equation (1) is a special
reduction of the Davey–Stewartson equation [20, 21].
Let Hψ = Hω = 0. Then via the Moutard transformation [1, 29, 30, 31]
U (z, z̄) −→ Û (z, z̄) = U (z, z̄) + 2∂ ∂¯ ln ω,
Z
1
¯ − ω ∂ψ)dz̄,
¯
ψ −→ θ =
(ψ∂ω − ω∂ψ)dz − (ψ ∂ω
ω
(3)
one can construct a new Schrödinger operator Ĥ = ∂z ∂¯z + Û and Ĥθ = 0. We remark that
the Moutard transformation (3) is utilized to construct the N -solitons solutions of the Tzitzeica
equation [15].
The extended Moutard transformation was established such that Û (t, z, z̄) and V̂ (t, z, z̄)
defined by [13, 25]
Û (t, z, z̄) = U (t, z, z̄) + 2∂ ∂¯ ln W (ψ, ω),
V̂ (t, z, z̄) = V (t, z, z̄) + 2∂∂ ln W (ψ, ω),
where the skew product (alternating bilinear form) W is defined by
Z
¯ − ω ∂ψ)dz̄
¯
W (ψ, ω) = (ψ∂ω − ω∂ψ)dz − (ψ ∂ω
+ ψ∂ 3 ω − ω∂ 3 ψ + ω ∂¯3 − ψ ∂¯3 ω
¯ − ∂ψ
¯ ∂¯2 ω + 3V (ψ∂ω − ω∂ψ)
+ 2 ∂ 2 ψ∂ω − ∂ψ∂ 2 ω − 2 ∂¯2 ψ ∂ω
¯ − ω ∂ψ)
¯ dt,
− 3V̄ (ψ ∂ω
(4)
will also satisfy the Novikov–Veselov equation.
In [2, 6, 7, 8], the rational solutions and line solitons of the Novikov–Veselov equation (1) are
constructed by the ∂-dressing method. To get these kinds of solutions, the scattering datum
have to be delta-type and the reality of U also puts some extra constraints on them. In [39],
the singular rational solutions are obtained using the extended Moutard transformation (4);
however, the non-singular rational solutions are constructed in [4].
Next, we construct Pfaffian-type solutions. Given any N wave functions ψ1 , ψ2 , ψ3 , . . . , ψN
(or their linear combinations) of (2) for fixed potential U (z, z̄, t), the N -step extended Moutard
transformation can be obtained in the Pfaffian [1, 31] (also see [12, 35])
(
Pf(ψ1 , ψ2 , ψ3 , . . . , ψN ) if N even,
P (ψ1 , ψ2 , ψ3 , . . . , ψN ) =
f 1 , ψ2 , ψ3 , . . . , ψN ) if N odd,
Pf(ψ
X
Pf(ψ1 , ψ2 , ψ3 , . . . , ψN ) =
(σ)Wσ1 σ2 Wσ3 σ4 · · · WσN −1 σN ,
(5)
σ
f 1 , ψ2 , ψ3 , . . . , ψN ) =
Pf(ψ
X
(σ)Wσ1 σ2 Wσ3 σ4 · · · WσN −2 σN −1 ψσN ,
(6)
σ
where Wσi σj = W (ψσ(i) , ψσ(j) ) is defined by the skew product (4). The summations σ in (5)
and (6) run from over the permutations of {1, 2, 3, . . . , N } such that σ1 < σ2 , σ3 < σ4 , σ5 < σ6 ,
. . . and σ1 < σ3 < σ5 < σ7 < · · · , with (σ) = 1 for the even permutations and (σ) = −1 for
the odd permutations. Then the solution U and V can be expressed as [1]
¯ P (ψ1 , ψ2 , ψ3 , . . . , ψN )],
U = U0 + 2∂ ∂[ln
V = V0 + 2∂∂[ln P (ψ1 , ψ2 , ψ3 , . . . , ψN )],
On the N -Solitons Solutions in the Novikov–Veselov Equation
3
and the corresponding wave function is
ϕ=
P (ψ1 , ψ2 , ψ3 , . . . , ψN , ϑ)
,
P (ψ1 , ψ2 , ψ3 , . . . , ψN )
(7)
where ϑ is an arbitrary wave function different from ψ1 , ψ2 , ψ3 , . . . , ψN .
The paper is organized as follows. In Section 2, we obtain the N -solitons solutions using
the extended Moutard transformation and the Pfaffian expansion. Several examples are given.
In Section 3, the N -solitonic wave function is derived using (7) and the Pfaffian expansion.
Section 4 is used to prove a special property to characterize the N -solitons wave function.
Section 5 is devoted to the concluding remarks.
2
N -solitons solutions
In this section, one uses successive iterations of the extended Moutard transformation (4) to
construct N -solitons solutions.
To obtain the N -solitons solutions, we assume that V = 0 in (1) and recall that ∂ ∂¯ = 14 4.
One considers U = − 6= 0, i.e.,
¯ = ϕ,
∂ ∂ϕ
ϕt = ϕzzz + ϕz̄ z̄ z̄ ,
(8)
where is non-zero real constant. The general solution of (8) can be expressed as
Z
ϕ(z, z̄, t) =
e
(iλ)z+(iλ)3 t+ iλ
z̄+
3
t
(iλ)3
ν(λ)dλ,
(9)
Γ
where ν(λ) is an arbitrary distribution and Γ is an arbitrary path of integration such that the
r.h.s. of (9) is well defined.
Next, using (5) and (9), one can construct the N -solitons solutions. Let’s take νm (λ) =
δ(λ − pm ) and νn (λ) = an δ(λ − qn ), where pm , an , qn are complex numbers. Then one defines
φm =
ϕ(pm )
1
√
= √ eF (pm ) ,
3
3
ϕ(qn )
an
ψn = an √ = √ eF (qn ) ,
3
3
where
F (λ) = (iλ)z + (iλ)3 t +
3
z̄ +
t.
iλ
(iλ)3
Then a direct calculation of the extended Moutard transformation (4) can yield
qn − pm F (pm )+F (qn )
pn − pm F (pm )+F (pn )
e
,
W (φm , φn ) = i
e
,
qn + p m
pn + pm
qn − qm F (qm )+F (qn )
W (ψm , ψn ) = iam an
e
.
qn + qm
W (φm , ψn ) = ian
The N -solitons solutions are defined by
U (z, z̄, t) = − + 2∂ ∂¯ ln τN (z, z̄, t),
and then
U (z, z̄, t) → −
as
z z̄ → ∞,
V (z, z̄, t) = 2∂∂ ln τN (z, z̄, t),
(10)
4
J.H. Chang
where t is fixed. The τ -functions are defined as follows. For simplicity, let’s denote
W (pm , qn ) = W (φm , ψn ),
W (pm , pn ) = W (φm , φn ),
W (qm , qn ) = W (ψm , ψn ), (11)
and notice that F (−λ) = −F (λ). The τN is defined as
τN (z, z̄, t) = Pf(−p1 , q1 , −p2 , q2 , −p3 , q3 , . . . , −pN , qN ),
(12)
where
(−pm , −pn ) = W (−pm , −pn ),
(−pm , qn ) = W (−pm , qn ) + δmn ,
(qm , qn ) = W (qm , qn ).
(13)
To get the expansion of (12), we use the following useful formula [14, 38]
Pf(A + B) =
s X
X
(−1)|α|−r Pf(Aα ) Pf(Bαc ),
(14)
m
r=0 α∈I2r
where A and B are m × m matrices and s = [m/2] is the integer part of m/2; moreover, we
denote by αc the complementary set of α in the subset {1, 2, 3, . . . , m} which is arranged in
increasing order, and |α| = α1 + α2 + · · · + α2r for α = (α1 , α2 , . . . , α2r ). For the case (12), one
has


0
(−p1 , q1 ) (−p1 , −p2 ) (−p1 , q2 ) · · · (−p1 , qN )
 (q1 , −p1 )
0
(q1 , −p2 )
(q1 , q2 ) · · · (q1 , qN ) 


(−p2 , −p1 ) (−p2 , q1 )
0
(−p2 , q2 ) · · · (−p2 , qN )
AN (z, z̄, t) = 



..
..
..
..


.
.
.
···
.
(qN , −p1 )
(qN , q1 )
(qN , −p2 )
(qN , q2 )
···
0
and

BN
0
−1

0


= 0
 ..
 .

0
0
1 0
0
0 0
0
0 0
1
0 −1 0
..
..
.
. ···
0 0
0
0 0 ···

0
0

0

0
,



· · · 1
−1 0
···
···
···
···
..
.
where AN and BN are 2N × 2N matrices. Hence by (14) one can have the expansion of (12),
i.e.,


N
X
X
X
Y

τN = 1 +
f` +
f`1 f`2 · · · f`m
P`j `k  ,
(15)
`=1
m=2
1≤`1 <`2 <···<`m ≤N
1≤j<k≤m
where
f` = ia`
p` + q` F (q` )−F (p` )
e
,
q` − p `
P`j `k =
(p`j − p`k )(q`j − q`k )(p`j + q`k )(q`j + p`k )
.
(p`j + p`k )(q`j + q`k )(p`j − q`k )(q`j − p`k )
Here we have utilized the formula that if C is a 2N × 2N matrix with (i, j)-th entry
one has the Schur identity [32, 36]
Y αi − αj Pf(C) =
.
αi + αj
1≤i<j≤2N
αi −αj
αi +αj ,
then
(16)
On the N -Solitons Solutions in the Novikov–Veselov Equation
5
Next, we illustrate the formula (15) (or 12) with several examples.
(1) One-soliton solution:
0
(−p1 , q1 )
0 1
A1 (z, z̄, t) =
,
B1 =
.
(q1 , −p1 )
0
−1 0
Then
τ1 = Pf(A1 + B1 ) = 1 + ia1
q1 + p1 F (q1 )−F (p1 )
e
.
q1 − p1
(2) Two-solitons solution:


0
(−p1 , q1 ) (−p1 , −p2 ) (−p1 , q2 )
 (q1 , −p1 )
0
(q1 , −p2 )
(q1 , q2 ) 
,
A2 (z, z̄, t) = 
(−p2 , −p1 ) (−p2 , q1 )
0
(−p2 , q2 )
(q2 , −p1 )
(q2 , q1 )
(q2 , −p2 )
0


0 1 0 0
−1 0 0 0

B2 = 
 0 0 0 1 .
0 0 −1 0
Then
p1 + q1 F (q1 )−F (p1 )
p2 + q2 F (q2 )−F (p2 )
e
+ ia2
e
q1 − p1
q2 − p 2
p1 + q1 p2 + q2 p2 − p1 q2 − q1 p1 + q2 p2 + q1 F (q1 )−F (p1 )+F (q2 )−F (p2 )
+ ia1 ia2
e
q1 − p1 q2 − p2 p2 + p1 q2 + q1 q2 − p1 p2 − q1
τ2 = Pf(A2 + B2 ) = 1 + ia1
or
τ2 = 1 + f1 + f2 + P12 f1 f2 ,
(17)
where
p2 + q2 F (q2 )−F (p2 )
p1 + q1 F (q1 )−F (p1 )
e
,
f2 = ia2
e
,
q1 − p1
q2 − p 2
p 1 − p 2 q1 − q2 p 1 + q2 q1 + p 2
=
.
p 1 + p 2 q1 + q2 p 1 − q2 q1 − p 2
f1 = ia1
P12
The τ2 soliton (17) is also found in [2] using the ∂-dressing method.
(3) Three-solitons solutin:


0
(−p1 , q1 ) (−p1 , −p2 ) (−p1 , q2 ) (−p1 , −p3 ) (−p1 , q3 )
 (q1 , −p1 )
0
(q1 , −p2 )
(q1 , q2 )
(q1 , −p3 )
(q1 , q3 ) 


(−p2 , −p1 ) (−p2 , q1 )
0
(−p2 , q2 ) (−p2 , −p3 ) (−p2 , q3 )

,
A3 (z, z̄, t) = 
(q2 , q1 )
(q2 , −p2 )
0
(q2 , −p3 )
(q2 , q3 ) 
 (q2 , −p1 )

(−p3 , −p1 ) (−p3 , q1 ) (−p3 , −p2 ) (−p3 , q2 )
0
(−p3 , q3 )
(q3 , −p1 )
(q3 , q1 )
(q3 , −p2 )
(q3 , q2 )
(q3 , −p3 )
0


0 1 0 0 0 0
−1 0 0 0 0 0


 0 0 0 1 0 0

.
B3 = 

0
0
−1
0
0
0


 0 0 0 0 0 1
0 0 0 0 −1 0
6
J.H. Chang
Then
τ3 = Pf(A3 + B3 )
= 1 + f1 + f2 + f3 + P12 f1 f2 + P13 f1 f3 + P23 f2 f3 + P12 P13 P23 f1 f2 f3 ,
(18)
where
p1 + q1 F (q1 )−F (p1 )
p2 + q2 F (q2 )−F (p2 )
e
,
f2 = ia2
e
,
q1 − p1
q2 − p 2
p3 + q3 F (q3 )−F (p3 )
p1 − p2 q1 − q2 p1 + q2 q1 + p2
f3 = ia3
e
,
P12 =
,
q3 − p3
p1 + p2 q1 + q2 p1 − q2 q1 − p2
p 1 + q3 q1 + p 3 p 1 − p 3 q1 − q3
p2 + q3 q2 + p3 p2 − p3 q2 − q3
P13 =
,
P23 =
.
p 1 − q3 q1 − p 3 p 1 + p 3 q1 + q3
p2 − q3 q2 − p3 p2 + p3 q2 + q3
f1 = ia1
3
The wave functions
In this section, one uses (7) to construct the corresponding wave function of the τ function (15).
From (7), one knows that the corresponding wave function of the N -solitons (15) can be
written as
) ϕ(q ) ϕ(−p ) ϕ(q )
ϕ(−pN ) ϕ(qN ) ϕ(λ) √ 1 , √1 , √ 2 , √2 , . . . ,
√
P ϕ(−p
, √3 , √3
3
3
3
3
3
.
ϕN =
τN
Using the notations in (11), (12) and (13), we can express ϕN as
ϕN =
P (−p1 , q1 , −p2 , q2 , . . . , −pN , qN , λ)
.
τN
(19)
But we notice that
(−pm , λ)] = W (−pm , λ),
(qm , λ)] = W (qm , λ),
(20)
where (·)] means there is no δmn here when compared with (13). Now, let’s compute P (−p1 , q1 ,
−p2 , q2 , . . . , −pN , qN , λ) using (14). In this case,
P (−p1 , q1 , −p2 , q2 , . . . , −pN , qN , λ) = Pf(MN + QN ),
where

MN
(−p1 , λ)




AN (z, z̄, t)


=



(λ, −p1 ) (λ, q1 ) · · ·

)
)
√ 1
√1
− ϕ(−p
− ϕ(q
···
3
3
and

QN


BN


=


0 0 · · ·
0 0 ···

0 0
0 0

.. .. 
. .
.
0 0

0 0 0
0 0 0
(q1 , λ)
..
.
(qN , λ)
(λ, qN )
)
√N
− ϕ(q
3
0
√
− ϕ(λ)
3
ϕ(−p1 )
√
3 
ϕ(q1 ) 
√
3 


..
.
ϕ(qN )
√
3
ϕ(λ)
√
3
0








On the N -Solitons Solutions in the Novikov–Veselov Equation
7
Using (16), a simple calculation yields
Pf(MN + QN )

= φ 1 +
N
X
h` (λ) +
N
X

X
h`1 h`2 · · · h`m

m=2
`=1

15`1 <`2 <`3 <···<`m 5N
Y
P`j `k 
15j<k5m
= φχ̂N (λ),
(21)
where
p` + q` p` + λ q` − λ F (q` )−F (p` )
p` + λ q` − λ
e
= f`
,
q` − p ` p ` − λ q` + λ
p` − λ q` + λ


N
N
X
X
X
Y

χ̂N (λ) = 1 +
h` (λ) +
h`1 h`2 · · · h`m
P`j `k  ,
ϕ(λ)
φ= √ ,
3
h` (λ) = ia`
`=1
m=2
15`1 <`2 <`3 <···<`m 5N
and P`j `k is defined in (15).
We give several examples here.
(1) The one-soliton wave function:

(−p1 , λ)

A1 (z, z̄, t)

(q1 , λ)

M1 = 
(λ, −p ) (λ, q )
0
1
1

ϕ(−p1 )
ϕ(q1 )
ϕ(λ)
√
√
− 3
− 3
− √3
ϕ(−p1 ) 
√
3

ϕ(q1 ) 
√
3 ,

ϕ(λ) 
√

3
0
15j<k5m

0
 B1 0
Q1 = 
0 0 0
0 0 0

0
0
.
0
0
Then
Pf(M1 + Q1 )
φ
P (−p1 , q1 , λ)
=
=
ϕ1 =
τ1
τ1
τ1
p1 + q1 p1 + λ q1 − λ F (q1 )−F (p1 )
e
.
1 + ia1
q1 − p 1 p 1 − λ q1 + λ
We remark that
p 1 + λ q1 − λ
φ
2p1
2q1
φ
1 + ia1
f1 =
1 + ia1
−1
− 1 f1
ϕ1 =
1 + f1
p 1 − λ q1 + λ
1 + f1
p1 − λ
q1 + λ
"
F (q )−F (p ) #
1
1
1
(1 + f1 ) + 2ia1 p1p−λ
− q1q+λ
e 1
=φ
1 + f1
"
F (q1 )−F (p1 ) #
p1
q1
e
= φ 1 − 2ia1
+
.
(22)
λ − p 1 λ + q1
τ1
This is the one-soliton wave function in [2, p. 9].
(2) The two-soliton wave function:

(−p1 , λ)


(q1 , λ)


A2 (z, z̄, t)


(−p2 , λ)
M2 = 


(q2 , λ)


(λ, −p1 ) (λ, q1 ) (λ, −p2 ) (λ, q2 )
0
 ϕ(−p )
ϕ(q1 )
ϕ(−p2 )
ϕ(q2 )
1
ϕ(λ)
− √3
− √3 − √3
− √3
− √3

ϕ(−p1 )
√
3 
ϕ(q1 ) 
√

3 
ϕ(−p2 ) 
√

3 ,
ϕ(q2 ) 
√

3 
ϕ(λ) 

√

3
0
8
J.H. Chang

0

0

B2

0
Q2 = 

0

0 0 0 0 0
0 0 0 0 0

0
0

0
.
0

0
0
Then using (21), one has
Pf(M2 + Q2 )
φ
P (−p1 , q1 , −p2 , q2 , λ)
=
= [1 + h1 (λ) + h2 (λ) + P12 h1 (λ)h2 (λ)]
τ2
τ2
τ2
p1 + q1 p1 + λ q1 − λ F (q1 )−F (p1 )
p2 + q2 p2 + λ q2 − λ F (q2 )−F (p2 )
φ
=
1 + ia1
e
+ ia2
e
τ2
q1 − p1 p1 − λ q1 + λ
q2 − p 2 p 2 − λ q2 + λ
p1 + q1 p1 + λ q1 − λ p2 + q2 p2 + λ q2 − λ F (q1 )+F (q2 )−F (p1 )−F (p2 )
+ ia1 ia2
e
.
q1 − p1 p1 − λ q1 + λ q2 − p2 p2 − λ q2 + λ
ϕ2 =
(3) Three-solitons wave function:

(−p1 , λ)


(q1 , λ)



(−p2 , λ)


A3 (z, z̄, t)


(q2 , λ)
M3 = 


(−p3 , λ)



(q3 , λ)


(λ, −p1 ) (λ, q1 ) (λ, −p2 ) (λ, q2 ) (λ, −p3 ) (λ, q3 )
0

ϕ(q1 )
ϕ(−p2 )
ϕ(q2 )
ϕ(−p3 )
ϕ(q3 )
ϕ(−p1 )
ϕ(λ)
− √3 − √3
− √3 − √3
− √3
− √3
− √3


0 0

0 0




0
0


B3

0 0

.
Q3 = 
0 0



0 0


0 0 0 0 0 0 0 0

ϕ(−p1 )
√
3 
ϕ(q1 ) 
√
3 

ϕ(−p2 ) 
√

3 
ϕ(q2 ) 
√

3 ,
ϕ(−p3 ) 
√

3 
ϕ(q3 ) 

√
3 
ϕ(λ) 

√

3
0
0 0 0 0 0 0 0 0
From (21), we get
Pf(M3 + Q3 )
P (−p1 , q1 , −p2 , q2 , −p3 , q3 , λ)
=
τ3
τ3
φ
= [1 + h1 (λ) + h2 (λ) + h3 (λ) + P12 h1 (λ)h2 (λ) + P13 h1 (λ)h3 (λ) + P23 h2 (λ)h3 (λ)
τ3
+ P12 P13 P23 h1 (λ)h2 (λ)h3 (λ)],
ϕ3 =
where
p1 + λ q1 − λ p1 + q1 F (q1 )−F (p1 )
e
,
p1 − λ q1 + λ q1 − p1
p3 + λ q3 − λ p3 + q3 F (q3 )−F (p3 )
h3 = ia3
e
p3 − λ q3 + λ q3 − p3
h1 = ia1
and P12 , P13 and P23 are defined in (18).
h2 = ia2
p2 + λ q2 − λ p2 + q2 F (q2 )−F (p2 )
e
,
p2 − λ q2 + λ q2 − p2
On the N -Solitons Solutions in the Novikov–Veselov Equation
4
9
A property of N -solitons wave function
In this section, we will express the wave function (19) as another form to generalize the equation (22) to N -solitons case.
Firstly, according to the Pfaffian expansion in [11], it is not difficult to see that
f 1 , b2 , b3 , b4 , . . . , b2n−1 , b2n , b2n+1 )
Pf(b
=
2n+1
X
f 1 , b2 , . . . , b̂j , . . . , b̂m , . . . , b2n , b2n+1 ),
(−1)j+m (bj , bm )Pf(b
(23)
m=1
for j = 1, 2, . . . , 2n + 1,
where b̂j and b̂m mean these two terms are omitted.
Secondly, noticing (10) and letting λ = −pα or λ = qα , α = 1, 2, . . . , n, we have
f
Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα , qα , −pα+1 , qα+1 , . . . , −pN , qN , −pα )
f
= Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα , −pα+1 , qα+1 , . . . , −pN , qN ) = φ(−pα )χ̂N (−pα ),
f
Pf(−p
1 , q1 , −p2 , q2 , . . . , qα−1 , −pα , qα , −pα+1 , qα+1 , . . . , −pN , qN , qα )
f
= Pf(−p
1 , q1 , −p2 , q2 , . . . , qα−1 , qα , −pα+1 , qα+1 , . . . , −pN , qN ) = aα φ(qα )χ̂N (qα ). (24)
They can be seen as follows. By (23), one has
f
Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα , qα , −pα+1 , qα+1 , . . . , −pN , qN , −pα )
f 1 , −p2 , q2 , . . . , −pα−1 , qα−1 , −pα , −pα+1 , . . . , −pN , qN , −pα )
= −(qα , −p1 )Pf(q
f
+ (qα , q1 )Pf(−p
1 , −p2 , q2 , . . . , −pα−1 , qα−1 , −pα , −pα+1 , . . . , −pN , qN , −pα ) − · · ·
f
− (qα , −pα )Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα−1 , qα−1 , −pα+1 , . . . , −pN , qN , −pα ) + · · ·
f
+ (qα , −pα )] Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα−1 , qα−1 , −pα , −pα+1 , . . . , −pN , qN )
f
= [−(qα ,−pα ) + (qα , −pα )] ]Pf(−p
1 , q1 ,−p2 , q2 , . . . ,−pα−1 , qα−1 ,−pα , −pα+1 , . . . ,−pN , qN )
f
= Pf(−p
1 , q1 , −p2 , q2 , . . . , −pα−1 , qα−1 , −pα , −pα+1 , qα+1 , . . . , −pN , qN ),
where (qα , −pα )] is defined in (20) and we know that
f . . , −pα , . . . , −pα ) = 0.
Pf(.
The second equation of (24) can be proved similarly.
Finally, from (23), one yields
f
f
Pf(−p
1 , q1 , −p2 , q2 , . . . , −pN , qN , λ) = (λ, −p1 )Pf(q1 , −p2 , q2 , . . . , −pN , qN )
f
f
− (λ, q1 )Pf(−p
1 , −p2 , q2 , . . . , −pN , qN ) + (λ, −p2 )Pf(q1 , −p1 , q2 , . . . , −pN , qN )
f 1 , −p1 , −p2 , . . . , −pN , qN ) + · · ·
− (λ, q2 )Pf(q
f
+ (λ, −pN )Pf(−p
1 , −p2 , q2 , . . . , −pN −1 , qN −1 , qN )
f
− (λ, qN )Pf(−p
1 , −p2 , q2 , . . . , −pN −1 , qN −1 , −pN ) + φ(λ)τN .
Also, the wave function (21) can be written as
ϕN (λ) = φ(λ)χN (λ),
10
J.H. Chang
where χN (λ) =
χ̂N (λ)
τN .
Therefore, using (24) and letting ∆Fn = F (qn ) − F (pn ), we get
p1 + λ
q1 − λ
ia1 e∆F1 χN (q1 ) −
ia1 e∆F1 χN (−p1 )
λ − p1
q1 + λ
q2 − λ
p2 + λ
ia2 e∆F2 χN (q2 ) −
−
ia2 e∆F2 χN (−p2 ) − · · ·
λ − p2
q2 + λ
pN + λ
qN − λ
−
iaN e∆FN χN (qN ) −
iaN e∆FN χN (−pN )
λ − pN
qN + λ
2q1
2p1
ia1 e∆F1 χN (q1 ) −
=1−
ia1 e∆F1 χN (−p1 )
λ − p1
q1 + λ
2q2
2p2
ia2 e∆F2 χN (q2 ) −
−
ia2 e∆F2 χN (−p2 ) − · · ·
λ − p2
q2 + λ
2qN
2pN
iaN e∆FN χN (qN ) −
iaN e∆FN χN (−pN )
−
λ − pN
qN + λ
+ −ia1 e∆F1 χN (q1 ) + ia1 e∆F1 χN (−p1 ) − ia2 e∆F2 χN (q2 )
χN (λ) = 1 −
+ ia2 e∆F2 χN (−p2 ) − · · · − iaN e∆FN χN (qN ) + iaN e∆FN χN (−pN ) .
(25)
τN
f
Since Pf(−p
1 , q1 , −p2 , q2 , . . . , −pN , qN , 0) = √3 , we have χ̂N (0) = τN . Then the last term in
[· · · ] of (25) is zero (or lim χN (λ) = 1 ). Hence one has
λ→∞
2p1
2q1
ia1 e∆F1 χN (q1 ) −
ia1 e∆F1 χN (−p1 )
λ − p1
q1 + λ
2p2
2q2
−
ia2 e∆F2 χN (q2 ) −
ia2 e∆F2 χN (−p2 ) − · · ·
λ − p2
q2 + λ
2qN
2pN
iaN e∆FN χN (qN ) −
iaN e∆FN χN (−pN ).
−
λ − pN
qN + λ
χN (λ) = 1 −
This formula is also obtained by the d-bar dressing method when the d-bar data is the degenerate
delta kernel [2, p. 6].
When n=1, we have (22). For n=2, from (21), one knows that
p2 + q2 p2 − p1 q2 + p1 4F2
χ2 (−p1 ) = 1 + ia2
e
/τ2 ,
q2 − p2 p2 + p1 q2 − p1
p2 + q2 p2 + q1 q2 − q1 4F2
e
/τ2 ,
χ2 (q1 ) = 1 + ia2
q2 − p 2 p 2 − q1 q2 + q1
p1 + q1 p1 − p2 q1 + p2 4F1
χ2 (−p2 ) = 1 + ia1
e
/τ2 ,
q1 − p1 p1 + p2 q1 − p2
p1 + q1 p1 + q2 q1 − q2 4F1
χ2 (q2 ) = 1 + ia1
e
/τ2 .
q1 − p 1 p 1 − q2 q1 + q2
Then
2p1
2q1
ia1 e∆F1 χ2 (q1 ) −
ia1 e∆F1 χ2 (−p1 )
λ − p1
q1 + λ
2p2
2q2
−
ia2 e∆F2 χ2 (q2 ) −
ia2 e∆F2 χ2 (−p2 ).
λ − p2
q2 + λ
χ2 (λ) = 1 −
We remark that this formula also appears in [2, p. 10], the parameters being different. For n=3,
by (21), we obtain
p2 + q2 p2 − p1 q2 + p1 4F2
p3 + q3 p3 − p1 q3 + p1 4F3
χ3 (−p1 ) = 1 + ia2
e
+ ia3
e
q2 − p2 p2 + p1 q2 − p1
q3 − p3 p3 + p1 q3 − p1
On the N -Solitons Solutions in the Novikov–Veselov Equation
11
p2 + q2 p2 − p1 q2 + p1 p3 + q3 p3 − p1 q3 + p1
4F2 +4F3
/τ3 ,
+ ia2 ia3
P23 e
q2 − p2 p2 + p1 q2 − p1 q3 − p3 p3 + p1 q3 − p1
p3 + q3 p2 + q1 q3 − q1 4F3
p2 + q2 p2 + q1 q2 − q1 4F2
e
+ ia3
e
χ3 (q1 ) = 1 + ia2
q2 − p 2 p 2 − q1 q2 + q1
q3 − p3 p2 − q1 q3 + q1
p2 + q2 p2 + q1 q2 − q1 p3 + q3 p2 + q1 q3 − q1
4F2 +4F3
+ ia2 ia3
/τ3 ,
P23 e
q2 − p 2 p 2 − q1 q2 + q1 q3 − p 3 p 2 − q1 q3 + q1
p1 + q1 p1 − p2 q1 + p2 4F1
p3 + q3 p3 − p2 q3 + p2 4F3
χ3 (−p2 ) = 1 + ia1
e
+ ia3
e
q1 − p1 p1 + p2 q1 − p2
p3 − q3 p3 + p2 q3 − p2
p1 + q1 p1 − p2 q1 + p2 p3 + q3 p3 − p2 q3 + p2
4F1 +4F3
/τ3 ,
+ ia1 ia3
P13 e
q1 − p1 p1 + p2 q1 − p2 p3 − q3 p3 + p2 q3 − p2
p1 + q1 p1 + q2 q1 − q2 4F1
p3 + q3 p3 + q2 q3 − q2 4F3
χ3 (q2 ) = 1 + ia1
e
+ ia3
e
q1 − p 1 p 1 − q2 q1 + q2
q3 − p3 p3 − q2 q3 + q2
p1 + q1 p1 + q2 q1 − q2 p3 + q3 p3 + q2 q3 − q2
4F1 +4F3
+ ia1 ia3
/τ3 ,
P13 e
q1 − p 1 p 1 − q2 q1 + q2 q3 − p 3 p 3 − q2 q3 + q2
p1 + q1 p1 − p3 q1 + p3 4F1
p2 + q2 p2 − p3 q2 + p3 4F2
χ3 (−p3 ) = 1 + ia1
e
+ ia2
e
p1 − q1 p1 + p3 q1 − p3
q2 − p2 p2 + p3 q2 − p3
p1 + q1 p1 − p3 q1 + p3 p2 + q2 p2 − p3 q2 + p3
4F1 +4F2
+ ia1 ia2
P12 e
/τ3 ,
p1 − q1 p1 + p3 q1 − p3 q2 − p2 p2 + p3 q2 − p3
p1 + q1 p1 + q3 q1 − q3 4F1
p2 + q2 p2 + q3 q2 − q3 4F2
χ3 (q3 ) = 1 + ia1
e
e
+ ia2
q1 − p 1 p 1 − q3 q1 + q3
q2 − p2 p2 − q3 q2 + q3
p1 + q1 p1 + q3 q1 − q3 p2 + q2 p2 + q3 q2 − q3
4F1 +4F2
P12 e
/τ3 ,
+ ia1 ia2
q1 − p 1 p 1 − q3 q1 + q3 q2 − p 2 p 2 − q3 q2 + q3
where P12 , P13 and P23 are defined in (18). Then
2p1
ia1 e∆F1 χ3 (q1 ) −
λ − p1
2q2
−
ia2 e∆F2 χ3 (−p2 ) −
q2 + λ
χ3 (λ) = 1 −
5
2q1
2p2
ia1 e∆F1 χ3 (−p1 ) −
ia2 e∆F2 χ3 (q2 )
q1 + λ
λ − p2
2p3
2q3
ia3 e∆F3 χ3 (q3 ) −
ia3 e∆F3 χ3 (−p3 ).
λ − p3
q3 + λ
Concluding remarks
In this paper, we have used the extended Moutard transformation to construct the N -solitons
solutions. The basic idea comes from the successive iterations of solitons solutions, as remains to
be the simple method to obtain the N -solitons solutions. Also, the corresponding wave functions
are constructed by the Pfaffian expansion of the sum of two anti-symmetric matrices (14) when
compared with the ∂-dressing method [2].
To obtain real N -solitons solutions of the Novikov–Veselov equation (1), one has to put
extra relations between −pi and qi [2]. It could be interesting to investigate these real solutions
for the Schrödinger operator (self-adjoint). On the other hand, the resonance of N -solitons
solutions of DKP or KP theory has been studied in [16, 17, 18, 19]. And then the resonance
of N -solitons solutions of Pfaffian type (15) deserves to be investigated. These issues will be
published elsewhere.
Acknowledgements
This work is supported in part by the National Science Council of Taiwan under Grant No.
100-2115-M-606-001.
12
J.H. Chang
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