Table Of ContentA NEW METHOD TO INTRODUCE ADDITIONAL SEPARATED
VARIABLES FOR HIGH-ORDER BINARY CONSTRAINED FLOWS
0
0 Yunbo Zeng1
0
2
Department of Mathematical Sciences, Tsinghua University
n
Beijing 100084, China
a
J
0
1
Abstract. Degrees of freedom for high-order binary constrained flows of soliton
]
I equations admitting 2×2 Lax matrices are 2N +k . It is known that N +k pairs of
S 0 0
. canonical separated variables for their separation of variables can be introduced directly
n
i via their Lax matrices. In present paper we propose a new method to introduce the ad-
l
n ditional N pairs of canonical separated variables and N additional separated equations.
[
The Jacobi inversion problems for high-order binary constrained flows and for soliton
1 equations are also established. This new method can be applied to all high-order binary
v
0 constrained flows admitting 2×2 Lax matrices.
1
0
Keywords: separation of variables, Jacobi inversion problem, high-order binary
1
0 constrained flow, Lax representation, factorization of soliton equations.
0
0
/
n
i
l
n Classification numbers: 02.90.+p
:
v
i
X
r
a
1E-mail:[email protected]
Typeset by AMS-TEX
1
2 YUNBO ZENG
1. Introduction.
For a finite-dimensional integrable Hamiltonian systems (FDIHS), let m denote the
number of degrees of freedom, and P ,i = 1,...,m, be functionally independent inte-
i
grals of motion in involution, the separation of variables means to construct m pairs of
canonical separated variables v ,u ,k = 1,...,m, [1,2,3]
k k
{u ,u } = {v ,v } = 0, {v ,u } = δ , k,l = 1,...,m, (1.1)
k l k l k l kl
and m functions f such that
k
f (u ,v ,P ,...,P ) = 0, k = 1,...,m, (1.2)
k k k 1 m
which are called separated equations. The equations (1.2) give rise to an explicit fac-
torization of the Liouville tori. For the FDIHSs with the Lax matrices admitting the
r-matrices of the XXX,XXZ and XYZ type, there is a general approach to their sep-
aration of variables [1-6]. The corresponding separated equations enable us to express
the generating function of canonical transformation in completely separated form as an
abelian integral on the associated invariant spectral curve. The resulted linearizing map
is essentially the Abel map to the Jacobi variety of the spectral curve, thus providing a
link with the algebro-geometric linearization methods given by [7-9].
The separation of variables for a FDIHS requires that the number of canonical sep-
arated variables u should be equal to the number m of degrees of freedom. In some
k
cases, the number of u resulted by the normal method may be less than m and one
k
needs to introduce some additional canonical separated variables. So far very few mod-
els in these cases have been studied. These cases remain to be a challenging problem
[3].
The separation of variables for constrained flows of soliton equations has been stud-
ied (see, for example, [4,10-14]). In recent years binary constrained flows of soliton
hierarchies have attracted attention (see, for example, [15-22]). However the separation
of variables for binary constrained flows has not been studied. The degree of freedom
A(λ) B(λ)
forhigh-orderbinaryconstrainedflowsadmitting2×2LaxmatricesM =
C(λ) −A(λ)
(cid:18) (cid:19)
is a natural number 2N+k . Via the Lax matrix M, N+k pairs of canonical separated
0 0
variables u ,...,u can be introduced by the set of zeros of B(λ) and v = A(u ),
1 N+k0 k k
and N +k separated equations can be found from the generation function of intrgrals
0
of motions. In previous papers [23,24] we pesented a method with two different ways for
determining additional N pairs of canonical separated variables and additional N sepa-
rated equations for first binary constrained flows with 2N degree of freedom. The main
idea in [23,24] is to construct two functions B(λ) and A(λ) and define u ,...,u
N+1 2N
by the set of zeros of B(λ) and v = A(u ). The ways for constructing B(λ)
N+k N+k
and A(λ) in [23] and [24] are some different. Inepresent peaper we propose a completely
different method from tehat in [23,24] to intreoduce the additional N separated variaebles
and Ne separated equations for high-order binary constrained flows with 2N + k de-
0
gree of freedom. It is observed that the introduction of v has some link with integrals
k
A METHOD TO INTRODUCE ADDITIONAL SEPARATED VARIABLES 3
of motion and should lead to the separated equations. We find that there are N in-
tegrals of motion P ,...,P among the 2N + k integrals of motion for the
N+k0+1 2N+k0 0
high-order binary constrained flows which commute with A(λ) and B(λ). This ob-
servation and property stimulate us to directly use the additional integrals of motion
to define both the N pairs of additional separated variables and N separated equa-
tions by v = P ,j = 1,...,N. Then we can find the conjugated variables
N+k0+j N+k0+j
u ,1,...,N,commuting with A(λ) and B(λ). In contrast to the method in [23,24],
N+k0+j
this method is easer to be applied to the high-order binary constrained flows.
Wewillalsopresenttheseparationofvariablesofsolitonequations. Thefirststepisto
factorize (1+1)−dimensional soliton equations into two commuting x− and t−FDIHSs
viahigh-orderbinaryconstrainedflows, namelythex−andt−dependences ofthesoliton
equations are separated by the x− and t−FDIHSs obtained from the x- and t-binary
constrained flows. The second step is to produce separation of variables for the x− and
t−FDIHDs. Finally, combining the factorization of soliton equations with the Jacobi
inversion problems for x− and t−FDIHSs enables us to establish the Jacobi inversion
problems for soliton equations. If the Jacobi inversion problem can be solved by the
Jacobi inversion technique [7], one can obtain the solution in terms of the Riemann-
theta function for soliton equations. We illustrate the method by KdV, AKNS and
Kaup-Newell (KN) hierarchies. The paper is organized as follows.
Insection2, wefirstrecallthehigh-orderbinaryconstrainedflowsandfactorizationof
the KdV hierarchy. Then propose the mehtod for introducing the N pairs of additional
separated variables. We illustrate the method by both first binary constrained flow
and second binary constrained flow. Finally we present the separation of variables for
KdV hierarchy. In section 3 and 4, the method is applied to the AKNS hierarchy and
KN hierarchy, respectively. In fact this method can be applied to all high-order binary
constrained flows and other soliton hierarchies admitting 2×2 Lax pairs.
2. Separation of variables for the KdV equations.
We first recall the high-order binary constrained flows of the KdV hierarchy.
2.1 High-order binary constrained flows of the KdV hierarchy.
Consider the Schro¨dinger equation [25]
φ +(λ+u)φ = 0
xx
which is equivalent to the following spectral problem
0 1 φ
1
φ = U(u,λ)φ, U(u,λ) = , φ = . (2.1)
x −λ−u 0 φ
(cid:18) (cid:19) (cid:18) 2(cid:19)
Take the time evolution law of φ as
φ = V(n)(u,λ)φ, (2.2)
t
n
4 YUNBO ZENG
where
n+1
a b 0 0
V(n)(u,λ) = i i λn+1−i + ,
c −a b 0
i i n+2
i=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
a = b = 0, c = −1, a = 0, b = 1,
0 0 0 1 1
1 1 1
b = Lb = − Lk−1u,, a = − b , c = − b −b −b u, k = 1,2,··· ,
k+1 k k k,x k k,xx k+1 k
2 2 2
1 1
L = − ∂2 −u+ ∂−1u , ∂ = ∂ , ∂−1∂ = ∂∂−1 = 1. (2.3)
x x
4 2
The compatibility condition of (2.1) and (2.2) gives rise to the n-th KdV equation
which can be written as an infinite-dimensional Hamiltonian system [25]
δH
u = −2b = ∂Lnu = ∂ n, (2.4)
tn n+2,x δu
with the Hamiltonian H = 4bn+3 and δHn = −2b .
n 2n+3 δu n+2
For n = 1 we have
1u λ− 1u
φ = V(1)(u,λ)φ, V(1) = 4 x 2 , (2.5)
t1 −λ2 − 1uλ+ 1u + 1u2 −1u
(cid:18) 2 4 xx 2 4 x (cid:19)
and the equation (2.4) for n = 1 is the well-known KdV equation
1
u = − (u +6uu ). (2.6)
t1 4 xxx x
The adjoint spectral problem reads
ψ
ψ = −UT(u,λ)ψ, ψ = 1 . (2.7)
x
ψ
(cid:18) 2(cid:19)
By means of the formula in [26], we have
δλ φ ψ φ ψ ∂U(u,λ)
= Tr[ 1 1 1 2 ] = −ψ φ .
δu φ ψ φ ψ ∂u 2 1
2 1 2 2
(cid:18) (cid:19)
According to [15-22], the high-order binary x-constrained flows of the KdV hierar-
chy (2.4) consist of the equations obtained from the spectral problem (2.1) and the
adjoint spectral problem (2.7) for N distinct real numbers λ and the restriction of the
j
variational
derivatives for the conserved quantities H (for any fixed k ) and λ :
k0 0 j
Φ = Φ , Φ = −ΛΦ −uΦ , (2.8a)
1,x 2 2,x 1 1
A METHOD TO INTRODUCE ADDITIONAL SEPARATED VARIABLES 5
Ψ = ΛΨ +uΨ , Ψ = −Ψ , (2.8b)
1,x 2 2 2,x 1
N
δH δλ
k0 − j = −2b + < Ψ ,Φ >= 0. (2.8c)
δu δu k0+2 2 1
j=1
X
Hereafter we denote the inner product in RN by < .,. > and
Φ = (φ ,··· ,φ )T, Ψ = (ψ ,··· ,ψ )T, i = 1,2, Λ = diag(λ ,··· ,λ ).
i i1 iN i i1 iN 1 N
Thebinary t -constrained flows of theKdV hierarchy (2.4)aredefined by thereplicas
n
of (2.2) and its adjoint system for N distinct real number λ
j
φ φ ψ ψ
1j = V(n)(u,λ ) 1j , 1j = −(V(n)(u,λ ))T 1j , j = 1,...,N,
j j
φ φ ψ ψ
(cid:18) 2j(cid:19)tn (cid:18) 2j(cid:19) (cid:18) 2j(cid:19)tn (cid:18) 2j(cid:19)
(2.9a)
as well as the n-th KdV equation itself (2.4) in the case of the higher-order constraint
for k ≥ 1
0
u = −2b . (2.9b)
tn n+2,x
(1) For k = 0, we have
0
1 1
b = − u = < Ψ ,Φ >, i.e., u = − < Ψ ,Φ > . (2.10)
2 2 1 2 1
2 2
By substituting (2.10), (2.8a) and (2.8b) becomes a finite-dimensional Hamiltonian sys-
tem (FDHS) [18]
∂F ∂F ∂F ∂F
1 1 1 1
Φ = , Φ = , Ψ = − , Ψ = − , (2.11)
1x 2x 1x 2x
∂Ψ ∂Ψ ∂Φ ∂Φ
1 2 1 2
1
F =< Ψ ,Φ > − < ΛΨ ,Φ > + < Ψ ,Φ >2 .
1 1 2 2 1 2 1
2
Under the constraint (2.10) and the x-FDHS (2.11), the binary t -constrained flow
1
obtained from (2.9a) with V(1) given by (2.5) can also be written as a t -FDHS
1
∂F ∂F ∂F ∂F
2 2 2 2
Φ = , Φ = , Ψ = − , Ψ = − , (2.12)
1,t1 ∂Ψ 2,t1 ∂Ψ 1,t1 ∂Φ 2,t1 ∂Φ
1 2 1 2
1
F = − < Λ2Ψ ,Φ > + < ΛΨ ,Φ > + < Ψ ,Φ >< ΛΨ ,Φ >
2 2 1 1 2 2 1 2 1
2
1 1
+ < Ψ ,Φ >< Ψ ,Φ > + (< Ψ ,Φ > − < Ψ ,Φ >)2.
2 1 1 2 2 2 1 1
2 8
6 YUNBO ZENG
The Lax representation for the x-constrained flow (2.8) and the t -constrained flow
n
(2.9) can be deduced from the adjoint representation of (2.1) and (2.2) by using the
method in [27,28]
M = [U,M], M = [V(n),M], (2.13)
x t
n
where U and V(n) are obtained from U and V(n) under the system (2.8), and the Lax
e e
matrix M is of the form
A(λ) B(λ)
e e
M = .
C(λ) −A(λ)
(cid:18) (cid:19)
The Lax matrix M for x-FDHS (2.11) and t -FDHS (2.12) is given by
1
N N
1 ψ φ −ψ φ 1 ψ φ
1j 1j 2j 2j 2j 1j
A(λ) = , B(λ) = 1+ ,
4 λ−λ 2 λ−λ
j j
j=1 j=1
X X
N
1 1 ψ φ
1j 2j
C(λ) = −λ+ < Ψ ,Φ > + . (2.14)
2 1
2 2 λ−λ
j
j=1
X
The generating function of integrals of motion for (2.11) and (2.12) yields
N P P2
A2(λ)+B(λ)C(λ) ≡ P(λ) = −λ+ [ j + N+j ], (2.15)
λ−λ (λ−λ )2
j j
j=1
X
where P ,...,P are independent integrals of motion for the FDHSs (2.11) and (2.12)
1 2N
1 1 1
P = ψ φ +(− λ + < Ψ ,Φ >)ψ φ
j 1j 2j j 2 1 2j 1j
2 2 4
1 1
+ [(ψ φ −ψ φ )(ψ φ −ψ φ )+4ψ φ ψ φ ], (2.16a)
1j 1j 2j 2j 1k 1k 2k 2k 1j 2j 2k 1k
8 λ −λ
j k
k6=j
X
1
P = (ψ φ +ψ φ ), j = 1,...,N. (2.16b)
N+j 1j 1j 2j 2j
4
It is easy to verify that
N N
F = 2 P , F = 2 (λ P +P2 ). (2.17)
1 j 2 j j N+j
j=1 j=1
X X
With respect to the standard Poisson bracket
N
∂f ∂g ∂f ∂g ∂f ∂g ∂f ∂g
{f,g} = ( + − − ), (2.18)
∂ψ ∂φ ∂ψ ∂φ ∂φ ∂ψ ∂φ ∂ψ
1j 1j 2j 2j 1j 1j 2j 2j
j=1
X
A METHOD TO INTRODUCE ADDITIONAL SEPARATED VARIABLES 7
by calculating the formulas like (2.31), it is easy to verify that
{A2(λ)+B(λ)C(λ),A2(µ)+B(µ)C(µ)} = 0, (2.19)
which implies that P ,..,P are in involution, (2.11) and (2.12) are FDIHSs and com-
1 2N
mutewitheachother. Theconstructionof(2.11)and(2.12)ensuresthatif(Ψ ,Ψ ,Φ ,Φ )
1 2 1 2
satisfies the FDIHSs (2.11) and (2.12) simultaneously, then u defined by (2.10) solves
the KdV equation (2.6).
Set
∞
A2(λ)+B(λ)C(λ) = λ F λ−k, (2.20)
k
k=0
X
e
where F ,k = 1,2,..., are also integrals of motion for both the x-FDHSs (2.11) and the
k
t -binary constrained flows (2.9). Comparing (2.20) with (2.15), one gets
n
e
N
F = −1, F = 0, F = [λk−2P +(k −2)λk−3P2 ], k = 2,3,.... (2.21)
0 1 k j j j N+j
j=1
X
e e e
By employing the method in [28,29], the t -FDIHS obtained from the t -binary con-
n n
strained flow (2.9) is found to be of the form
∂F ∂F ∂F ∂F
n+1 n+1 n+1 n+1
Φ = , Φ = , Ψ = − , Ψ = − , (2.22a)
1,tn ∂Ψ 2,tn ∂Ψ 1,tn ∂Φ 2,tn ∂Φ
1 2 1 2
n
1 α
F = ( )m−1 m F ...F , (2.22b)
n+1 2 m+1 l1 lm+1
mX=0 l1+...+Xlm+1=n+2
e e
where l ≥ 1,...,l ≥ 1,α = 1,α = 1,α = 3, and [28,29]
1 m+1 0 1 2 2 2
m−2 m−1
1
α = 2α + α α − α α , m ≥ 3. (2.22c)
m m−1 l m−l−1 l m−l
2
l=1 l=1
X X
The n-th KdV equation (2.4) is factorized by the x-FDIHS (2.11) and the t -FDIHS
n
(2.22).
(2) For k = 1, one gets
0
1 1
b = (u +3u2) = < Ψ ,Φ > . (2.23)
3 xx 2 1
8 2
By introducing q = u,p = 1u , (2.8a), (2.8b) and (2.23) can be written as a x-FDHS
4 x
∂F ∂F ∂F ∂F
1 1 1 1
Φ = , Ψ = − , i = 1,2, q = , p = − , (2.24)
ix ix x x
∂Ψ ∂Φ ∂p ∂q
i i
8 YUNBO ZENG
1
F = − < ΛΨ ,Φ > + < Ψ ,Φ > −q < Ψ ,Φ > +2p2 + q3.
1 2 1 1 2 2 1
4
Under the constraint (2.23), V(1) becomes
p λ− 1q
V(1) = 2 . (2.25)
−λ2 − 1qλ+ < Ψ ,Φ > −1q2 −p
(cid:18) 2 2 1 4 (cid:19)
e
Under the constraint (2.23) and the x-FDHS (2.24), the binary t -constrained flow
1
consists of (2.9a) with V(1) replaced by V(1) and (2.9b) given by (2.6) can also be
written as a t -FDHS
1
e
∂F ∂F ∂F ∂F
2 2 2 2
Φ = , Ψ = − , i = 1,2, q = , p = − , (2.26)
it1 ∂Ψ it1 ∂Φ t1 ∂p t1 ∂q
i i
1 1
F = − < Λ2Ψ ,Φ > + < ΛΨ ,Φ > − q < ΛΨ ,Φ > − q < Ψ ,Φ >
2 2 1 1 2 2 1 1 2
2 2
1 1
+p < Ψ ,Φ > −p < Ψ ,Φ > + < Ψ ,Φ >2 − q2 < Ψ ,Φ > .
1 1 2 2 2 1 2 1
2 4
The Lax representations for the x-FDHS (2.24) and the t -FDHS (2.26), which can
1
can be deduced from the adjoint representation of (2.1) and (2.2), are given by (2.13)
with V(1) defined by (2.25) and U obtained from U by using q instead of u as well as
M given by
e e
N N
1 ψ φ −ψ φ 1 1 ψ φ
1j 1j 2j 2j 2j 1j
A(λ) = p+ , B(λ) = λ− q + ,
4 λ−λ 2 2 λ−λ
j j
j=1 j=1
X X
N
1 1 1 1 ψ φ
C(λ) = −λ2 − qλ+ < Ψ ,Φ > − q2 + 1j 2j. (2.27)
2 1
2 2 4 2 λ−λ
j
j=1
X
The generating function of integrals of motion for (2.24) and (2.26) yields
N P P2
A2(λ)+B(λ)C(λ) ≡ P(λ) = −λ3 +P + [ j + N+j ], (2.28)
0 λ−λ (λ−λ )2
j j
j=1
X
where P ,...,P are independent integrals of motion for the FDHSs (2.24) and (2.26)
0 2N
and P = 1F ,
0 2 1
1 1 1 1
P = − λ2ψ φ + λ ψ φ − λ qψ φ − qψ φ
j 2 j 2j 1j 2 j 1j 2j 4 j 2j 1j 4 1j 2j
1 1 1
+ p(ψ φ −ψ φ )+ (< Ψ ,Φ > − q2)ψ φ
1j 1j 2j 2j 2 1 2j 1j
2 4 2
A METHOD TO INTRODUCE ADDITIONAL SEPARATED VARIABLES 9
1 1
+ [(ψ φ −ψ φ )(ψ φ −ψ φ )+4ψ φ ψ φ ], (2.29a)
1j 1j 2j 2j 1k 1k 2k 2k 1j 2j 2k 1k
8 λ −λ
j k
k6=j
X
1
P = (ψ φ +ψ φ ), j = 1,...,N. (2.29b)
N+j 1j 1j 2j 2j
4
We have
N
F = 2P , F = 2 P . (2.30)
1 0 2 j
j=1
X
Similarly, it can be shown that (2.24) and (2.26) are FDIHSs and commute with each
other. The KdV equation (2.6) is factorized by x-FDIHS (2.24) and t -FDIHS (2.26). If
1
(Ψ ,Ψ ,p,Φ ,Φ ,q) satisfies the FDIHSs (2.24) and (2.26) simultaneously, then u = q
1 2 1 2
solves the KdV equation (2.6).
2.2 The separation of variables for the KdV equations.
(1) For the case k = 0, we first consider the separation of variables for FDIHSs
0
(2.11) and (2.12). With respect to the standard Poisson bracket (2.18), it is found that
for the A(λ) and B(λ) given by (2.14) we have
1
{A(λ),A(µ)} = {B(λ),B(µ)} = 0, {A(λ),B(µ)} = [B(µ)−B(λ)]. (2.31)
2(λ−µ)
An effective way to introduce the separated variables v ,u and to obtain the separated
k k
equations is to use the Lax matrix M and the generating function of integrals of motion.
The commutator relations (2.31) and the equation (2.15) enable us to define the first
N pairs of the canonical variables u ,...,u by the set of zeros of B(λ) [1-3]
1 N
N
1 ψ φ R(λ)
2j 1j
B(λ) = 1+ = , (2.32a)
2 λ−λ K(λ)
j
j=1
X
where
N N
R(λ) = (λ−u ), K(λ) = (λ−λ ),
k k
k=1 k=1
Y Y
and v ,...,v by
1 N
v = 2A(u ), k = 1,...,N. (2.32b)
k k
The commutator relations (2.31) guarantee that u ,...,u and v ,...,v satisfy the
1 N 1 N
canonical conditions (1.1) [1-3]. Then substituting u into (2.15) gives rise to the first
k
N separated equations
N P P2
j N+j
v = 2A(u ) = 2 P(u ) = 2 −u + [ + ], k = 1,...,N.
k k k k
v u −λ (u −λ )2
u j=1 k j k j
p u X
t (2.33)
10 YUNBO ZENG
The FDIHSs (2.11) and (2.12) have 2N degrees of freedom, we need to introduce the
other N pairs of canonical variables v ,u ,k = N +1,...,2N. Notice that P given
k k N+j
by (2.16b) are integrals of motion for the FDIHSs (2.11) and (2.12), and satisfy
{B(λ),P } = {A(λ),P } = 0. (2.34)
N+j N+j
Thus we may define
1
v = 2P = (ψ φ +ψ φ ), j = 1,...,N, (2.35a)
N+j N+j 1j 1j 2j 2j
2
which also give rise to the separated equations. It is easy to see that if we take
φ
1j
u = ln , j = 1,...,N, (2.35b)
N+j
ψ
2j
then
{v ,u } = δ , {v ,v } = {u ,u } = 0, j,k = 1,...,N, (2.36)
N+j N+k jk N+j N+k N+j N+k
{B(λ),u } = {A(λ),u } = {B(λ),v } = {A(λ),v } = 0. (2.37)
N+j N+j N+j N+j
We have the following proposition.
Proposition 1. Assume that λ ,φ ,ψ ∈ R,i = 1,2,j = 1,...,N. Introduce the
j ij ij
separated variables u ,...,u and v ,...,v by (2.32) and (2.35). If u ,...,u , are
1 2N 1 2N 1 N
single zeros of B(λ), then v ,...,v and u ,...,u are canonically conjugated, i.e.,
1 2N 1 2N
they satisfy (1.1).
Proof. By following the similarmethod in [1-6,23,24], it is easy toshow that v ,...,v
1 N
and u ,...,u satisfy (1.1). Notice B′(u ) 6= 0. Hereafter the prime denotes the differ-
1 N k
entiation with respect to λ. It follows from (2.36) and (2.37) that
0 = {u ,B(u )} = B′(u ){u ,u }+{u ,B(µ)}| = B′(u ){u ,u },
N+k l l N+k l N+k µ=ul l N+k l
{v ,u } = 2{A(u ),u }
k N+l k N+l
= 2A′(u ){u ,u }+{A(λ),u }| = 2A′(u ){u ,u }, (2.38)
k k N+l N+l λ=uk k k N+l
which leads to {u ,u } = {u ,v } = 0. Similarly we can show that {v ,u } =
N+k l N+k l N+k l
{v ,v } = 0. These together with (2.36) complete the proof.
N+k l
It follows from (2.32a) and (2.35b) that
N
u = − < Ψ ,Φ >= 2 (u −λ ), (2.39)
2 1 j j
j=1
X
