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2
4
’v1,t ’ v1,x ’ m1 u2 = »(u2 + v4 )u1 ,
2
4
u2,t ’ u2,x ’ m1 v1 = ’»(u2 + v3 )v2 ,
2
3
’v2,t + v2,x ’ m1 u1 = ’»(u2 + v3 )u2 ,
2
3
u3,t + u3,x ’ m2 v4 = ’»(u2 + v2 )v3 ,
2
2
’v3,t ’ v3,x ’ m2 u4 = ’»(u2 + v2 )u3 ,
2
2
u4,t ’ u4,x ’ m2 v3 = »(u2 + v2 )v4 ,
2
2
’v4,t + v4,x ’ m2 u3 = »(u2 + v2 )u4 .
2
(3.127)
2

We introduce functions R1 , . . . , R4 by

R1 = u2 + v1 ,
2
R2 = u 2 + v 2 ,
2
1 2
R3 = u2 + v3 ,
2
R4 = u 2 + v 4 .
2
3 4

We ¬rst rewite the Federbush model as a Hamiltonian system, i.e.,
du
= „¦’1 δH, (3.128)
dt
7. SYMMETRIES OF THE FEDERBUSH MODEL 141

where „¦ is a symplectic operator, H is the Hamiltonian and δH is the
Fr´chet derivative1 of H, u = (u1 , v1 , . . . , u4 , v4 ). In (3.128) we have
e
« 
J000
¬0 J 0 0· 01
„¦=¬ ·, J= ,
0 0 J 0 ’1 0
000J
and

1
u1x v1 ’u1 v1x ’u2x v2 +u2 v2x +u3x v3 ’u3 v3x ’u4x v4 +u4 v4x dx
H=
’∞ 2
» »
’ m1 (u1 u2 + v1 v2 ) ’ m2 (u3 u4 + v3 v4 ) ’ R1 R4 + R2 R3 .
2 2
By de¬nition, to each Hamiltonian symmetry Y (also called canonical sym-
metry) there corresponds a Hamiltonian F (Y ), where

F(Y ) dx,
F (Y ) = (3.129)
’∞
F(Y ) being the Hamiltonian density, such that
Y = „¦’1 δF (Y ), (3.130)
and the Poisson bracket of F (Y ) and H vanishes.
Suppose that Y1 , Y2 are two Hamiltonian symmetries. Then [Y1 , Y2 ] is a
Hamiltonian symmetry and
F ([Y1 , Y2 ]) = {F (Y1 ), F (Y2 )}, (3.131)
where {·, ·} is the Poisson bracket de¬ned by
{F (Y1 ), F (Y2 )} = δF (Y1 ), Y2 , (3.132)
·, · denoting the contraction of a 1-form and a vector ¬eld:
d
H(x + y)| =0 = δH, y . (3.133)
d
The Hamiltonians F (X) associated to the Hamiltonian densities F(X) are
de¬ned by (3.134):

F(X) dx.
F (X) = (3.134)
’∞
From these de¬nitions it is a straightforward computation that the symme-
tries Y0+ , Y1+ , Y’1 , Y0’ , Y1’ , Y’1 obtained sofar are all Hamiltonian, where

+

the Hamiltonian densities are given by
1
F(Y0+ ) = (R1 + R2 ),
2
1 » 1
F(Y1+ ) = ’ (u2x v2 ’ u2 v2x ) + R34 R2 ’ m1 (u1 u2 + v1 v2 ),
2 4 2
1
By the Fr´chet derivative the components of the Euler“Lagrange operator are
e
understood.
142 3. NONLOCAL THEORY

1 » 1
+
F(Y’1 ) = ’ (u1x v1 ’ u1 v1x ) + R34 R1 + m1 (u1 u2 + v1 v2 ),
2 4 2
1
F(Y0’ ) = (R3 + R4 ),
2
1 » 1
F(Y1’ ) = ’ (u4x v4 ’ u4 v4x ) ’ R12 R4 ’ m2 (u3 u4 + v3 v4 ),
2 4 2
1 » 1

F(Y’1 ) = ’ (u3x v3 ’ u3 v3x ) ’ R12 R3 + m2 (u3 u4 + v3 v4 ), (3.135)
2 4 2
whereas the densities F(Yi± ), i = ’2, 2, are given by
1 » 1
F(Y2+ ) = ’ (u2 + v2x ) + R34 (u2x v2 ’ u2 v2x ) ’ m1 (u2x v1 ’ u1 v2x )
2
2 2x 2 2
122 1 12
’ » R34 R2 + m1 »R34 (u1 u2 + v1 v2 ) ’ m1 R12 ,
8 4 8
1 » 1
+
F(Y’2 ) = ’ (u2 + v1x ) + R34 (u1x v1 ’ u1 v1x ) + m1 (u1x v2 ’ u2 v1x )
2
1x
2 2 2
1 1 1
’ »2 R34 R1 ’ m1 »R34 (u1 u2 + v1 v2 ) ’ m2 R12 ,
2
81
8 4
1 » 1
F(Y2’ ) = ’ (u2 + v4x ) ’ R12 (u4x v4 ’ u4 v4x ) ’ m2 (u4x v3 ’ u3 v4x )
2
2 4x 2 2
1 1 1
’ »2 R12 R4 ’ m2 »R12 (u3 u4 + v3 v4 ) ’ m2 R34 ,
2
82
8 4
1 » 1

F(Y’2 ) = ’ (u2 + v3x ) ’ R12 (u3x v3 ’ u3 v3x ) + m2 (u3x v4 ’ u4 v3x )
2
2 3x 2 2
122 1 12
’ » R12 R3 + m2 »R12 (u3 u4 + v3 v4 ) ’ m2 R34 , (3.136)
8 4 8
and the densities associated to Y3+ , Y’3 are given by
+


F(Y3+ ) = ’(u2xx v2x ’ v2xx u2x ) ’ »R34 (u2xx u2 + v2xx v2 )
»
+ R34 (u2 + v2x )
2
2x
2
3
’ m1 (u1x u2x + v1x v2x ) ’ »2 R34 (u2x v2 ’ u2 v2x )
2
4
1
+ m1 »R34 (u1x v2 ’ u1 v2x + u2x v1 ’ u2 v1x )
2
1 1 1
’ m2 (u1x v1 ’ u1 v1x ) ’ m2 (u2x v2 ’ u2 v2x ) ’ m3 (u1 u2 + v1 v2 )
1 1
41
4 2
1 1 1
+ »3 R34 R2 ’ m1 »2 R34 (u1 u2 + v1 v2 ) + m2 »R34 (R1 + 2R2 ),
3 2
81
8 4
»
+
F(Y’3 ) = u1xx v1x ’ v1xx u1x + »R34 (u1xx u1 + v1xx v1 ) + R34 (u2 + v1x )
2
1x
2
3
’ m1 (u1x u2x + v1x v2x ) + »2 R34 (u1x v1 ’ u1 v1x )
2
4
1
+ m1 »R34 (u1x v2 ’ u1 v2x + u2x v1 ’ u2 v1x )
2
7. SYMMETRIES OF THE FEDERBUSH MODEL 143

1 1 1
+ m2 (u1x v1 ’ u1 v1x ) + m2 (u2x v2 ’ u2 v2x ) ’ m3 (u1 u2 + v1 v2 )
21 41 41
1 1 1
’ »3 R34 R1 ’ m1 »2 R34 (u1 u2 + v1 v2 ) ’ m2 »R34 (2R1 + R2 ).
3 2
81
8 4
’ ’
+ +
The vector ¬elds Z’1 , Z1 , Z’1 , Z1 are Hamiltonian vector ¬elds too,
and the associated densities are given by
F(Z0 ) = x F(Y1+ ) ’ F(Y’1 ) + t F(Y1+ ) + F(Y’1 ) ,
+ + +

1 1
F(Z1 ) = x F(Y2+ ) ’ m2 F(Y0+ ) + t F(Y2+ ) + m2 F(Y0+ ) ,
+
41 41
1 1
F(Z’1 ) = x ’ F(Y’2 ) + m2 F(Y0+ ) + t F(Y’2 ) + m2 F(Y0+ ) ,
+ + +
1
41
4
F(Z0 ) = x F(Y1’ ) ’ F(Y’1 ) + t F(Y1’ ) + F(Y’1 ) ,
’ ’ ’

1 1
F(Z1 ) = x F(Y2’ ) ’ m2 F(Y0’ ) + t F(Y2’ ) + m2 F(Y0’ ) ,

2
42
4
1 1
F(Z’1 ) = x(’F(Y’2 ) + m2 F(Y0’ ) + t F(Y’2 ) + m2 F(Y0’ ) .
’ ’ ’
2
42
4
We now arrive at the following remarkable fact: The vector ¬elds Y + (2, 0)
and Y ’ (2, 0) are again Hamiltonian vector ¬elds, the corresponding Hamil-
tonian densities being given by
1
F(Y ’ (2, 0)) = x2 F(Y2’ ) ’ m2 F(Y0’ ) + F(Y’2 )’
2
2
’ ’
+ 2xt F(Y2 ) ’ F(Y’2 )
1
+ t2 F(Y2’ ) + m2 F(Y0’ ) + F(Y’2 )

2
2
1
= (x + t)2 F(Y2’ ) ’ m2 (x + t)(x ’ t)F(Y0’ )
22

+ (x ’ t)2 F(Y’2 ), (3.137)
and similarly
1
F(Y + (2, 0)) = (x + t)2 F(Y2+ ) ’ m2 (x + t)(x ’ t)F(Y0+ )
21
+
+ (x ’ t)2 F(Y’2 ), (3.138)
’ ’
+ +
Now the Hamiltonians F (Z1 ), F (Z’1 ), F (Z1 ), F (Z’1 ) act as cre-
ating and annihilating operators on the (x, t)-independent Hamiltonians
F (Y’3 ), . . . , F (Y3+ ) and F (Y’3 ), . . . , F (Y3’ ), by the action of the Poisson

+

bracket: for example
{F (Z1 ), F (Y0+ )} = 0,
+


1 1
+ +
(R1 + R2 ) = m2 F (Y0+ ),
{F (Z1 ), F (Y’1 )} = m2
41 41
’∞
{F (Z1 ), F (Y1+ )} = ’F (Y2+ ).
+
144 3. NONLOCAL THEORY

In the next subsection we give a formal proof for the existence of in-
¬nite number of hierarchies of higher symmetries by proving existence of
in¬nite number of hierarchies of Hamiltonians, thus leading to those for the
symmetries.
7.5.3. The in¬nity of the hierarchies. We shall prove here a lemma con-
cerning the in¬niteness of the hierarchies of Hamiltonians for the Federbush
model. From this we obtain a similar result for the associated hierarchies of
Hamiltonian vector ¬elds.
r r
Lemma 3.12. Let Hn (u, v) and Kn (u, v) be de¬ned by

r
xr (u2 + vn ),
2
Hn (u, v) = n
’∞

r
xr (un+1 vn ’ vn+1 un ),
Kn (u, v) = (3.139)
’∞
whereas in (3.139) r, n = 0, 1, . . . , and r, n are such that the degrees of
r r
Hn (u, v) and Kn (u, v) are positive.
Let the Poisson bracket of F and L, denoted by {F, L}, be de¬ned as

δF δL δF δL
{F, L} = ’ . (3.140)
’∞ δv δu δu δv
Then the following results hold
1 r r
{H1 , Hn } = 4(n ’ r)Kn ,
1 r r r’2
{H1 , Kn } = (4(n ’ r) + 2)Hn+1 + r(r ’ 1)(r ’ n ’ 1)Hn ,
2 r r+1
{H1 , Hn } = 4(2n ’ r)Kn ,
r+1
{H1 , Kn } = (2n + 1 ’ r)(4Hn+1 ’ r2 Hn ),
2 r r’1
(3.141)
r, n = 0, 1, . . .
Proof. We shall now prove the third and fourth relation in (3.141), the
proofs of the other two statements running along similar lines.
r r
Calculation of the Fr´chet derivatives of Hn , Kn yields
e
r
δHn
= (’Dx )n (2xr un ),
δu
r
δHn
= (’Dx )n (2xr vn ),
δv
r
δKn
= (’Dx )n+1 (xr vn ) ’ (’Dx )n (xr vn+1 ),
δu
r
δKn
= ’(’Dx )n+1 (xr un ) + (’Dx )n (xr un+1 ). (3.142)
δv
Substitution of (3.142) into the third relation of (3.141) yields

2 r
’Dx (2x2 v1 ) · (’1)n Dx (2xr un )
n
{H1 , Hn } =
’∞
+ Dx (2x2 u1 ) · (’1)n Dx (2xr vn )
n
7. SYMMETRIES OF THE FEDERBUSH MODEL 145


2n’1
Dx (2x2 u1 )Dx (2xr vn ) ’ Dx (2x2 v1 )Dx (2xr un )
n n
= (’1)
’∞

(x2 un+1 + 2nxun + n(n ’ 1)un’1 )(xr vn+1 + rxr’1 vn )
= ’4
’∞
’ (x vn+1 + 2nxvn + n(n ’ 1)vn’1 )(xr un+1 + rxr’1 un )
2


rxr+1 (un+1 vn ’ vn+1 un ) ’ 2nxr+1 (un+1 vn ’ vn+1 un )
= ’4
’∞
+ n(n ’ 1)x (vn+1 un’1 ’ un+1 vn’1 ) + n(n ’ 1)rxr’1 (vn un’1 ’ un vn’1 )
r

r+1
= 4(2n ’ r)Kn , (3.143)
which proves the third relation in (3.141).
The last equality in (3.143) results from the fact that the last two terms
are just constituting a total derivative of
n(n ’ 1)xr (vn un’1 ’ un vn’1 ). (3.144)
In order to prove the fourth relation in (3.141), we substitute (3.142),
which leads to

2 r
’Dx (2x2 v1 ) · (’1)n+1 Dx (xr vn ) ’ (’1)n Dx (xr vn+1 )
n+1 n
{H1 , Hn } =
’∞

+ Dx (2x2 u1 ) · (’1)n+1 Dx (xr un ) ’ (’1)n Dx (xr un+1 ) . (3.145)
n+1 n


Integration, n times, of the terms in brackets leads to

2 r
Dx (x2 v1 ) · (Dx (xr vn ) + xr vn+1 )
n+1
{H1 , Hn } =2
’∞
+ Dx (x2 u1 ) · (Dx (xr un ) + xr un+1 )
n+1


(x2 vn+2 + 2(n + 1)xvn+1 + n(n + 1)vn )(2xr vn+1 + rxr’1 vn )
=2
’∞

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