Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 661–676
Research Article
Algebro-geometric solutions for the generalized
nonlinear Schrödinger hierarchy
Qian Lia , Tiecheng Xiaa,∗, Chao Yueb,
a
Department of Mathematics, Shanghai University, Shanghai, 200444 China.
b
College of Information Engineering, Taishan Medical University, Taian, 271016 China.
Communicated by B. G. Sidharth
Abstract
This paper is dedicated to provide explicit theta function representation of algebro-geometric solutions for
the generalized nonlinear Schrödinger hierarchy. Our main tools include zero-curvature equation to derive
the generalized nonlinear Schrödinger hierarchy, the hyper-elliptic curve with genus of N , the Abel-Jacobi
coordinates, the meromorphic function, the Baker-Akhiezer functions, and the Dubrovin-type equations for
auxiliary divisors. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the whole generalized
c
nonlinear Schrödinger hierarchy. 2016
All rights reserved.
Keywords: Algebro-geometric solutions, Abel-Jacobi coordinates, meromorphic function, Dubrovin-type
equations, generalized nonlinear Schrödinger hierarchy.
2010 MSC: 37K10, 58F07, 35Q35.
1. Introduction
Algebro-geometric solution (also called quasi-periodic solution or finite gap solution), as an important
feature of integrable system, was originally researched on the KdV equation based on the inverse spectral
theory [1, 3, 4, 5], subsequently, algebro-geometric method developed in literatures such as [1, 3, 4, 5, 19,
20, 25], in the late 1970s. As a degenerated case of the algebro-geometric solution, the elliptic function
solution and multi-soliton solution may be worked out [1, 22, 25]. The inverse spectral theory has been
extended to the whole (1+1) dimensional integrable hierarchy by Gesztesy and Holden, such as the AKNS
∗
Corresponding author
Email address: xiatc@shu.edu.cn (Tiecheng Xia)
Received 2015-06-26
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
662
hierarchy, the Camassa-Holm hierarchy, and so on [9, 10, 11, 12]. In recent years, Fan and his co-authors
[15, 16, 17, 18, 29, 30, 31] investigated the Ruijsenaars-Toda Hierarchy, the Degasperis-Procesi hierarchy,
derivative Burgers hierarchy, etc., and obtained their algebro-geometric solutions.
As is well-known, the nonlinear Schrödinger equation is
iut + uxx + 2|u|2 u = 0,
(1.1)
it arises from a wide variety of fields, such as plasma physics, the theory of deep water waves, and nonlinear
fibre optics, etc. The discovery of the integrability of the equation by the inverse scattering transformation
approach has promoted the understanding of the generality of this method to a large extent. So far, there
has been much research on the Eq. (1.1). such as its soliton solutions, conserved quantities, Darboux
transformation, Bäcklund transformation, and others have been studied in [2, 6, 14, 21, 24].
In this paper, we are going to investigate the generalized nonlinear Schrödinger (GNLS) hierarchy [7],
and the first non-trival member is GNLS equation (t1 = t)
ipt = 21 pxx − p2 q + 2iγ(pq)x p + 2γ 2 p3 q 2 ,
(1.2)
iqt =
− 12 qxx
+
pq 2
+ 2iγ(pq)x q −
2γ 2 p2 q 3 ,
where γ is a constant.
Geng discussed the soliton solutions of the equation (1.2) by Darboux transformation [8], Yan studied
its N -Hamiltonian structures and finite-dimensional involutive systems and integrable systems [27], Qin
investigated algebro-geometric solutions for equation (1.2) by variable separation method [26]. However, to
the best of authors knowledge, the algebro-geometric solutions for the whole GNLS hierarchy are still not
presented yet. The main aim of the present paper is to uniformly construct algebro-geometric solutions of
the entire GNLS hierarchy.
The present paper is organized as follows. In Section 2, based on the Lenard gradient and the stationary
zero-curvature equation, we obtain the GNLS hierarchy associated with a 2 × 2 matrix spectral problem.
In Section 3, we introduce a Lax matrix and an algebraic curve KN with genus N , by which we discuss
nonlinear recursion relations of the corresponding homogeneous coefficients, and a direct relation between
the elliptic variables and the potentials is established. Then the hierarchy is decomposed into solvable
ordinary differential equations. In Section 4, all the flows of the GNLS hierarchy are straighten out under
the Abel-Jacobi coordinates. The meromorphic function φ and the Baker-Akhiezer function ψ are introduced
on the hyperelliptic curve KN . Then we study the meromorphic function φ such that φ satisfies nonlinear
differential equations, and discuss the properties of the Baker-Akhiezer function ψ. We present the explicit
theta function representations for the meromorphic function and the Baker-Akhiezer function. In particular,
we give the algebro-geometric solutions for the whole GNLS hierarchy.
2. The GNLS hierarchy
In this section, we shall derive the hierarchy associated with the following spectral problem [7],
iλ − iγpq
p
ψ1
ψx = U ψ, U =
, ψ=
,
q
−iλ + iγpq
ψ2
(2.1)
where p and q are two potentials, γ is a constant,λ is a constant spectral parameter. We now introduce the
Lenard gradient sequence {Sj }, j = 0, 1, 2 . . . to derive the hierarchy
KSj−1 = JSj , j = 1, 2, 3, . . . , Sj |(p,q)=0 = 0, S−1 = (−1, 0, 0)T , JS−1 = 0,
where Sj = (aj , bj , cj ) and operators (∂ = ∂/∂x)




2p ∂ + 2iγpq
0
0 2i 0
0
∂ − 2iγpq  , J =  0 0 −2i  .
K =  −2q
∂
q
−p
∂ q −p
After direct calculation from the recursion relation (2.2) we get
(2.2)
(2.3)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
663



0
− 21 pq
S0 =  ip  , S1 =  12 px + iγp2 q  ,
iq
− 12 qx + iγpq 2


i
2 2
4 (px q − pqx ) − γp q
S2 =  − 4i pxx + 2i p2 q + 23 γppx q + 12 γp2 qx + iγ 2 p3 q 2  .
− 4i qxx + 2i pq 2 − 32 γpqx q − 21 γq 2 px + iγ 2 p2 q 3

(2.4)
(2.5)
Consider the auxiliary problem:
(r)
ψtr = V (r) ψ,
V (r) =
where
(r)
V11 = 2iγar − i
(r)
r+1
P
(r)
j=0
r+1
P
V12 = −i
V21 = −i
(r)
V11
V12
(r)
(r)
V21 −V11
r+1
P
!
,
(2.6)
aj−1 λr+1−j ,
j=0
bj−1 λr+1−j ,
(2.7)
cj−1 λr+1−j .
j=0
(r)
Then the compatibility condition of Eq. (2.1) and Eq. (2.6) is Utr − Vx + [U, V (r) ] = 0, which is equivalent
to
ptr = (2 − 4iγp∂ −1 q)br+1 + 4iγp∂ −1 pcr+1 ,
(2.8)
qtr = 4iγq∂ −1 qbr+1 + (−2 − 4iγq∂ −1 p)cr+1 .
The first non-trival member is GNLS equation
pt1 = − 2i pxx + ip2 q + 2γ(pq)x p − 2iγ 2 p3 q 2 ,
(2.9)
qt1 = 2i qxx − ipq 2 + 2γ(pq)x q + 2iγ 2 p2 q 3 .
which is just Eq. (1.2) when t1 = t.
3. Evolution of elliptic variables
Assume that (2.1) and (2.6) have two basic solutions χ = (χ1 , χ2 )T and Φ = (Φ1 , Φ2 )T . Then we define
a matrix W of three functions g, f, h by
1
g f
0 −1
T
T
,
(3.1)
W = (Φχ + χΦ )σ =
, σ=
h −g
1 0
2
which satisfies the Lax equations
Wx = [U, W ], Wtr = [V (r) , W ].
(3.2)
Therefore, ∂x detW = 0, ∂tr detW = 0. Then Eq. (3.1) and Eq. (3.2) can be written as
fx = 2i(λ − γpq)f − 2pg,
gx = ph − qf,
hx = 2qg − 2i(λ − γpq)h,
and
ftr
gtr
htr
(r)
(3.3)
(r)
= 2f V11 − 2gV12 ,
(r)
(r)
= hV12 − f V21 ,
(r)
(r)
= 2gV21 − 2hV11 .
Supposing that the functions g, f, h are finite-order polynomials in λ:
(3.4)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
f=
N
+1
X
N +1−j
fj−1 λ
, g=
N
+1
X
j=0
gj−1 λ
N +1−j
664
, h=
j=0
N
+1
X
hj−1 λN +1−j .
(3.5)
j=0
Substituting (3.5) into (3.3) yields:
f−1 = 0, h−1 = 0, g−1 = −1,
fj,x = −2pgj − 2iγpqfj + 2ifj+1 , hj,x = 2qgj + 2iγpqhj + 2hj+1 , −1 ≤ j ≤ N − 1,
gj,x = phj − qfj , −1 ≤ j ≤ N, fN,x = 0, gN,x = 0,
(3.6)
which is equivalent to
KQj−1 = JQj (j = 0, 1, 2, . . . , N ), JQ−1 = 0, KQN = 0, Qj = (gj , fj , hj )T .
(3.7)
It is easy to see that JQ−1 = 0 has the general solution
Q−1 = α0 S−1 ,
(3.8)
where α0 is constant of integration. Without loss of generality, let α0 = 1, and acting with the operator
(J −1 K)k+1 upon (3.8), we obtain from (2.4) and (3.7) that:
Qk =
k+1
X
αj Sk−j , k = 0, 1, . . . , N,
(3.9)
j=0
where α0 , α1 , . . . , αN +1 are integral constants. The first few members in (3.9) are


−α1
Q0 =  ip  ,
iq


− 21 pq − α2
Q1 =  12 px + iγp2 q + ipα1  ,
− 12 qx + iγpq 2 + iqα1
(3.10)
(3.11)
where {αl }l∈N denote integration constants. For subsequent use we introduce the homogeneous coefficients
ĝk , fˆk and ĥk defined by the vanishing of the integration constants {αl } for l = 1, 2, . . . , k(k ∈ N0 ),
ĝ−1 = g−1 = −1, ĥ−1 = h−1 = 0, fˆ−1 = f−1 = 0,
ĝk = gk |αl =0,l=1,...,k , fˆk = fk |αl =0,l=1,...,k , ĥk = hk |αl =0,l=1,...,k .
(3.12)
Defining α0 = 1, then we obtain
gk =
k
X
αk−j ĝj , fk =
j=−1
k
X
αk−j fˆj , hk =
k
X
αk−j ĥj .
(3.13)
j=0
j=0
Next, we consider the function detW , which is a (2N+2) th-order polynomial in λ with constant coefficients
of the x-flow and tr -flow, we have
2
− detW = g + f h =
2N
+1
Y
(λ − λj ) = R(λ).
(3.14)
j=0
We introduce the hyperelliptic curve KN of arithmetic genus N defined by
2
KN : y =
2N
+1
Y
(λ − λj ) = R(λ).
j=0
(3.15)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
665
The curve KN can be compactified by joining two points at infinity P∞± , where P∞+ 6= P∞− . And
the compactification is also denoted by KN for notational simplicity. Here we assume that the zeros λj of
R(λ) in (3.15) are mutually distinct, which means λj 6= λk , when j 6= k, 0 ≤ j, k ≤ 2N + 1. Then the hyper
elliptic curve KN becomes nonsingular. We can determine the integration constants αk (0 ≤ k ≤ N ) by the
constants λ0 , . . . , λ2N +1 from the following theorem. We shall give some elementary results before giving
the theorem.
−1
Let {λj }j=0,1,...,2N +1 ⊂ C for some N ∈ N0 and ξ ∈ C, |ξ| < min{λ−1
0 , . . . , λ2N +1 }, and abbreviate
λ = (λ0 , . . . , λ2N +1 ), then
2N
+1
∞
Y
X
1
(1 − λj ξ)− 2 =
ĉk (λ)ξ k ,
(3.16)
j=0
k=0
where
ĉ0 (λ) = 1, ĉ1 (λ) =
P
ĉk (λ) =
2N
P+1
1
2
λj , ĉ2 (λ) =
1
4
P
λj λk +
3
8
2N
P+1
j=0
k=0
j<k
j2N +1
(2j0 )!···(2j2N +1 )!
j0
λ · · · λ2N +1 .
22k (j0 !)2 ···(j2N +1 !)2 0
λ2j ,
(3.17)
j0 ,...,j2N +1 =0
j0 +...+j2N +1 =k
Similarly, we have
2N
+1
Y
1
(1 − λj ξ) 2 =
j=0
where
c0 (λ) = 1, c1 (λ) = − 21
∞
X
ck (λ)ξ k ,
2N
P+1
λj , c2 (λ) =
k=0
1
4
P
λj λk −
j<k
j
P
ck (λ) =
j0 ,...,j2N +1 =0
j0 +...+j2N +1 =k
(3.18)
k=0
1
8
2N
P+1
j=0
(3.19)
j
2N +1
(2j0 )!···(2j2N +1 )!λ00 ···λ2N
+1
.
2k
2
2
2 (j0 !) ···(j2N +1 !) (2j0 −1)···(2j2N +1 −1)
A comparison of the coefficients of ξ k in the following equation

 

!
2N
+1
2N
+1
∞
Y
Y
X
1
1
−
k
1=
(1 − λj ξ) 2  · 
(1 − λj ξ) 2  =
ck (λ) ξ
·
j=0
λ2j ,
j=0
k=0
∞
X
!
ĉk (λ) ξ
k
(3.20)
k=0
we get
k
X
cj (λ)ĉk−j (λ) = δk,0 k ∈ N0 .
(3.21)
αl = cl (λ), l = 0, . . . , N.
(3.22)
j=0
Theorem 3.1.
Proof. It will be convenient to introduce the notion of a degree, deg(.), to effectively distinguish between
homogeneous and nonhomogeneous quantities. One defines
deg(p) = 1, deg(q) = 1, deg(∂x ) = 2,
(3.23)
deg(fˆk ) = 2k + 1, deg(ĥk ) = 2k + 1, deg(ĝk ) = 2k, k ∈ N0 ,
(3.24)
implying
1
by using the linear recursion relation (3.6) and induction on k. Next, dividing f (λ), h(λ), g(λ) by R(λ) 2
near infinity respectively, one obtains
f (λ)
1
R(λ) 2
h(λ)
1
R(λ) 2
=
∞
P
=
j=0
∞
P
j=0
ĉj (λ)λ−N −1−j
ĉj (λ)λ−N −1−j
NP
+1
j=0
NP
+1
j=0
fj−1 λN +1−j =
hj−1 λN +1−j =
∞
P
f˘l−1 λ−l ,
l=0
∞
P
l=0
(3.25)
h̆l−1 λ−l ,
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
g(λ)
1
R(λ) 2
=
∞
P
ĉj (λ)λ−N −1−j
j=0
NP
+1
gj−1 λN +1−j =
j=0
666
∞
P
ğl−1 λ−l ,
l=0
for some coefficients fˇl , ǧl , ȟl to be determined next. Noticing (3.3) and (3.24), we get
f˘k,x = −2pğk − 2iγpq f˘k + 2if˘k+1 ,
h̆k,x = 2qğk + 2iγpq h̆k + 2h̆k+1 ,
ğk,x = ph̆k − q f˘k , k = −1, 0, 1, . . . .
(3.26)
Here we have chosen that f˘−1 = fˆ−1 = 0, ğ−1 = ĝ−1 = −1, h̆−1 = ĥ−1 = 0, f˘0 = fˆ0 = ip, ğ0 = ĝ0 = 0 and
h̆0 = ĥ0 = iq. Moreover, one can prove inductively that
deg(f˘k ) = 2k + 1, deg(h̆k ) = 2k + 1, deg(ğk ) = 2k.
(3.27)
Hence, ğl , f˘l and h̆l are equal to ĝl , fˆl and ĥl , respectively, for all l ∈ N0 . Thus we proved
f (λ)
1
R(λ) 2
h(λ)
1
=
∞
∞
P
P
fˆl−1 λ−l =
fˆl λ−l−1 ,
=
l=0
∞
P
=
l=0
∞
P
R(λ) 2
g(λ)
1
R(λ) 2
ĥl−1
λ−l
=
l=0
∞
P
ĥl λ−l−1 ,
(3.28)
l=0
ĝl−1 λ−l .
l=0
Using (3.20), we compute that
k
P
m=0
k−l
k
m
k
P
P
P
P
ck−m (λ)fˆm =
ck−m ĉk−s−l (λ)ĉs (λ) = fk ,
ck−m (λ)
fl ĉm−l (λ) =
fl
m=0
l=0
(3.29)
s=0
l=0
where k = 0, . . . , N, which together with (3.13) implies (3.21) holds. Hence Theorem 3.1 is proved.
If we write f and h as finite products which take the following form
f = ip
N
Q
h = iq
j=1
N
Q
(λ − µj ),
(3.30)
(λ − νj ),
j=1
N
where the roots {µj }N
j=1 and {νj }j=1 are called elliptic variables, from which we obtain
g|λ=µk =
p
p
R(µk ), g|λ=νk = R(νk ).
(3.31)
Noticing (3.3), (3.29) and (3.30), we get
fx |λ=µk = −2pg|λ=µk = −ipµk,x
N
Y
(µk − µj ),
(3.32)
j=1
j6=k
hx |λ=νk = 2qg|λ=νk = −iqνk,x
N
Y
j=1
j6=k
which means
(νk − νj ),
(3.33)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
µk,x
667
p
p
R(µk )
R(νk )
, νk,x = 2i N
, 1 ≤ k ≤ N.
= −2i N
Q
Q
(µk − µj )
(νk − νj )
j=1
j6=k
(3.34)
j=1
j6=k
In a way similar to the calculation of (3.33), we obtain the evolution of {µj } and {νj } along the tr − f low :
µk,tr
p
p
(r)
(r)
R(µk )V12
R(νk )V21
, νk,tr = 2i N
.
= −2i N
Q
Q
(µk − µj )
(νk − νj )
p
q
j=1
j6=k
(3.35)
j=1
j6=k
where 1 ≤ k ≤ N .
4. Algebro-geometric solutions
In the section, we shall construct algebro-geometric solutions for the whole GNLS hierarchy (2.8). First
we will recall the hyper-elliptic curve KN of arithmetic genus N defined by
KN : y 2 − R(λ) = 0.
(4.1)
N
We can lift the roots {µj }N
j=1 and {νj }j=1 to KN by introducing
µ
bj (x, tr ) = (µj (x, tr ), −G(µj (x, tr ), x, tr )),
(4.2)
νbj (x, tr ) = (νj (x, tr ), G(νj (x, tr ), x, tr )),
(4.3)
where j = 1, · · · , N, (x, tr ) ∈ R2 . Now we introduce the Baker-Akhiezer function ψ = (ψ1 , ψ2 )T by
ψx (P, x, x0 , tr , t0,r ) = U (λ, x, tr )ψ(P, x, x0 , tr , t0,r ),
ψtr (P, x, x0 , tr , t0,r ) = V (r) (λ, x, tr )ψ(P, x, x0 , tr , t0,r ),
W (λ, x, tr )ψ(P, x, x0 , tr , t0,r ) = y(P )ψ(P, x, x0 , tr , t0,r ),
ψ(P, x0 , x0 , t0,r , t0,r ) = 1, (x, tr ) ∈ R2 .
(4.4)
Closely associated with ψ(P, x, x0 , tr , t0,r ) is the meromorphic function φ(P, x, tr ) on KN , defined by
φ(P, x, tr ) =
ψ1 (P, x, x0 , tr , t0,r )
, P ∈ KN ,
ψ2 (P, x, x0 , tr , t0,r )
(4.5)
y−g
h
=
,
f
y+g
(4.6)
which implies from (4.4) that
φ(P, x, tr ) =
where P = (λ, y) ∈ KN \ {P∞+ , P∞− }, (x, tr ) ∈ R2 . Hence the divisor of φ(·, x, tr ) reads [10]
φ(·, x, tr ) = DP∞+ ,bν (x,tr ) − DP∞− ,bµ(x,tr ) ,
where
Dµb(x,tr ) =
N
X
µ̂j (x, tr ), Dbν (x,tr ) =
j=1
N
X
ν̂j (x, tr ).
j=1
The holomorphic sheet exchange map ∗ is defined by
∗ : KN → KN , P = (λ, y) → P ∗ = (λ, −y), P, P ∗ ∈ KN .
(4.7)
(4.8)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
668
With these preparations, we can compute that the meromorphic function φ satisfies the Riccati-type nonlinear differential equations
(r)
(r)
(r)
φx − 2iγpqφ + pφ2 = q − 2iλφ, φtr = V21 − 2V11 φ − V12 φ2 ,
(4.9)
φ(P )φ(P ∗ ) = − fh , φ(P ) + φ(P ∗ ) = −2 hg , φ(P ) − φ(P ∗ ) = 2 fy .
(4.10)
as well as
After direct calculation, we can derive the properties of ψ(P, x, x0 , tr , t0,r )
R
x
ψ1 (P, x, x0 , tr , t0,r ) = exp x0 (iλ − iγp (x0 , tr ) q (x0 , tr ) + p (x0 , tr ) φ (P, x0 , tr )) dx0
R tr (r)
(r)
V11 (λ, x0 , s) + V12 (λ, x0 , s)φ(P, x0 , s) ds
+ t0,r
h
i1
R x y(P )p(x0 ,tr ) 0 R tr y(P )V12(r) (λ,x0 ,s)
2
f (λ,x,tr )
= f (λ,x0 ,t0,r ) · exp x0 f (λ,x0 ,tr ) dx + t0,r
ds ,
f (λ,x0 ,s)
(4.11)
and
ψ1 (P, x, x0 , tr , t0,r )ψ1 (P ∗ , x, x0 , tr , t0,r ) =
f (λ,x,tr )
f (λ,x0 ,t0,r ) ,
h(λ,x,tr )
,
ψ2 (P, x, x0 , tr , t0,r )ψ2 (P ∗ , x, x0 , tr , t0,r ) = − f (λ,x
0 ,t0,r )
(4.12)
(4.13)
g(λ, x, tr )
, (4.14)
f (λ, x0 , t0,r )
y
. (4.15)
ψ1 (P, x, x0 , tr , t0,r )ψ2 (P ∗ , x, x0 , tr , t0,r ) − ψ1 (P ∗ , x, x0 , tr , t0,r )ψ2 (P, x, x0 , tr , t0,r ) = −2
f (λ, x0 , t0,r )
ψ1 (P, x, x0 , tr , t0,r )ψ2 (P ∗ , x, x0 , tr , t0,r ) + ψ1 (P ∗ , x, x0 , tr , t0,r )ψ2 (P, x, x0 , tr , t0,r ) = −2
Next, let us introduce the Riemann surface Γ of the hyperelliptic curve KN and equip Γ with canonical basis
cycles: a1 , . . . , aN ; b1 , . . . , bN , which are independent and have intersection numbers as follows
ai ◦ aj = 0, bi ◦ bj = 0, ai ◦ bj = δij .
For the present, we will choose our basis as the following set
λl−1 dλ
ω
el = p
, 1 ≤ l ≤ N,
R(λ)
(4.16)
which are N linearly independent homomorphic differentials on Γ. By using the cycles a and b, the period
matrices A and B can be constructed from
Z
Z
Aij =
ω
ei , Bij =
ω
ei .
aj
bj
It is possible to show that matrices A and B are invertible [13, 23]. Now we define the matrices C and τ
by C = A−1 , τ = A−1 B. The matrix τ can be shown to be symmetric (τij = τji ) and it has positive definite
imaginary part (Imτ > 0). If we normalize ω
el into the new basis ωj ,
ωj =
N
X
Cjl ω
el , 1 ≤ j ≤ N,
l=1
then we have
Z
ωj =
ai
N
X
l=1
Z
Cjl
ai
Z
ω
el = δji ,
ωj = τji .
bi
(4.17)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
669
Define the Abel-Jacobi coordinates
(1)
ρj (x, tr )
=
N Z
X
k=1
(2)
ρj (x, tr )
=
µ
bk (x,tr )
ωj =
P0
N Z
X
k=1
N X
N
X
νbk (x,tr )
Cjl
λ(P0 )
k=1 l=1
N X
N
X
ωj =
P0
µk
Z
Z
νk
Cjl
λ(P0 )
k=1 l=1
λl−1 dλ
p
,
R(λ)
(4.18)
λl−1 dλ
p
,
R(λ)
(4.19)
where 1 ≤ j ≤ N . Without loss of generality, we choose the branch point P0 = (λj0 , 0), j0 ∈ {0, . . . , 2N + 1},
as a convenient base point, and λ(P0 ) is its local coordinate. From (3.33), (3.34), (4.18) and (4.19) we obtain
the following two lemmas.
Lemma 4.1. (Straightening Out of the x-Flow).
(1)
∂x ρj
(2)
= −2iCjN , ∂x ρj
= 2iCjN , 1 ≤ j ≤ N,
(4.20)
where
(i)
(i)
C N = (C1N , . . . , CN N ), ρ(i) = (ρ1 , . . . , ρN ), i = 1, 2.
Proof. It can be calculated from (3.33), (4.18), (4.19) that
(1)
∂x ρj
=
N X
N
X
N
N
µl−1 µk,x X X −2iCjl µl−1
k
,
=
Cjl pk
N
Q
R(µ
)
k
l=1 k=1
k=1 l=1
(µk − µj )
(4.21)
j=1
j6=k
(2)
∂x ρj
N X
N
X
N
N
X X 2iCjl ν l−1
ν l−1 νk,x
k
=
Cjl pk
=
,
N
Q
R(ν
)
k
k=1 l=1
l=1 k=1
(νk − νj )
(4.22)
j=1
j6=k
which implies
(1)
∂x ρj
(2)
= −2iCjN , ∂x ρj
= 2iCjN , 1 ≤ j ≤ N,
(4.23)
with the help of the following equalities
N
X
k=1
µl−1
k
N
Q
(µk − µj )
=

 δlN ,

P
i1 +···+iN =l−N,ij ≥0
µi11
· · · µiNN ,
1 ≤ l ≤ N,
l > N.
(4.24)
j=1
j6=k
Thus we complete the proof of the lemma.
Lemma 4.2. (Straightening Out of the tr -Flow).
∂tr ρ(1) = 2
r
X
γl C N −r+l ,
(4.25)
l=0
∂tr ρ(2) = −2
r
X
γl C N −r+l ,
l=0
where
(i)
(i)
C j = (C1j , . . . , CN j ), ρ(i) = (ρ1 , . . . , ρN ), 1 ≤ j ≤ N,
(4.26)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
670
and the recursive formula:
γ0 = −i, γ1 = iα1 , γ2 = −i(α12 − α2 ), γk = −
k
X
αj γk−j .
j=1
Proof. From (3.9) we obtain
− ifk =
k
X
(r)
αj V12,k−j ,
(4.27)
j=0
which implies
(r)
V12,k
=
k
X
γl fk−l .
(4.28)
j=0
From (3.34), (4.18) and (4.28),we have
(1)
∂tr ρj
=
=
=
=
=
N P
N
P
N P
N
P
(r)
−2iCjl µl−1
k V12 (µk )
N
Q
R(µ
)
k
l=1 k=1
k=1 l=1
p
(µk −µj )
j=1
j6=k
N P
N
r
P
P
−2iCjl µl−1
(r)
k
(
V12,s µr−s
N
k )
Q
s=0
l=1 k=1 p
(µk −µj )
j=1
j6=k
r P
s
N P
N
P
P
−2iCjl µl−1
k
(
γt fs−t )µr−s
N
k
Q
l=1 k=1 p
(µk −µj ) s=0 t=0
j=1
j6=k
r
r
r−s
P
P
−2iγt P
fs−t
Cj,N −(r−s)+l Υl
p
s=t
t=0
l=0
r−t k
r
P
−2iγt P P
Cj,N −(r−t)+k fl Υk−l ,
p
t=0
k=0 l=0
l−1
µk
Cjl √
µk,tr
=
(4.29)
with
µj11 · · · µjNN , k ≥ 1.
X
Υ0 = 1, Υk =
(4.30)
j1 +···+jN =k,ji ≥0
Then we get
(1)
∂tr ρj
=
r
X
−2iγl
l=0
p
f0 Cj,N −r+l = 2
r
X
γl Cj,N −r+l
(4.31)
l=0
with the help of the formula [28]
X
Υj1 Υj2 = 0, 1 ≤ k ≤ r,
(4.32)
j1 +j2 =k,ji ≥0
where
f0 = ip, f1 = −ip
N
X
µl , fk = (−1)k ip
l=1
X
j1 <···<jk ,ji ≥1
Thus the proof of lemma is completed.
From the above two lemmas, we have the following theorem.
Υj1 Υj2 = 0, 1 ≤ k ≤ N.
(4.33)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
Theorem 4.3.
671
(1)
ρ(1) = −2iC N x + Ωr tr + ρ0 ,
(2)
ρ(2) = 2iC N x − Ωr tr + ρ0 ,
where Ωr = 2
r
P
l=0
(4.34)
(i)
γl C j,N −r+l , r = 0, . . . , N − 1, constants ρ0 ∈ RN , i = 1, 2.
Let T be the lattice generated by 2N vectors δj , τj , where δj = (0, . . . , 0, 1, 0, . . . , 0) and τj = τ δj . The
| {z } | {z }
N −j
j−1
complex torus J = CN /T is called Jacobian variety of Γ. Now we introduce the Abel map A(P ) : DiV (Γ) →
J
!
Z P
X
X
ω, A
A(P ) =
nk Pk =
nk A(Pk ),
(4.35)
P0
k
k
where P, Pk ∈ KN , ω = (ω1 , ω2 , . . . , ωN )> . Consider two special divisors
N
P
k=1
N
X
A
!
(i)
Pk
=
k=1
(1)
with Pk
(2)
=µ
bk (x, tr ) and Pk
N
X
A
(i)
Pk
=
N Z
X
k=1
k=1
(i)
Pk , i = 1, 2, and
(i)
Pk
ω = ρ(i) ,
P0
= νbk (x, tr ), whose components are
N Z
X
k=1
(i)
Pk
P0
(i)
ωj = ρj , 1 ≤ j ≤ N, i = 1, 2.
The Riemann theta function is defined as [10, 13, 23]
θ(P, D) = θ(Λ − A(P ) + A(D),
(4.36)
where P ∈ KN , D ∈ Div(Γ), and Λ = (Λ1 , . . . , ΛN ) is defined by
Z
Z P
N
X
1
Λj = (1 + τjj ) −
ωi
ωj , j = 1, . . . , N.
2
ai
Q0
i=1,i6=j
Then according to (4.35) and the definition of Riemann theta function in (4.36), we have
θ(P, Dµb(x,tr ) ) = θ(Λ − A(P ) + ρ(1) ),
(4.37)
θ(P, Dbν (x,tr ) ) = θ(Λ − A(P ) + ρ(2) ).
(4.38)
Lemma 4.4. Suppose that p(x, tr ), q(x, tr ) ∈ C∞ (R2 ) satisfy the hierarchy of nonlinear Eq. (2.8). Let
λj ∈ C\{0}, (0 ≤ j ≤ 2N + 1), and P = (λ, y) ∈ KN \(P∞+ , P∞− ). Then
(
− iq2 ζ + O(ζ 2 ) as P → P∞+ ,
φ =
ζ = λ−1 .
(4.39)
2i −1
+ ppx2 + 2iγq + O(ζ) as P → P∞− ,
ζ→0
pζ
Proof. Introducing the local coordinate ζ = λ−1 near P∞± , from Theorem 3.1 we have
y
2N
Q+1
1
(λ − λj ) 2 = ∓ζ −N −1
2N
Q+1
1
=
∓
=
∓ζ −N −1 (1 + α1 ζ + α2 ζ 2 + α3 ζ 3 + O(ζ 4 )) as P → P∞± .
j=0
ζ→0
(1 − ζλj ) 2 ,
j=0
(4.40)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
672
From (3.5) and (3.6), we can derive
f −1
=
=
(f0 λN + f1 λN −1 + · · · + fN )−1
f0−1 ζ N [1 − f0−1 f1 ζ + O(ζ 2 )]
=
− pi ζ N (1
ζ→0
ζ→0
g
+
x
( ip
2p
− γpq − α1 )ζ +
(4.41)
O(ζ 2 ))
as P → P∞± ,
g−1 λN +1 + g0 λN + g1 λN −1 + g2 λN −2 + O(λN −3 )
ζ −N −1 (g−1 + g0 ζ + g1 ζ 2 + g2 ζ 3 + O(ζ 4 )) as P → P∞± .
=
=
ζ→0
(4.42)
Then according to the definition of φ in (4.6), we finally obtain that
φ(P, x, tr )
=
=
ζ→0
=
ζ→0
y−g
f
i −1
[∓(1
pζ
+ α1 ζ + α2 ζ 2 + α3 ζ 3 + O(ζ 4 ))
+(g−1 + g0 ζ + g1 ζ 2 + g2 ζ 3 + O(ζ 4 ))]
2
x
·(1 + ( ip
2p − γpq − α1 )ζ + O(ζ ))
(
− iq2 ζ + O(ζ 2 ) as P → P∞+ ,
2i −1
+ ppx2 + 2iγq + O(ζ) as P → P∞− .
pζ
(4.43)
Thus proves the lemma.
Lemma 4.5. Under the same supposition in Lemma 4.4, in the special case of (3.9) when α0 = 1,
αk = 0, 1 ≤ k ≤ N, we have that
exp[iζ −1 (x − x0 ) + iζ −r−1 (tr − t0,r ) + O(1)] as P → P∞+ ,
ψ1 (P, x, x0 , tr , t0,r ) =
exp[−iζ −1 (x − x0 ) − iζ −r−1 (tr − t0,r ) + O(1)] as P → P∞− .
ζ→0
Proof. In the special case of (3.9) when α0 = 1, αk = 0, 1 ≤ k ≤ N, we derive that fj = bj , hj = cj , −1 ≤ j.
Introducing the local coordinate ζ = λ−1 near P∞± , we obtain from (4.39) that
Rx
0
0
0
0
0
exp( 
x0 (iλ −Riγp(P, x , tr )q(P, x , tr ) + p(x , tr )φ(P, x , tr ))dx
x
exp[ x0 (iζ −1 − iγp(P, x0 , tr )q(P, x0 , tr )




+p(P,R x0 , tr )(− iq2 ζ + O(ζ 2 ))dx0 ] as P → P∞+
=
x
exp[ x0 (iζ −1 − iγp(P, x0 , tr )q(P, x0 , tr )
ζ→0 
(4.44)


 +p(P, x0 , tr )(− 2i0 ζ −1 + px + 2iγq + O(ζ)))dx0 ] asP → P∞−
2
p(P,x ,tr )
p
(
exp(iζ −1 (x − x0 ) + O(1)) as P → P∞+ ,
=
p(x,tr )
−1 (x − x ) + O(1)) as P → P
ζ→0
0
∞− .
p(x0 ,tr ) exp(−iζ
Then in the special case of (3.9), combining (2.8), (3.4), (3.29), (4.6), (4.40) and (4.41) yields
R tr
(r)
(r)
(λ, x , s) + V12 (λ, x0 , s)φ(P, x0 , s))ds)
t0,r (V11
hR 0
i
(r)
tr
y
1 ftr (λ,x0 ,s)
= exp t0,r
V
(λ,
x
,
s)
+
ds
0
2 f (λ,x0 ,s)
f (λ,x0 ,s) 12

 P
 
r
bj (ζ −1 ,x0 ,s)ζ j−r
R tr
−1
±i j=0
= exp 
+ ftr (ζ−1 ,x0 ,s) + O(ζ) ds
∞
P
t0,r
ζ→0
fˆl (ζ −1 ,x0 ,s)ζ l+1
2f (ζ
,x0 ,s)
l=0


r
P
+
bj (ζ −1 ,x0 ,s)ζ j
R tr
j=0
= exp  t0,r ±iζ −r−1 P
+
∞
fˆl (ζ −1 ,x0 ,s)ζ l
ζ→0

ptr (x0 ,s)
2p(x0 ,s)

+ O(ζ) ds
j=0
= exp
ζ→0
−1 ,x
hR
(ζ
±iζ −r−1 (1 − ζ r+1 br+1
ˆ −1
hR
±iζ −r−1
tr
t0,r
tr
t0,r
0 ,s)
)+
i
+ O(ζ) ds
i
+ O(ζ) ds
ptr (x0 ,s)
2p(x0 ,s)
f0 (ζ ,x0 ,s)
br+1 (ζ −1 ,x0 ,s)
tr (x0 ,s)
i ˆ −1
) + p2p(x
0 ,s)
f0 (ζ ,x0 ,s)
= exp
∓
exp(iζ −r−1 (tr − t0,r ) + O(1)) as P → P∞+ ,
=
exp(−iζ −r−1 (tr − t0,r ) + O(1)) as P → P∞− .
ζ→0
ζ→0
(4.45)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
673
Hence, combining with the definition of ψ1 in (4.11), we can arrive at (4.38). Next, we shall derive the
(3)
representation of φ, ψ, p and q in terms of the Riemann theta function. Let ωP∞+ ,P∞− be the normal
differential of the third kind holomorphic on KN \{P∞+ , P∞− } with simple poles at P∞+ and P∞− and
residues 1 and -1, respectively, which can be expressed as
(3)
ωP∞+ ,P∞− =
N
1Y
(λ − βj )dλ,
y
(4.46)
j=1
where βj ∈ C, j = 1, . . . , N are constants that are determined by
Z
(3)
ωP∞+ ,P∞− = 0, j = 1, . . . , N.
aj
(3)
If the local coordinate near P∞± is given by ζ = λ−1 , then we have the asymptotic expansions of ωP∞+ ,P∞−
near P∞± :
2N
N
Q+1
Q
1
(3)
ωP∞+ ,P∞− = ∓ζ N +1
(1 − ζβj )dζ
(1 − ζλj )− 2 · (−ζ −N −2 )
ζ→0
±ζ −1
=
ζ→0
j=0
2N
Q+1
j=1
(1 − ζλj )
− 12
·
N
Q
(1 − ζβj )dζ
(4.47)
j=1
j=0
±(ζ −1 + O(1)) as P → P∞± .
=
ζ→0
Therefore,
Z
P
(3)
ωP∞+ ,P∞− = ±(lnζ − lnω0 + O(ζ)) as P → P∞±
ζ→0
P0
(4.48)
(2)
for some constant ω0 ∈ C. Next, let ωP∞±,r , r ∈ N0 , be normalized differentials of the second kind with a
unique pole at P∞± and principal part near P∞± is −ζ −2−r dζ, and satisfying
Z
(2)
ωP∞±,r = 0, j = 1, . . . , N.
(4.49)
(2)
(2)
Then we can define Ω0 and Ωr−1 by
(2)
(2)
(2)
Ω0 = ωP∞+,0 − ωP∞−,0 ,
(2)
Ωr−1
=
r−1
X
(4.50)
(2)
(2)
αr−1−s (s + 1)(ωP∞+,s − ωP∞−,s ),
(4.51)
s=0
where αr−1−s are the integral constants in (3.9). Therefore,
Z
Z
(2)
(2)
Ω0 = 0,
Ωr−1 = 0, j = 1, . . . , N.
aj
Z
P
P0
Z
P
P0
(2)
Ω0
(2)
Ωr−1
= ±(ζ −1 + e0,0 + e0,1 ζ + O(ζ 2 )) as P → P∞± ,
ζ→0
= ±
ζ→0
(4.52)
aj
r−1
X
(4.53)
!
αr−1−s ζ
−1−s
+ er−1,0 + O(ζ)
as P → P∞±
(4.54)
s=0
for some constants e0,0 , e0,1 , er−1,0 ∈ C. If Dµb(x,tr ) and Dbν (x,tr ) in (4.7) are assumed to be nonspecial [13],
then according to the Riemanns vanishing theorem [10, 13], the definition and asymptotic properties of the
meromorphic function φ(P, x, tr ), φ has the expressions of the following type:
Z P
θ(P, Dµb(x,tr ) )
(3)
φ(P, x, tr ) = C(x, tr )
exp
ωP∞+ ,P∞− ,
(4.55)
θ(P, Dbν (x,tr ) )
P0
where C(x, tr ) is independent of P ∈ KN .
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
674
Theorem 4.6. Let P = (λ, y) ∈ KN \(P∞+ , P∞− ), (x, tr ), (x0 , t0,r ) ∈ M where M ⊆ R2 is open and
connected. Suppose p, q ∈ C ∞ (M ) satisfy the hierarchy of nonlinear equations (2.8), and assume that
λj , 0 ≤ j ≤ 2N + 1 in (3.15) satisfy λj ∈ C\{0} and λj 6= λk as j 6= k. Moreover, suppose that Dµb(x,tr ) , or
equivalently, Dbν (x,tr ) is nonspecial for (x, tr ). Then φ, ψ1 , ψ2 admit the following representation:
φ(P, x, tr ) = C0
θ(P∞+ , Dµb(x,tr ) )θ(P, Dbν (x,tr ) )
θ(P∞− , Dbν (x,tr ) )θ(P, Dµb(x,tr ) )
ψ1 (P, x, tr , x0 , tr,0 ) =
Z
P
· exp
P0
θ(P∞+ , Dµb(x0 ,t0,r ) )θ(P, Dµb(x,tr ) )
(3)
ωP∞+ ,P∞− + 2ie0,0 x + 2ier−1,0 tr ,
Z
· exp i
θ(P∞+ , Dµb(x,tr ) )θ(P, Dµb(x0 ,t0,r ) )
Z P
(2)
,
· exp i
iΩr−1 − er−1,0 )(tr − t0,r
P
P0
(2)
Ω0
(4.56)
− e0,0 )(x − x0
(4.57)
P0
θ(P∞+ , Dµb(x0 ,t0,r ) )θ(P, Dbν (x,tr ) )
ψ2 (P, x, tr , x0 , t0,r ) = C0
θ(P∞− , Dbν (x,tr ) )θ(P, Dµb(x0 ,t0,r ) )
Z P
Z
(2)
· exp i
Ω0 + e0,0 (x − x0 ) + i
P0
Z
P
· exp
P0
(4.58)
P
P0
(3)
ωP∞+ ,P∞−
(2)
Ωr−1
+ er−1,0 (tr − t0,r )
+ 2ie0,0 x0 + 2ier−1,0 t0,r ,
where
C0 =
θ(P∞− , Dµb(x0 ,t0,r ) )
−2i
exp(−2ie0,0 x0 − 2ier−1,0 t0,r ),
q(x0 , t0,r )ω0 θ(P∞+ , Dµb(x0 ,t0,r ) ))
p(x, tr ) = p(x0 , t0,r )
q(x, tr ) = q(x0 , t0,r )
θ(P∞+ , Dµb (x0 , t0,r ))θ(P∞− , Dµb(x,tr ) )
θ(P∞− , Dµb(x0 ,t0,r ) )θ(P∞+ , Dµb(x,tr ) )
exp(−2ie0,0 x0 − 2ier−1,0 t0,r ),
θ(P∞− , Dbν (x0 , t0,r ))θ(P∞+ , Dbν (x,tr ) )
exp(2ie0,0 x0 + 2ier−1,0 t0,r ),
θ(P∞+ , Dbν (x0 ,t0,r ) )θ(P∞− , Dbν (x,tr ) )
p(x0 , t0,r )q(x0 , t0,r ) =
4 θ(P∞+ , Dbν (x0 , t0,r ))θ(P∞− , Dµb(x0 ,t0,r ) )
·
.
ω02 θ(P∞− , Dbν (x0 ,t0,r ) )θ(P∞+ , Dµb(x0 ,t0,r ) )
(4.59)
(4.60)
(4.61)
(4.62)
Proof. First, we shall consider the theta function representation (4.57) for ψ1 . Without loss of generality, in
the following, we only consider the special case of (3.9) for α0 = 1, αk = 0, 1 ≤ k ≤ N + 1. We temporarily
f for appropriate M
f ⊆ M and define the right-hand
assume that µj (x, tr ) 6= µj 0 (x, tr ) as j 6= j 0 and (x, tr ) ∈ M
side of (4.57) to be ψe1 . In order to prove ψ1 = ψe1 , we investigate the local zeros and poles of ψ1 defined by
(4.11). From (3.33), (3.34), (4.2), and (4.6), we have
v(x0 , tr )φ(P, x0 , tr )
(r)
=
P →b
µj (x0 ,tr )
V12 (λ, x0 , s)φ(P, x0 , s)
=
−∂x0 ln(λ − µj (x0 , tr )) + O(1),
P →b
µj (x0 ,s)
−∂s ln(λ − µj (x0 , s)) + O(1),

P →µ
bj (x, tr ) 6= µ
bj (x0 , tr,0 ),
 (λ − µj (x, tr ))O(1),
O(1),
P →µ
bj (x, tr ) = µ
bj (x0 , tr,0 ),
ψ1 (P, x, tr , x0 , tr,0 ) =

(λ − µj (x0 , tr,0 ))−1 O(1), P → µ
bj (x0 , tr,0 ) 6= µ
bj (x, tr ),
(4.63)
(4.64)
(4.65)
Q. Li, T. C. Xia, C. Yue, J. Nonlinear Sci. Appl. 9 (2016), 661–676
675
f, and O(1) 6= 0. Hence ψ1 and ψe1 share the same singularities
with P = (λ, y) ∈ KN , (x, tr ), (x0 , t0,r ) ∈ M
and zeros on KN \(P∞+ , P∞− ), which are all simple by hypothesis (4.63). Next, we study the behavior of
ψ1 and ψe1 near P∞± , taking into account (4.38), (4.53), (4.54) and (4.57), then shows that ψ1 and ψe1 have
identical exponential behavior up to order O(1) near P∞± . Thus, ψ1 and ψe1 share the same singularities
and zeros. Then the Riemann-Roch-type uniqueness proves that ψ1 = ψe1 . Hence (4.57) holds subject to
(4.63). Inserting (4.48) into (4.55) and comparing with (4.39) one finds
q = C(x, tr )
θ(P∞− , Dµb(x,tr ) )
2i θ(P∞+ , Dbν (x,tr ) )
2i
,p = −
.
ω0 θ(P∞+ , Dµb(x,tr ) )
C(x, tr )ω0 θ(P∞− , Dbν (x,tr ) )
(4.66)
Re-examining the asymptotic behavior of ψ1 near P∞− yields
ψ1 (P, x, tr , x0 , t0,r ) =
=
p(x,tr )
−1 (x − x ) + O(1)) · p(x0 ,tr ) exp(−iζ −1−r (t
0
r
p(x0 ,tr ) exp(−iζ
p(x0 ,t0,r )
p(x,tr )
−1
−r−1
(x − x0 ) − iζ
(tr − t0,r ) + O(1)].
p(x0 ,t0,r ) exp[−iζ
− t0,r ) + O(1))
(4.67)
A comparison of (4.57) and (4.68) then proves (4.60). Inserting (4.60) into the second equation of (4.67),
we have
C(x, tr ) =
θ(P∞+ , Dµb(x,tr ) )θ(P∞− , Dµb(x0 ,t0,r ) )
−2i
· exp(2ie0,0 (x − x0 ) + 2ier−1,0 (tr − t0,r )), (4.68)
q(x0 , t0,r )ω0 θ(P∞− , Dbν (x,tr ) )θ(P∞+ , Dbν (x0 ,t0,r ) )
together with the first equation of (4.67) yields (4.61), (4.62). Given C(x, tr ), we can determine φ in (4.5)
f. Finally the
from (4.55) and ψ2 in (4.58) from ψ2 = φψ1 . So we complete the prove of the theorem on M
f to M then follows by continuity of the Abel map A(P ) and nonspecialty
extension of all these results from M
of Dµb(x,tr ) on M. Hence, we obtain the algebro-geometric solutions of the whole GNLS hierarchy.
Acknowledgements:
This Work is in part supported by the Natural Science Foundation of China (Grant No. 11271008), the
First-class Discipline of University in Shanghai and the Shanghai Univ. Leading Academic Discipline Project
(A.13-0101-12-004), and the Natural Science Foundation of Shandong Province (Grant Nos. ZR2012AM021
and ZR2012AQ011).
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