Neumann and Bargmann systems associated with an extension of the coupled KdV hierarchy

8 Pages·0.1 MB·English

by  Zhimin Jiang

Journal of Nonlinear MathematicalPhysics 1999, V.6, N 1, 5–12. Letter
Neumann and Bargmann Systems Associated
with an Extension of the Coupled KdV Hierarchy
Zhimin JIANG
9
9 Department of Mathematics, Shangqiu Teachers College, Shangqiu 476000, China
9
1
Received October 16, 1998; Accepted December 03, 1998

n
a
Abstract
J

1 An eigenvalue problem with a reference function and the corresponding hierarchy

of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the
]
I hierarchy is established by using the trace identity. The isospectral problem is non-
S
linearized as to be finite-dimensional completely integrable systems in Liouville sense
.
n under Neumann and Bargmann constraints.
i
l
n
[ 1 Introduction

1
v
A major difficulty in theory of integrable systems is that there is to date no completely
1
0 systematic method for choosing properly an isospectral problem ψx = Mψ so that the
2 zero-curvature representation M −N +[M,N]= 0 is nontrivial. By inserting a reference
t x
1
function into AKNS and WKI isospectral problems, we have obtained successfully two
0
9 new hierarchies [1, 2].
9 The coupled KdV hierarchy associated with the isospectral problem
/
n
1 1
i
l − λ+ u −v
n 2 2
: ψx = Mψ, M =   (1.1)
v 1 1
i  1 λ− u 
X  2 2 
 
r
a is discussed by D. Levi, A. Sym and S. Wojciechowsk [3]. The isospectral problem (1.1)
has been nonlinearized as finite-dimensional completely integrable systems in Liouville
sense [4].
In this paper, we introduce the eigenvalue problem
1 1
− λ+ u −v
2 2
ψx = Mψ, M =  , (1.2)
1 1
f(v) λ− u
 
2 2
 
 
Copyright (cid:13)c 1999 by Z. Jiang
6 Z. Jiang
where u and v are two scalar potentials, λ is a constant spectral parameter and f(v)
called reference function is an arbitrary smooth function. The bi-Hamiltonian structure
of the corresponding hierarchy is established by using the trace identity [5, 6]. Since the
reference function f(v) in (1.2) can be chosen arbitrarily, many new hierarchies and their
Hamiltonian forms are obtained.When f = (−v)β (β ≥ 0), the isospectral problem (1.2) is
nonlinearizedasfinite-dimensionalcompletely integrable sysstemsinLiouvillesenseunder
Neumann and Bargmann constraints between the potentials and eigenfunctions.
2 Preliminaries
Consider the adjoint representation of (1.2)
∞
a b a b
N = MN −NM, N = = i i λ−j (2.1)
x c −a c −a
i i
(cid:18) (cid:19) j=0(cid:18) (cid:19)
X
which leads to
1
c = b = 0, a = − α (constant), (2.2)
0 0 0
2
c = αf(v), b = −αv, a = 0, (2.3)
1 1 1
c = α(f′(v)v +uf(v)), b = α(v −uv), a = −αvf(v), (2.4)
2 x 2 x 2
a = −∂−1(vc +f(v)b ), (2.5)
j j j
c f(v) c c
1 = α , j+1 = L j , j = 1,2,..., (2.6)
b −v b b
1 j+1 j
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
d
where ∂ = , ∂∂−1 = ∂−1∂ =1,
dx
∂+u+2f∂−1v 2f∂−1f
L = .
−2v∂−1v −∂+u−2v∂−1f !
It is easy from (1.2) and (2.1) to calculate that
∂M ∂M ∂M
tr N = −a, tr N = a, tr N = −c+f′(v)b.
∂λ ∂u ∂v
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Noticing the trace identity [5, 6]
δ δ ∂
, (−a)= (a,−c+f′(v)b),
δu δv ∂λ
(cid:18) (cid:19)
hence we deduce that
δ δ a
(1) (2) j+1
, H = G ,G , H = , (2.7)
δu δv j j−2 j−2 j
(cid:18) (cid:19)
(cid:16) (cid:17)
where
G(1) = a , G(2) = −c +f′(v)b . (2.8)
j−2 j j−2 j j
Neumann and Bargmann Systems and the Coupled KdV Hierarchy 7
3 The hierarchy and its Hamiltonian structure
Let ψ satisfy the isospectral problem (1.2) and the auxiliary problem
A B
ψ = Nψ, N = , (3.1)
t C −A
(cid:18) (cid:19)
where
m−1 m m
A = A + a λm−j, B = b λm−j, C = c λm−j.
m j j j
j=0 j=1 j=1
X X X
The compatible condition ψ = ψ between (1.1) and (3.1) gives the zero-curvature
xt tx
representation M −N +[M,N]= 0, from which we have
t x
A = w(∂ +u)c +wf′(v)(∂ −u)b ,
m m m
u c c
t = θ L m = θ m+1 , (3.2)
v 0 b 0 b
t m m+1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1
where w = (vf′(v)+f)−1,
2
2∂w −2∂wf′(v)
θ = . (3.3)
0 2wv 2wf
(cid:18) (cid:19)
By (2.6) we know that Eqs.(3.2) are equivalent to the hierarchy of nonlinear evolution
equations
u αf(v)
t = θ Lm , m = 1,2,.... (3.4)
v 0 −αv
t
(cid:18) (cid:19) (cid:18) (cid:19)
Let the potentials u and v in (1.2) belong to the Schwartz space S(−∞,+∞) over
(−∞,+∞). Noticing (2.5) and (2.8) we get
(1)
cj = θ Gj−2 , θ = −2wf′(v)∂ −2wf . (3.5)
(cid:18) bj (cid:19) 1 G(j2−)2  1 (cid:18) −2w∂ 2wv (cid:19)
 
Then the recursion relations (2.5), (2.6) and the hierarchy (3.2) can be written as
1
G = − α(1,0)T, G = −α(0,vf′(v)+f)T, G = −α(vf,uf +uvf′(v))T,
−2 −1 0
2
KG = JG , (3.6)
j−1 j
(u ,v )T = JG = KG , (3.7)
t t m−1 m−2
where J = θ θ and K =θ Lθ are two skew-symmetric operators,
0 1 0 1
0 −2∂w K K
J = , K = 11 12 ,
−2w∂ 0 K K
21 22
(cid:18) (cid:19) (cid:18) (cid:19)
8 Z. Jiang
in which
K = −2∂−4∂w(∂f′(v)+f′(v)∂)w∂,
11
K = −2∂wu+4∂w(f′(v)∂v−∂f)w,
 12
 K21 = −2wu∂ +4w(f∂ −v∂f′(v))w∂,

K = −4w(v∂f +f∂v)w.
22



From (2.7) we obtain the desired bi-Hamiltonian form of (3.7)
δ δ
ut = J δu H = K δu H . (3.8)
v  δ  m+1  δ  m
t
(cid:18) (cid:19)
 δv   δv 
   
4 Nonlinearization of the isospectral problem
Let λ and ψ(x) = (q (x),p (x))T be eigenvalue and the associated eigenfunction of (1.2).
j j j
δλ δλ
j j
Through direct verification we know that the functional gradient ∇ λ = ,
(u,v) j δu δv
(cid:18) (cid:19)
satisfies
∇ λ = q p ,−p2−f′(v)q2 , (4.1)
(u,v) j j j j j
(cid:0) (cid:1)
p2 p2 p2
θ ∇λ = j , L j = λ j (4.2)
1 j −q2 −q2 j −q2
(cid:18) j (cid:19) (cid:18) j (cid:19) (cid:18) j (cid:19)
in view of (1.2). Substituting the first expression of (4.2) into the second expression and
acting with θ upon once, we have
0
K∇λ = λ J∇λ . (4.3)
j j j
So, the Lenard operator pair K, J and their gradient series G satisfy the basic conditions
j
(3.6) and(4.3) given inRefs.[7, 8]forthenonlinearization oftheeigenvalue problem(1.2).
Proposition 4.1. When f(v) = (−v)β (β ≥ 0), the isospectral problem (1.2) can be
nonlinearized as to be a Neumann system.
N
In fact, the Neumann constraint G | = ∇λ gives
−1 α=1 j
j=1
P
hq,pi = 0,hp,pi = (β +1)(−v)β +β(−1)β−1hq,qi. (4.4)
By differentiating (4.4) with respect to x and using (1.2), we have
1 hΛp,pi hΛq,qi
u= +β ,
β+1 hp,pi hq,qi
(4.5)
 (cid:18) (cid:19)

 v = hq,qi.


Neumann and Bargmann Systems and the Coupled KdV Hierarchy 9
Substituting (4.5) into the equations for the eigenfunctions
1 1
− λ + u −v
qjx = 2 j 2 qj , j = 1,...,N, (4.6)
p  1 1  p
(cid:18) jx (cid:19) (−v)β λ − u (cid:18) j (cid:19)
j
 2 2 
 
we obtain the Neumann system
1 1 hΛp,pi hΛq,qi
q = − Λq−hq,qip+ +β q,
x
2 2(β +1) hp,pi hq,qi
 (cid:18) (cid:19)
 1 1 hΛp,pi hΛq,qi (4.7)
 px = Λp+hp,piq− +β p,
 2 2(β +1) hp,pi hq,qi

(cid:18) (cid:19)
 hp,pi = (−1)βhq,qiβ, hq,pi = 0.




wherep = (p ,...,p )T, q = (q ,...,q )T, Λ = diag(λ ,...,λ ), and h,i stands for the
1 N 1 N 1 N
canonical inner product in RN.
Proposition 4.2. When f(v) = (−v)β (β ≥ 0), the isospectral problem (1.2) can be
nonlinearized as to be a Bargmann system.
N
In fact, the Bargmann constraint G | = ∇λ gives
0 α=1 j
j=1
P
1 − β β − 1
u= hp,pihq,pi β+1 − hq,qihq,pi β+1,
β+1 β +1
(4.8)

 1
 v = −hq,piβ+1.
Substituting (4.8) into (4.6), we obtain the finite-dimensional Hamiltonian system
1 1 1 − β
qx = − Λq+hq,piβ+1p+ hp,pihq,pi β+1q
2 2(β +1)

 − β hq,qihq,pi−β+11q = ∂H,
 2(β +1) ∂p

 (4.9)

 px = 1Λp− 1 hp,pihq,pi−ββ+1p+hq,piββ+1
2 2(β +1)

 + β hq,qihq,pi−β+11p = −∂H.
 2(β +1) ∂q



The Hamiltonian is

1 1 1 1 β
H = − hΛq,pi+ hp,pihq,piβ+1 − hq,qihq,piβ+1.
2 2 2
5 Integrability of the Neumann system
The Poisson brackets of two functions in symplectic space (R2N,dp∧dq) are defined as
N
∂F ∂G ∂F ∂G
(F,G) = − = hF ,G i−hF ,G i.
q p p q
∂q ∂p ∂p ∂q
j j j j
j=1(cid:18) (cid:19)
X
10 Z. Jiang
The functions defined by (m = 0,1,2,...)
1 1 hΛiq,qi hΛiq,pi
F = − hΛm+1q,pi−
m 2 2 (cid:12) hΛjp,qi hΛjp,pi (cid:12)
i+Xj=m(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
are in involution in pairs (see, [9]). (cid:12) (cid:12)
Consider the Moser constraint on the tangent bundle
1
TSN−1 = (p,q) ∈ R2N|F = hq,pi = 0, G = (hp,pi−(−1)βhq,qiβ)= 0 .
2(β +1)
(cid:26) (cid:27)
Through direct calculations we have
(F,F ) = 0, (F,G) = hp,pi,
m
1
(F ,G) = − hΛm+1p,pi+(−1)ββhq,qiβ−1hΛm+1q,qi .
m
2(β +1)
(cid:16) (cid:17)
Thus the Lagrangian multipliers are
(F ,G) 1 hΛm+1p,pi hq,qiβ−1
µ = m = − +(−1)ββ hΛm+1q,qi .
m
(F,G) (β+1) hp,pi hp,pi
(cid:18) (cid:19)
Since F = 0 on the tangent bundle TSN−1, the restriction of the canonical equation of
H∗ = F −µ F on TSN−1 is
0 0
qx = F0,p−µ0Fp|TSN−1,
( px = −F0,q +µ0Fq|TSN−1
which is exactly the Neumann system (4.7).
Theorem 5.1. The Neumann system (4.7) (TSN−1,dp∧dq|TSN−1,H∗ = F0 −µ0F) is
completely integrable in Liouville sense.
Proof. Let F∗ = F −µ F, m = 1,...,N −1, then it is easy to verify (F∗,F∗) = 0 on
m m m k l
TSN−1. Hence {F∗} is an involutive system.
m
6 Integrability of the Bargmann system
Let
N B2
kj
Γ = , (6.1)
k
λ −λ
k j
jX=1
j6=k
where B = p q −p q , we have (see Refs. [9, 10])
kj k j j k
Lemma 6.1.
hq,pi,p2 = 2p2, hq,pi,q2 = −2q2, (6.2)
l l l l
(cid:0) (cid:1) (cid:0) (cid:1)
Neumann and Bargmann Systems and the Coupled KdV Hierarchy 11
−4B −4B
p2,Γ = lk p p , q2,Γ = lk q q ,
k l λ −λ k l k l λ −λ k l
l k l k
(6.3)
(cid:0) (cid:1) (cid:0) (cid:1)
−2B
lk
(q p ,Γ )= (p q +q p ).
k k l k l k l
λ −λ
l k
Lemma 6.2.
(Γ ,Γ ) = (hq,pi,Γ ) = (hq,pi,q p )= 0, (6.4)
k l l l l
p2,p2 = q2,q2 =(q p ,q p ) = 0, (6.5)
k l k l k k l l
(cid:0)q p ,p(cid:1)2 =(cid:0) 2p p(cid:1)δ , q2,p2 = 4q p δ , q2,p q =2q q δ . (6.6)
k k l k l kl k l k l kl k l l k l kl
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Proposition 6.1. Let
1 1 1 β 1 1
Ek = 2hq,piβ+1p2k − 2hq,piβ+1qk2− 2λkqkpk − 2Γk,
the E ,...,E constitute an N-involutive system.
1 N
Proof. Obviously (E ,E ) = 0 for k = l. Suppose k 6= l, in virtue of (6.4)–(6.6) and the
k l
property of Poisson bracket in (R2N,dp∧dq), we have
1 1−β 1 1−β
4(Ek,El) = β+1p2khq,piβ+1 hq,pi,p2l + β +1p2lhq,piβ+1 p2k,hq,pi
(cid:0) (cid:1) (cid:0) (cid:1)
1 β 1
− β+1p2k hq,pi,ql2 − β+1ql2 p2k,hq,pi −hq,piβ+1 p2k,Γl
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1 β 1
−hq,piβ+1 Γk,p2l − β+1qk2 hq,pi,p2l − β +1p2l qk2,hq,pi
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
β β−1 β β−1
+ q2hq,piβ+1 hq,pi,q2 + q2hq,piβ+1 q2,hq,pi
β+1 k l β +1 l k
(cid:0) (cid:1) (cid:0) (cid:1)
β β
+hq,piβ+1 qk2,Γl +hq,piβ+1 Γk,ql2 +λk(qkpk,Γl)+λl(Γk,qlpl).
Substituting (6.2) and (6.3) i(cid:0)nto th(cid:1)e above equa(cid:0)tion yi(cid:1)elds (Ek,El)= 0.
Consider a bilinear function Q (ξ,η) on RN:
z
N ∞
ξ η
Q (ξ,η) = h(z−Λ)−1ξ,ηi = k k = z−m−1hΛmξ,ηi.
z
z−λ
k
k=1 m=0
X X
The generating function of Γ is (see, [9, 10])
k
N
Qz(q,q) Qz(q,p) Γk
= .
(cid:12) Qz(p,q) Qz(p,p) (cid:12) z−λk
(cid:12) (cid:12) Xk=1
(cid:12) (cid:12)
Hence(cid:12)the generating funct(cid:12)ion of Ek is
(cid:12) (cid:12)
1 1 1 β 1
hq,piβ+1Qz(p,p)− hq,piβ+1Qz(q,q)− Qz(Λq,p)
2 2 2
(6.7)
N
1 Qz(q,q) Qz(q,p) Ek
− = .
2(cid:12) Qz(p,q) Qz(p,p) (cid:12) z−λk
(cid:12) (cid:12) Xk=1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
12 Z. Jiang
Substituting the Laurent expansion of Q and
z
∞
(z−λ )−1 = z−m−1λm
k k
m=0
X
in to both sides of (6.7) respectively, we have
Proposition 6.2. Let
N
F = λmE , m = 0,1,2,...
m k k
k=1
X
then
1 1 1 β 1
F0 = hq,piβ+1hp,pi− hq,piβ+1hq,qi− hΛq,pi,
2 2 2
1 1 1 β
Fm = hq,piβ+1hΛmp,pi− hq,piβ+1hΛmq,qi
2 2
1 1 m hΛj−1q,qi hΛj−1q,pi
− hΛm+1q,pi− .
2 2 (cid:12) hΛm−jp,qi hΛm−jp,pi (cid:12)
Xj=1(cid:12) (cid:12)
(cid:12) (cid:12)
Moreover, (F ,F ) = 0. (cid:12) (cid:12)
k l (cid:12) (cid:12)
Hence we arrive at the following theorem.
Theorem 6.1. The Bargmann system defined by (4.9) is completely integrable in Liouville
sense in the symplectic manifold (R2N,dp∧dq).
Acknowledgement
I am very grateful to Professor Cao Cewen for his guidance. This project is supported by
the Natural Science Fundation of China.
References
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