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SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION
1999,19( 5):505-511
.A~ctaG'citntia
1~¥AJI~tIl SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION
1
Xu Xixiang ( ~&~f ) Department of Basic Courses, Shandong Mining Institute, Taian 271019, China Abstract In this paper,the translation of the Lax pairs of the Levi equations is presented.Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems, Finally , the involutive solutions of the Levi equations are presented.
Key words Zero curvature representation, Lax pair, nonlinearization method, Liouville integrable system 1991 MR Subject Classification
1
35Q51, 58F05
Introduction It is well known that finding new finite-dimensional completely integrable systems in the Li-
ouville sense is importantlll. The mono-nonlinearization method is a great success on generating finite-dimensional completely integrable systems[2].It constrain (1+1) - dimensional integrable systems to finite-dimensional Liouville integrable systems. Recently, the mono-nonlinearisation method have successfully been extended[3.4,5]. The corresponding method is called a binary nonlinearization method because it involves the Lax pairs and the adjoint Lax pairs,through which a few interesting classical' integrable systems have successfully been obtained. However the binary nonlinearization method can not directly applied for some Lax pairs and the abjoint Lax pairs of soliton equations. In order to apply binary nonlinearization method, the Lax pairs must be first translated. In this paper the translation of Lax pairs of Levi hierarchy is presented. This paper is organized as follows. In Section 2, in order to obtain binary nonlinearization of the Levi hierarchy, its Lax pairs are translated. This is different from what is discussed in [3.4.5.]. Afterwards we consider the Bargmann constraint problem of the translated pairs and its adjoint Lax pairs of the Levi equations. Under the Bargmann constraint the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrained as finite-dimensional Liouville integrable Hamiltonian systems. In Section 3, It is proved that the translated Lax pairs and its adjoint Lax pairs of the Levi equations when binary nonlinearized become commutable flows of the finite-dimensional completely integrable systems in the Liouville sense. Furthermore, the 1 Received
Nov.29,1996; revised Nov.ll ,1998. This work is supported by the Natural Science Foundation of
Shandong provice,
506
ACTA MATHEMATICA SCIENTIA
Vo1.19
solutions of the Levi equations are obtained by using the involutive solutions of the commutable flows in the completely integrable.
2
A Symmetry Constraint of the Levi Equations We consider the spectral problem of the Levi equation[6.71.
In order to derive the Levi hierarchy by using zero curvature equation, we first solve the adjoint representation of Eq. (2.1) : (2.2) setting
v=
(
b ) _
ac
-a
f: (
n=O
an Cn
n
b -an
)
A-no
It is easy to see that Eq.(2.2) read as
a,r {
bo = Co = 0,
= -qc + b,
= -b(A - 1') +2. qa , x = -2a+ (A - r)c
C
= -qc
+ bn ,
2:: 0 bn x = rb.; - bn +i + 2qan , n 2:: () Cn,' = -rCn + Cn+i - 2a,,, n 2:: 0 an x
or
b,r
n
n
(2.3)
we choose ao = 1 and assume an 1,,=0 = 0, n ?: 1. In this way, the recursion relation (2.3) uniquely gives a series of polynomial functions with respect tou, (l.r, For example, we have 0000
=
=
ai 0, bi 2q, Ci Lax pairs[7,8]
= 2.a2 = -2q, b2 = 2qr -
2qx, C2
= 2r,···. The compatibility condition of
u(n) _ Vi
-
determine a hierarchy of the Levi system
tttn
=(
q) __ J ( r
~
-Cn+i
an+i
=JLn-i (-2r) ) -2q
= JbH n+i,n2:: 1, btt
(2.4)
where the Hamiltonian operator J and recursion operator L read as
J=
a a= -.a-ia = es:' = 1. ax
In order to obtain the binary nonlinearization of the Levi' hierarchy, the Lax pairs of the Levi' hierarchy must be translated.
507
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
No.5
Set
1 e = -(-A +r) ( 1 0 ) ,In = 2
0
1
1 --Cn+l 2
(1
0
o1 )
,U=Ul+e,V(n) =Vl (n) +In'
It is easy to verify that the compatibility condition of the Lax pairs I{Jx
= UI{J, I{Jm = v(n)1{J
(2.5)
also determine the Levi' hierarchy (2.4) . and Eq . Vx = [U, V] is equivalent to (2.3) . Consider spectral problem 1
-q 1
2A - 2r
) , I{J
=(
'Pl )
(2.6)
,
1{J2
and its. adjoint spectral problem
(2.7) where T means transposition of the matrix. From[5], through direct calculations we can obtain from (2.6) and (2.7)
0..\ = ~ (
ou
E
1Pl1{J2 lh'P2-!/J\'P\
2
),E =
1+
00
-00
1{J21P2
-l{Jl'fhdx.
(2.8)
2
Let us consider nonlinearization problem of the Lax pairs and the adjoint Lax pairs of the Levi systems. For N distinct eigenvalues AI, A2,' ", AN, we have
when lim Ixl->oo
l{Ji
= lim 1Pi = 0, i = 1,2, a direct calculation gives Ixl->oo
(2.11) therefore (1{J21Pl, 'f 2tf\; 'f\1P! ) T belongs to the invariant space of the recursion operator L. Letoo Ixl->oo diang(..\1,..\2,···,..\n). we impose the following Bargmann constraint[3]
(2.12)
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ACTA MATHEMATIC A SCIENTIA
Vol.19
which engenders an explicit symmetry constraint
By substituting (2.13) into (2.9) we obtain the nonlinearized Lax pairs
t,p1
( 1j ) 1It ( 1It?' -J
j)
t,p~
= ui«. Aj)IB ( t,p1j ) , x
t,p~
= -UT(u, Aj)!B ( 1It
1j
1It?' -J
X
)
.i : :;: 1,2"", N,
(2.14)
where the subscript B means substitution of (2.13) into the expression and (2.14) may be expressed as the Hamiltonian form
(2.15) with the Hamiltonian function
) ( ) _ (A~z, 1It z) - (A~1l1It1) (~1l1It1) - (~z, 1Itz) ( 2 4 ~z, 1It1 - ~ll1Itz .
H -
By using (2.11) under the constraint (2.13) we have
(2.16)
=qCn+1 + anHx = _(An-1~1l1Itz),n 2: l. From (2.2) we can get (VZ)x = [U, V Z]. Therefore bn+1
Fx
(2.17)
= ('21 trV z)x = dxd (a z + bc) = 0
i. e. F is a generating function of integrals of motion for (2.15). After setting F =
l: n~O
we have
Fo = 1, F1
r;
= Fz = 0, F3 = 2H,
n-1
= 2:)Uian-i + bicn-i) + 2a
n
;=1
= (An-Z~z, 1It z) - (An-2~1l 1It1) -
+
2(An-3~1l 1It 2)
(~2,1It2) - (~1' 1It 1) (An-3~? 1It ) 2
-,
2
n-4
+~ ~)((Ai~1' 1It 1) - (A;~2' 1Itz)((An-4-i~1l1It1) 4
i=O
_(An-4-i~z, 1It z))
+ 4(Ai~1l1Itz)(An-4-;~Z.1It1))' n 2:
4.
Fn)..-n
509
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
No.5
In the following we consider the temporal parts of the Lax pairs and the adjoint Lax pairs by substituting (2.13) into (2.10) we obtain the nonlinearized Lax pairs
,¢J1
j) ,J=1,2, . .. ·,N,
(2.18)
( l/J2j
where the subscript B means substitution of (2.13) into the expression and (2.18) may be expressed as the Hamiltonian forms (2.19) We consider the involution problem of F n under poisson bracket defined by N
?
{f y} ,
-
~'" ( of ~ _ of ~) LJ LJ 0.1... a,n·· a,n·· a.f,.· . ;=1 j=1 If') '1-'1)
1"1)
Through tedious calculations we can get {Fn +2 , F m + 2 } functions F" are in involution in pairs. In addition. it is easy to show that
VF,,+2
-
(2.20)
1"1)
= O. n
2: 1. i. e the polynomial
= (( aF n+2) T ( aFn+2)T ( aFn+2)T ( aFn+2)T) 0'1'1' 0'1'2 ' a 1 ' a2
1<
< 2N
• - n -
,
are linearly independent by observing that
and that the Vandermode determinant V(A1'>'2"",>'N) # O. Hence (2.14) and (2.18) are all finite-dimensional integrable system, in the Liouville sense.
3
The Involutive Solutions of the Levi Hierarchy Consider the canonical systems of H-flow and Hm-flow , respectively. 8H
1
8'Ii,
8H 8~,
(H)
8H
= IVH,
(3.1)
- 81>,
8H
:r
8H m
1
(H m )
8~,
8H m
2
8'1',
_8H m
'1'1 '1'2
- 81>,
81>,
tn>
-
8H m 81>,
= IVHn" I = (
0
-u»
I?N) 0
(3.2)
510
ACTA MATHEMATICA SCIENTIA
VaLl 9
where 12 N is the 2N x 2N unit matrix and H", = -Fm +2 ' Let g'1, gt'~, be the Hamiltonian phase flows associated with the canonical systems (3.1) and (3.2) , repeetively. Since H, H", are in involution, we arrive at the following propositions[1].
Proposition 1 (1) Canonical systems (H), (H m ) compatible. (2) The Hamiltonian phase flow g'H, gil~ commute. il>1(X,t",)
il>1(0,O)
il>2(X, t",)
x t =YHYH
w1(x, t-, )
il>2(0,0) m
W2(X,t",)
(3.3)
W1(0,O) W2(0,0)
with il>i(O, 0), W(O, 0), i = 1,2, being arbitrary constant vector .. The commutativity of y'H and yk m implied that (3.3) is a smooth function of (x, t",),. which is called the involutive solution of the consistent systems of equation (H),(H",). Proposition 2 Let (il>1(X,t",)T,il>2(X,t",)T, W1(X,t",)T, W2(x,t",)Tf be an involutive solution of the consistent system (H),(H",). Then
q(x, t",) =
(il>1(X, t",), W1(X,t m)) - (il>2(X, t",), W2(:l;. t TII 4
) )
_ (il>2(X, t",), W1(X, tm)) r (x,t", ) - 2
(3.4)
is a solution of the nonlinear evolution equation
. _(q) _.J (-C"'+1) /Lt", -
l'
-
tm.
(Lm+1
(3.5)
Proof By using (2.16),(2.17)and (2.18),it is easy to verify that
~:, = -(Am- 1il>1( X, t",), W2(X, t m)) tm aar
q(x, t",)(A",-1il>2(X. till)' Wdx, t-. )),
(3.6)
= (A"'-1il>2(X,t m), W2(X,t",)) - (A"'-1il>1(X,t m), W1(X,t m)) +(A mil>2(X, t m), W1(X,t m)) - r(x, tm)(Am- 1il>2(X, t m), W1(X,t m)).
(3.7)
On the other hand , we have
(Am_l.p2("'.tm),q;2(X'tm)}~;(Am_'<}dx,tm).q;d:r.tm)},.) (
(3.8)
(A"'-1il>2(X. t",),W1(X. tn> )):r
Observing (2.14) , one can find that (3.6) , (3.7) and (3.8) imply (3.5) . Therefore the mth Levi equation in hierarchy (3.5) possesses special solution (3.4) . The above involutive representation of solutions to integrable systems exhibits both the interrelation between (1+1)-dimensional integrable systems and finite-dimensional integrable systems. Moreover (3.4) provides a kind of separation of variables for Levi equations, i. e, we can separably solve the Hamiltonian systems (2.15) and (2.19) to find solutions of Levi hierarchy.
No.5
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
Acknowledgments
The author would like to express his sincere thanks to Dr.
511 Ma
wenxiu for his help and enthusiastic encouragement. References Arnold V 1. Mathematical methods of classical mechanice. Berlin: Springer-vcrlag,1978 2 Cao Cewen, Geng Xianguo. Classical integrable systems generated through nonlinea.rization of eigenvalue problems.in Nonlinear Physics. In:Gu H C et al ed. Research Reports in Physics. Berlin:Springer-Verlag, 1990. 68-78 3 Ma Wenxiu, Strampp W. An explicit symmetry constraint for the Lax pairs mHI the adjont Lax pairs of AKNS systems. Physics Letter A. 1994,185:277-286 4 Ma Wenxiu, New finite-dimensional integrable systems by symmetry constraint of the Kdv Equat.ions. Journal of the physical' society of Japan, 1995,64:1085-1091 5 Ma Wenxiu. Symmetry constraint of Mkdv equations by binary uoulinearizntiou. Physica A, 1995, 219:1083-1091 6 Levi D, Sym A, Swojciechowsy. A hierarchy of coupled korteweg-devries equations and the normalisntiou condition of the Hilbert-Riemann Problem. J Phys A:Math Gen, 1983,16:2423-2432 7 Ma Wenxiu. A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chinese Jomnal of contemporary Mathematics, 1992,13:79-89 8 Newell A C. Solitons in mathematics and physics. Philadelphia: SIAM, 1985
.A~ctaG'citntia
1~¥AJI~tIl SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION
1
Xu Xixiang ( ~&~f ) Department of Basic Courses, Shandong Mining Institute, Taian 271019, China Abstract In this paper,the translation of the Lax pairs of the Levi equations is presented.Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems, Finally , the involutive solutions of the Levi equations are presented.
Key words Zero curvature representation, Lax pair, nonlinearization method, Liouville integrable system 1991 MR Subject Classification
1
35Q51, 58F05
Introduction It is well known that finding new finite-dimensional completely integrable systems in the Li-
ouville sense is importantlll. The mono-nonlinearization method is a great success on generating finite-dimensional completely integrable systems[2].It constrain (1+1) - dimensional integrable systems to finite-dimensional Liouville integrable systems. Recently, the mono-nonlinearisation method have successfully been extended[3.4,5]. The corresponding method is called a binary nonlinearization method because it involves the Lax pairs and the adjoint Lax pairs,through which a few interesting classical' integrable systems have successfully been obtained. However the binary nonlinearization method can not directly applied for some Lax pairs and the abjoint Lax pairs of soliton equations. In order to apply binary nonlinearization method, the Lax pairs must be first translated. In this paper the translation of Lax pairs of Levi hierarchy is presented. This paper is organized as follows. In Section 2, in order to obtain binary nonlinearization of the Levi hierarchy, its Lax pairs are translated. This is different from what is discussed in [3.4.5.]. Afterwards we consider the Bargmann constraint problem of the translated pairs and its adjoint Lax pairs of the Levi equations. Under the Bargmann constraint the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrained as finite-dimensional Liouville integrable Hamiltonian systems. In Section 3, It is proved that the translated Lax pairs and its adjoint Lax pairs of the Levi equations when binary nonlinearized become commutable flows of the finite-dimensional completely integrable systems in the Liouville sense. Furthermore, the 1 Received
Nov.29,1996; revised Nov.ll ,1998. This work is supported by the Natural Science Foundation of
Shandong provice,
506
ACTA MATHEMATICA SCIENTIA
Vo1.19
solutions of the Levi equations are obtained by using the involutive solutions of the commutable flows in the completely integrable.
2
A Symmetry Constraint of the Levi Equations We consider the spectral problem of the Levi equation[6.71.
In order to derive the Levi hierarchy by using zero curvature equation, we first solve the adjoint representation of Eq. (2.1) : (2.2) setting
v=
(
b ) _
ac
-a
f: (
n=O
an Cn
n
b -an
)
A-no
It is easy to see that Eq.(2.2) read as
a,r {
bo = Co = 0,
= -qc + b,
= -b(A - 1') +2. qa , x = -2a+ (A - r)c
C
= -qc
+ bn ,
2:: 0 bn x = rb.; - bn +i + 2qan , n 2:: () Cn,' = -rCn + Cn+i - 2a,,, n 2:: 0 an x
or
b,r
n
n
(2.3)
we choose ao = 1 and assume an 1,,=0 = 0, n ?: 1. In this way, the recursion relation (2.3) uniquely gives a series of polynomial functions with respect tou, (l.r, For example, we have 0000
=
=
ai 0, bi 2q, Ci Lax pairs[7,8]
= 2.a2 = -2q, b2 = 2qr -
2qx, C2
= 2r,···. The compatibility condition of
u(n) _ Vi
-
determine a hierarchy of the Levi system
tttn
=(
q) __ J ( r
~
-Cn+i
an+i
=JLn-i (-2r) ) -2q
= JbH n+i,n2:: 1, btt
(2.4)
where the Hamiltonian operator J and recursion operator L read as
J=
a a= -.a-ia = es:' = 1. ax
In order to obtain the binary nonlinearization of the Levi' hierarchy, the Lax pairs of the Levi' hierarchy must be translated.
507
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
No.5
Set
1 e = -(-A +r) ( 1 0 ) ,In = 2
0
1
1 --Cn+l 2
(1
0
o1 )
,U=Ul+e,V(n) =Vl (n) +In'
It is easy to verify that the compatibility condition of the Lax pairs I{Jx
= UI{J, I{Jm = v(n)1{J
(2.5)
also determine the Levi' hierarchy (2.4) . and Eq . Vx = [U, V] is equivalent to (2.3) . Consider spectral problem 1
-q 1
2A - 2r
) , I{J
=(
'Pl )
(2.6)
,
1{J2
and its. adjoint spectral problem
(2.7) where T means transposition of the matrix. From[5], through direct calculations we can obtain from (2.6) and (2.7)
0..\ = ~ (
ou
E
1Pl1{J2 lh'P2-!/J\'P\
2
),E =
1+
00
-00
1{J21P2
-l{Jl'fhdx.
(2.8)
2
Let us consider nonlinearization problem of the Lax pairs and the adjoint Lax pairs of the Levi systems. For N distinct eigenvalues AI, A2,' ", AN, we have
when lim Ixl->oo
l{Ji
= lim 1Pi = 0, i = 1,2, a direct calculation gives Ixl->oo
(2.11) therefore (1{J21Pl, 'f 2tf\; 'f\1P! ) T belongs to the invariant space of the recursion operator L. Let
(2.12)
508
ACTA MATHEMATIC A SCIENTIA
Vol.19
which engenders an explicit symmetry constraint
By substituting (2.13) into (2.9) we obtain the nonlinearized Lax pairs
t,p1
( 1j ) 1It ( 1It?' -J
j)
t,p~
= ui«. Aj)IB ( t,p1j ) , x
t,p~
= -UT(u, Aj)!B ( 1It
1j
1It?' -J
X
)
.i : :;: 1,2"", N,
(2.14)
where the subscript B means substitution of (2.13) into the expression and (2.14) may be expressed as the Hamiltonian form
(2.15) with the Hamiltonian function
) ( ) _ (A~z, 1It z) - (A~1l1It1) (~1l1It1) - (~z, 1Itz) ( 2 4 ~z, 1It1 - ~ll1Itz .
H -
By using (2.11) under the constraint (2.13) we have
(2.16)
=qCn+1 + anHx = _(An-1~1l1Itz),n 2: l. From (2.2) we can get (VZ)x = [U, V Z]. Therefore bn+1
Fx
(2.17)
= ('21 trV z)x = dxd (a z + bc) = 0
i. e. F is a generating function of integrals of motion for (2.15). After setting F =
l: n~O
we have
Fo = 1, F1
r;
= Fz = 0, F3 = 2H,
n-1
= 2:)Uian-i + bicn-i) + 2a
n
;=1
= (An-Z~z, 1It z) - (An-2~1l 1It1) -
+
2(An-3~1l 1It 2)
(~2,1It2) - (~1' 1It 1) (An-3~? 1It ) 2
-,
2
n-4
+~ ~)((Ai~1' 1It 1) - (A;~2' 1Itz)((An-4-i~1l1It1) 4
i=O
_(An-4-i~z, 1It z))
+ 4(Ai~1l1Itz)(An-4-;~Z.1It1))' n 2:
4.
Fn)..-n
509
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
No.5
In the following we consider the temporal parts of the Lax pairs and the adjoint Lax pairs by substituting (2.13) into (2.10) we obtain the nonlinearized Lax pairs
,¢J1
j) ,J=1,2, . .. ·,N,
(2.18)
( l/J2j
where the subscript B means substitution of (2.13) into the expression and (2.18) may be expressed as the Hamiltonian forms (2.19) We consider the involution problem of F n under poisson bracket defined by N
?
{f y} ,
-
~'" ( of ~ _ of ~) LJ LJ 0.1... a,n·· a,n·· a.f,.· . ;=1 j=1 If') '1-'1)
1"1)
Through tedious calculations we can get {Fn +2 , F m + 2 } functions F" are in involution in pairs. In addition. it is easy to show that
VF,,+2
-
(2.20)
1"1)
= O. n
2: 1. i. e the polynomial
= (( aF n+2) T ( aFn+2)T ( aFn+2)T ( aFn+2)T) 0'1'1' 0'1'2 ' a 1 ' a2
1<
< 2N
• - n -
,
are linearly independent by observing that
and that the Vandermode determinant V(A1'>'2"",>'N) # O. Hence (2.14) and (2.18) are all finite-dimensional integrable system, in the Liouville sense.
3
The Involutive Solutions of the Levi Hierarchy Consider the canonical systems of H-flow and Hm-flow , respectively. 8H
1
8'Ii,
8H 8~,
(H)
8H
= IVH,
(3.1)
- 81>,
8H
:r
8H m
1
(H m )
8~,
8H m
2
8'1',
_8H m
'1'1 '1'2
- 81>,
81>,
tn>
-
8H m 81>,
= IVHn" I = (
0
-u»
I?N) 0
(3.2)
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ACTA MATHEMATICA SCIENTIA
VaLl 9
where 12 N is the 2N x 2N unit matrix and H", = -Fm +2 ' Let g'1, gt'~, be the Hamiltonian phase flows associated with the canonical systems (3.1) and (3.2) , repeetively. Since H, H", are in involution, we arrive at the following propositions[1].
Proposition 1 (1) Canonical systems (H), (H m ) compatible. (2) The Hamiltonian phase flow g'H, gil~ commute. il>1(X,t",)
il>1(0,O)
il>2(X, t",)
x t =YHYH
w1(x, t-, )
il>2(0,0) m
W2(X,t",)
(3.3)
W1(0,O) W2(0,0)
with il>i(O, 0), W(O, 0), i = 1,2, being arbitrary constant vector .. The commutativity of y'H and yk m implied that (3.3) is a smooth function of (x, t",),. which is called the involutive solution of the consistent systems of equation (H),(H",). Proposition 2 Let (il>1(X,t",)T,il>2(X,t",)T, W1(X,t",)T, W2(x,t",)Tf be an involutive solution of the consistent system (H),(H",). Then
q(x, t",) =
(il>1(X, t",), W1(X,t m)) - (il>2(X, t",), W2(:l;. t TII 4
) )
_ (il>2(X, t",), W1(X, tm)) r (x,t", ) - 2
(3.4)
is a solution of the nonlinear evolution equation
. _(q) _.J (-C"'+1) /Lt", -
l'
-
tm.
(Lm+1
(3.5)
Proof By using (2.16),(2.17)and (2.18),it is easy to verify that
~:, = -(Am- 1il>1( X, t",), W2(X, t m)) tm aar
q(x, t",)(A",-1il>2(X. till)' Wdx, t-. )),
(3.6)
= (A"'-1il>2(X,t m), W2(X,t",)) - (A"'-1il>1(X,t m), W1(X,t m)) +(A mil>2(X, t m), W1(X,t m)) - r(x, tm)(Am- 1il>2(X, t m), W1(X,t m)).
(3.7)
On the other hand , we have
(Am_l.p2("'.tm),q;2(X'tm)}~;(Am_'<}dx,tm).q;d:r.tm)},.) (
(3.8)
(A"'-1il>2(X. t",),W1(X. tn> )):r
Observing (2.14) , one can find that (3.6) , (3.7) and (3.8) imply (3.5) . Therefore the mth Levi equation in hierarchy (3.5) possesses special solution (3.4) . The above involutive representation of solutions to integrable systems exhibits both the interrelation between (1+1)-dimensional integrable systems and finite-dimensional integrable systems. Moreover (3.4) provides a kind of separation of variables for Levi equations, i. e, we can separably solve the Hamiltonian systems (2.15) and (2.19) to find solutions of Levi hierarchy.
No.5
Xu: SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS
Acknowledgments
The author would like to express his sincere thanks to Dr.
511 Ma
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