-w

ε=0
R Schmid, T Xu and Z Li
(3)
−δw δu δv δw
.
Notice that Am−1 is an arbitrary function. Xm ≡ JLm−1 is the mth order vector field of the evolution equation hierarchy w u = Xm , m = 1, 2, . . . . (4) v t
262 Lemma 1. The commutator [M, Nj ] = M Nj − Nj M satisfies: [M, Nj ] = Njx + M∗ (KLj ) − ζM∗ (JLj ), where M∗ (δw, δu, δv ) = ∂ ∂ε M (w +εδw, u+εδu, v +εδv ) = ζ
Journal of Nonlinear Mathematical Physics
2001, V.8, Supplement, 261–265
Proceedings: NEEDS’99
A Finite Dimensional Completely Integrable System Associated with the WKIand Heisenberg Hierarchies
Lemma 2. Let
q p
be an involutive solution of (H0 ) and (Hm )
−1
(u, v )T = Aq, p
( Aq, q , − Ap, p )T = f (q, p),
then there exist constants α0 , α1 , . . . , αm−2 such that
2 −ζj y1 j −1
γj =
Ω N
2 −vy1 j
− wy1j y2j +
2 uy2 j
dx
.
Let G0 =
j =1
grad ζj . We get the constraint
α α α w = − Aq, p , u = Aq, q , v = − Ap, p , r r r where p = (y11 , .√ . . , y1N )T , q = (y21 , . . . , y2N )T and A = diag (ζ1 , . . . , ζN ). In the case of Heisenberg, r = uv + w2 = 1, the constraint becomes u = Aq, q , v = − Ap, p . (6) √ √ In the case of WKI, w = 1, r = w2 + uv = 1 + uv , we have the following constraint: u = Aq, p
m−1 j =0
Aj +1 q, q Am−j p, q
Aj +1 q, p Am−j p, p
+ Am q, p H0
= Gm + Am q, p H0 ,
m = 2, 3, . . . , N.
where {Gm } is a confocal involutive system, Cao [2] , Moser [5]. The level sets are Mf = (q, p) ∈ R2N |H0 (q, p) = 0, Hm (q, p) = fm , m = 2, 3, . . . N .
−1
w = − Aq, p ,
Aq, q ,
v = Aq, p
−1
Ap, p .
(7)
Now we nonlinearize the Lax pairs by plugging the constraint (7) into the Lax pairs. Then (1) becomes the restricted flow px = −wAp + uAp, px = −Ap + Aq, p
Theorem 2. The Hamiltonian system (10) is completely integrable (in the Liouville sense) with the following N functions in involution: H0 , H2 , . . . HN Hm = − 1 2
Abstract We consider the following spectral problem y1 y2 =ζ
x
‡ 1 2
−w v

(1)
where u, v , w are smooth functions. It produces a hierarchy of evolution equations with an arbitrary function Am−1 . This hierarchy includes the WKI [8] and Heisenberg [7] hierarchies by properly selecting the special function Am−1 . We derive this new evolution equations, and give the finite dimensional completely integrable systems (FDCIS) associated with theses equations.
m
From Lemma 1 and the fact that KL0 = 0 we get the following:
Theorem 1. Let {Lj }, j = 0, 1, 2, . . . be a Lenard sequence. Each of the vector fields Xm (m = 1, 2, . . .) has a commutator representation: M∗ (Xm ) = Vmx + [Vm , M ], where
be the smooth matrix functions. Set the pair 1 0 v −u K = 2u −2w 0 , J = 0 0 −2v 0 2w
j = 0, 1, 2, . . . m − 2.
Copyright c 2001 by R Schmid, T Xu and Z Li
264
R Schmid, T Xu and Z Li
3
The relation between the higher order WKI and Heisenberg equations and the system (H0 )
q p q (x, tm ) p(x, tm ) q (0, 0) p(0, 0) q (0, 0) p(0, 0)
Rudolf SCHMID
† †1 ,
Taixi XU
†2
and Zhongding LI
‡
Department of Mathematics and Computer Science, Emory University Atlanta, GA 30322, USA Department of Basic Courses, Shijiazhuang Railway Institute, China E-mail: rudolf@ E-mail: taixi@
Define the involutive solution of H0 and Hm : ≡
x tm = g0 ◦ gm
,
∈ Mf ,
x and g tm are respectively the phase flows with Hamiltonian function (H ) where g0 0 m and (Hm ).
1
Let
The hierarchy and the Lax pair
−w u v w Aj Cj Bj −Aj of Lenard’s operators as 0 0 1 0 ∂x , 0 1
M =ζ
,
Nj =
and define the Lenard sequence {Lj } (j = 0, 1, 2, . . . , m − 1) recursively: Lj = (−Aj , Bj , Cj )T with w w α u ≡ α u , L0 = √ 2 r w + uv v v JLj = KLj +1 , (α = 0 constant), (2)
−1
qx = vAp + wAq. Aq, q Aq, qx = Aq − Aq, p
−1
(8)
Now plugging in the constraint (7) into (8), we get Ap, p Ap. (9)
The equation (9) is the nonlinearization of the eigenvalue problem (1), which can be written in canonical Hamiltonian form ∂H0 ∂H0 , px = − (10) qx = ∂p ∂q where H0 = − Aq, p + sequence (2). α2 + Aq, q Ap, p with α = 0 the constant given by the Lenard
m−1
Vm =
j =0
Nj ζ m−j ,
ζtm ≡ 0.
Corollary. Each evolution equation (4) has a Lax pair: φx = M φ, φtm = Vm φ, ζtm = 0. (5)