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+ 2m2 u2 (R1 + R2 ) ’ mu1 (R2 + 4R1 R2 + R1 )
2 2

3 2 2
’ 4mu2 R(R2 + 2R1 ) + u2 (R1 + 6R1 R2 + 3R1 R2 )). (3.60)

The vector ¬eld associated to •8 = (•u1 , •v1 , •u2 , •v2 ) can be derived from
8 8 8 8
•7 by the transformation
±
 u 1 ’ u 2 , v1 ’ v 2 , u 2 ’ u 1 , v2 ’ v 1 ,

T : ‚/‚x ’ ’‚/‚x, (3.61)


R1 ’ R2 , R2 ’ R1 , R ’ R

in the following way:

•u1 = ’T (•u2 ), •v1 = ’T (•v2 ),
8 7 8 7
•u2 = ’T (•u1 ), •v2 = ’T (•v1 ). (3.62)
8 7 8 7
120 3. NONLOCAL THEORY

The Lie bracket of vector ¬elds can be computed by calculation of the
Jacobi bracket of the associated generating functions:
[Xi , Xj ]l = Xi (Xj ) ’ Xj (Xil ),
l
l = u 1 , . . . , v2 ; i, j = 1, . . . , 8, (3.63)
where Xi = •i , which results in the following nonzero commutators:
[ •1 , •3 ] = •1 ,
=’
[ •2 , •3 ] •2 ,

m2
[ •3 , •5 ] = ’2 •5 ’ •4 ,
2
m2
[ •3 , •6 ] = 2 •6 ’ •4 ,
2
m2
[ •3 , •7 ] = ’3 •7 + ( •1 + •2 ),
2
m2
[ •3 , •8 ] = 3 •8 ’ ( •1 + •2 ). (3.64)
2
Transformation of the vector ¬elds •1 , . . . , •8 by
Y1 = •1 ,
Y2 = •2 ,
Y3 = •3 ,
Y4 = •4 ,

m2
Y 5 = •5 + •4 ,
4
m2
Y 6 = •6 ’ •4 ,
4
m2 m2
Y 7 = •7 ’ •1 ’ •2 ,
2 4
m2 m2
Y 8 = •8 ’ •1 ’ •2 , (3.65)
4 2
then leads to the following commutator table presented on Fig. 3.1.
Note that from (3.64) and (3.65) we see that [Yi , Yj ] = 0, i, j = 1, 2, 5, 6,
7, 8, while Y3 is the scaling symmetry.

6.2. Nonlocal symmetries. Here we shall discuss nonlocal symme-
tries of the massive Thirring model [41]. In order to ¬nd nonlocal variables
for the system
‚u1 ‚u1
= mv2 ’ (u2 + v2 )v1 ,
2
’ + 2
‚x ‚t
‚u2 ‚u2
= mv1 ’ (u2 + v1 )v2 ,
2
+ 1
‚x ‚t
‚v1 ‚v1
= mu2 ’ (u2 + v2 )u1 ,
2
’ 2
‚x ‚t
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 121

[—, —] Y 1 Y2 Y3 Y4 Y5 Y6 Y7 Y8
Y1 00 Y1 0 0 0 0 0
0 0 ’Y2 0
Y2 0 0 0 0
0 ’2Y5 2Y6 ’3Y7 3Y8
Y3 00 0
Y4 00 0 0 0 0 0 0
Y5 00 0 0 0 0 0 0
Y6 00 0 0 0 0 0 0
Y7 00 0 0 0 0 0 0
Y8 00 0 0 0 0 0 0

Figure 3.1. Commutator table for symmetries of the mas-
sive Thirring model

‚v2 ‚v2
= mu1 ’ (u2 + v1 )u2 ,
2
’ ’ (3.66)
1
‚x ‚t
we ¬rst have to construct conservation laws, i.e., sets (Ax , At ) satisfying the
i i
condition
Dt (Ax ) = Dx (At ),
i i

from which we can introduce nonlocal variables.
6.2.1. Construction of nonlocal symmetries. To construct conservation
laws, we take great advantage of the grading of system (3.66).
Since
deg(x) = deg(t) = ’2,
we start from two arbitrary polynomials Ax , At with respect to the variables
u1 , . . . , v2 , u1,1 , . . . , v2,1 , . . . such that the degree with respect to the grading
is just k, k = 1, . . .
It should be noted here that to get rid of trivial conservation laws, we
are making computations modulo total derivatives: this means in practice
that we start from a general polynomial Ax of degree k ’ 2 (with respect to
0
the grading), and eliminate resulting constants in Ax by equating terms in
0
the expression
Ax ’ Dx (Ax ).
0

to zero. This procedure is quite e¬ective and has been used in several
applications. Another way to arrive at conservation laws here, is to start
from symmetries and to apply the N¨ther theorem (Theorem 2.23).
o
The result is the following number of conservation laws, (Ax , At ), i =
i i
1, . . . , 4:
1
Ax = (u1 v1,1 ’ u1,1 v1 + u2 v2,1 ’ u2,1 v2 ),
1
2
1
At = (u1 v1,1 ’ u1,1 v1 ’ u2 v2,1 + u2,1 v2 + R1 R2 ),
1
2
122 3. NONLOCAL THEORY

1
Ax = (u1 v1,1 ’ u1,1 v1 ’ u2 v2,1 + u2,1 v2 + R1 R2 ’ 2mR),
2
2
1
At = (u1 v1,1 ’ u1,1 v1 + u2 v2,1 ’ u2,1 v2 ),
2
2
1
Ax = (R1 + R2 ),
3
2
1
At = (R1 ’ R2 ),
3
2
1
Ax = x(u1 v1,1 ’ u1,1 v1 ’ u2 v2,1 + u2,1 v2 + R1 R2 ’ 2mR)
4
2
1
+ t(u1 v1,1 ’ u1,1 v1 + u2 v2,1 ’ u2,1 v2 ),
2
1
At = x(u1 v1,1 ’ u1,1 v1 + u2 v2,1 ’ u2,1 v2 )
4
2
1
+ t(u1 v1,1 ’ u1,1 v1 ’ u2 v2,1 + u2,1 v2 + R1 R2 ), (3.67)
2
where in (3.67) we have

R1 = u2 + v1 ,
2
R2 = u 2 + v 2 ,
2
R = u 1 u 2 + v 1 v2 .
1 2

We now formally introduce variables the p0 , p1 , p2 by
1
Ax dx =
p0 = (R1 + R2 ) dx,
3
2
1
(Ax + Ax ) dx = (u1 v1,1 ’ u1,1 v1 + R1 R2 ’ mR) dx,
p1 = 1 2
2
1
(Ax ’ Ax ) dx = (u2 v2,1 ’ u2,1 v2 ’ R1 R2 + mR) dx.
p2 = (3.68)
1 2
2
Note that p0 , p1 , p2 are of degree 0, 2, 2 respectively (see (3.56)).
We now arrive from these nonlocal variables to the following augmented
system of partial di¬erential equations
’u1,1 + u1t = mv2 ’ (u2 + v2 )v1 ,
2
2
u2,1 + u2t = mv1 ’ (u2 + v1 )v2 ,
2
1
v1,1 ’ v1t = mu2 ’ (u2 + v2 )u1 ,
2
2
’v2,1 ’ v2t = mu1 ’ (u2 + v1 )u2 ,
2
1
1
(p0 )x = (R1 + R2 ),
2
1
(p0 )t = (R1 ’ R2 ),
2
1
(p1 )x = u1 v1,1 ’ u1,1 v1 + R1 R2 ’ mR,
2
1
(p1 )t = u1 v1,1 ’ u1,1 v1 + R1 R2 ,
2
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 123

1
(p2 )x = u2 v2,1 ’ u2,1 v2 ’ R1 R2 + mR,
2
1
(p2 )t = ’u2 v2,1 + u2,1 v2 + R1 R2 . (3.69)
2
We want to construct nonlocal higher symmetries of (3.53) which are just
higher symmetries of (3.69) (see Section 2). In e¬ect we shall just con-
struct the shadows of nonlocal symmetries, as discussed in Section 2. For
a more detailed exposition of the construction we refer to the construction
the nonlocal symmetries of the KdV equation in Section 5.
To construct nonlocal symmetries of (3.53), we start from a vertical
vector ¬eld Z of degree 2 and of polynomial degree one with respect to
x, t. So the generating functions Z u1 , . . . , Z v2 are of degree 3. The total
derivative operators D x , Dt are given by (3.70):
‚ ‚ ‚
Dx = Dx + (p0 )x + (p1 )x + (p2 )x ,
‚p0 ‚p1 ‚p2
‚ ‚ ‚
Dt = Dt + (p0 )t + (p1 )t + (p2 )t , (3.70)
‚p0 ‚p1 ‚p2
while the symmetry condition for the generating functions Z u1 , . . . , Z v2 is
’Dx (Z u1 ) + Dt (Z u1 ) = mZ v2 ’ v1 (2u2 Z u2 + 2v2 Z v2 ) ’ R2 Z v1 ,
Dx (Z u2 ) + Dt (Z u2 ) = mZ v1 ’ v2 (2u1 Z u1 + 2v1 Z v1 ) ’ R1 Z v2 ,
Dx (Z v1 ) ’ Dt (Z v1 ) = mZ u2 ’ u1 (2u2 Z u2 + 2v2 Z v2 ) ’ R2 Z u1 ,
’Dx (Z v2 ) ’ Dt (Z v2 ) = mZ u1 ’ u2 (2u1 Z u1 + 2v1 Z v1 ) ’ R1 Z u2 . (3.71)
Application of these conditions does lead to a number of equations for the
generating functions Z u1 , . . . , Z v2 .
The result is the existence of two nonlocal higher symmetries Z1 and
u1 v1 u2 v2
Z2 , where the generating functions Z1 = (Z1 , Z1 , Z1 , Z1 ) and
u v u v
Z2 = (Z2 1 , Z2 1 , Z2 2 , Z2 2 ) are given by
1
Z1 1 = v1 p2 + x(’2¦u1 ’ m2 v1 ) + t(2¦u1 ) + mu2 ,
u
5 5
2
1
Z1 1 = ’u1 p2 + x(’2¦v1 + m2 u1 ) + t(2¦v1 ) + mv2 ,
v
5 5
2
3
Z1 2 = v2 p2 + x(’2¦u2 ’ m2 v2 ) + t(2¦u2 ) + mu1 + 3v2,1 ,
u
5 5
2
3 1
’ R1 u2 ’ R2 u2 ,
2 2
3
Z1 2 = ’u2 p2 + x(’2¦v2 + m2 u2 ) + t(2¦v2 ) + mv1 ’ 3u2,1 ,
v
5 5
2
3 1
’ R1 v2 ’ R2 v2 ,
2
3
Z2 1 = v1 p1 + x(’2¦u1 + m2 v1 ) + t(’2¦u1 ) + mu2 ’ 3v1,1
u
6 6
2
124 3. NONLOCAL THEORY

3 1
’ R2 u1 ’ R1 u1 ,
2 2
3
Z2 1 = ’u1 p1 + x(’2¦v1 ’ m2 u1 ) + t(’2¦v1 ) + mv2 + 3u1,1
v
6 6
2
3 1
’ R2 v1 ’ R1 v1 ,
2 2
1
Z2 2 = v2 p1 + x(’2¦u2 + m2 v2 ) + t(’2¦u2 ) + mu1 ,
u
6 6
2
1
Z2 2 = ’u2 p1 + x(’2¦v2 ’ m2 u2 ) + t(’2¦v2 ) + mv1 .
v
(3.72)
6 6
2
p0 p2 p0 p2
The components Z1 , . . . , Z1 , Z2 , . . . , Z2 can be obtained from the invari-
ance of the equations
1
(p0 )x = (R1 + R2 ),
2
1
(p1 )x = u1 v1,1 ’ u1,1 v1 + R1 R2 ’ mR,
2
1
(p2 )x = u2 v2,1 ’ u2,1 v2 ’ R1 R2 + mR. (3.73)
2
6.2.2. Action of nonlocal symmetries. In order to derive the action of
the nonlocal symmetries Z1 , Z2 on the symmetries •1 , . . . , •6 , we have to
extend the Lie bracket of vector ¬elds in a way analogous to (3.52). This
is in e¬ect, as has been demonstrated for the KdV equation in previous
Section 5, where we extended the Jacobi bracket to the nonlocal variables,
i.e., u versus u, p, in this situation from u1 , v1 , u2 , v2 to p1 , p2 . Since the
nonlocal variable p0 does not take part in the presentation of the vector
¬elds •1 , . . . , •6 , Z1 , Z2 , we discard in this subsection the nonlocal variable
p0 , see (3.68).
The extended Lie bracket of the evolutionary vector ¬elds Zi , i =
1, 2, and •1 , . . . , •6 is obtained from the extended Jacobi bracket for the
generating functions, which is given by
{Zi , •j }w = w w

Zi (•j ) •j (Zi ), (3.74)
where in (3.74), i = 1, 2, j = 1, . . . , 6, w = u1 , . . . , v2 .
Since the generating functions •w are local, we do not need to compute
j
p1 p2 p1 p2
the components Z1 , Z1 , Z2 , Z2 , in order to calculate the ¬rst term
in the right-hand side of (3.74)). The calculation of the second term in
the righ-thand side of (3.74) however does require the components •p1 , 1
p2 p1 p2
•1 , . . . , •6 , •6 . These components result from the invariance of the partial
di¬erential equations (3.73) for the variables p1 , p2 , leading to the equations
1
Dx (•p1 ) = u1 v1,1 ’ u1,1 v1 + R1 R2 ’ mR ,
•j
j
2
1
Dx (•p2 ) = u2 v2,1 ’ u2,1 v2 ’ R1 R2 + mR . (3.75)
•j
j
2
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 125

From this we obtain the generating functions in the nonlocal, augmented
setting u1 , v1 , u2 , v2 , p1 , p2 :

1
¦u1 = (’mv2 + v1 (u2 + v2 )),
2
2
1
2
1
¦v1 = (mu2 ’ u1 (u2 + v2 )),
2
2
1
2
1
¦u2 = (2u2,1 ’ mv1 + v2 (u2 + v1 )),
2
1
1
2
1
¦v2 = (2v2,1 + mu1 ’ u2 (u2 + v1 )),
2
1
1
2
1
¦p1 = ’ mR,
1
2
1 1
¦p2 = ’v2 u2,1 + u2 v2,1 + mR ’ R1 R2 ,
1
2 2
1
¦u1 = (2u1,1 + mv2 ’ v1 (u2 + v2 )),
2
2
2
2
1
¦v1 = (2v1,1 ’ mu2 + u1 (u2 + v2 )),
2
2
2
2
1
¦u2 = (mv1 ’ v2 (u2 + v1 )),
2
1
2
2
1
¦v2 = (’mu1 + u2 (u2 + v1 )),2
1
2
2
1 1
¦p1 = ’v1 u1,1 + u1 v1,1 ’ mR + R1 R2 ,
2
2 2
1
¦p2 = mR,
2
2
1
¦u1 = u1,1 (x + t) + mv2 x + u1 ’ v1 (u2 + v2 )x,
2
2
3
2
1
¦v1 = v1,1 (x + t) ’ mu2 x + v1 + u1 (u2 + v2 )x,
2
2
3
2
1
¦u2 = u2,1 (’x + t) + mv1 x ’ u2 ’ v2 (u2 + v1 )x,
2
1
3
2
1
¦v2 = v2,1 (’x + t) + mu1 x ’ v2 + u2 (u2 + v1 )x,
2
1
3
2
1
¦p1 = (x + t)(2u1 v1,1 ’ 2v1 u1,1 + R1 R2 ) ’ tmR + p1 ,
3
2
1
¦p2 = (x + t)(’2u2 v2,1 + 2v2 u2,1 + R1 R2 ) + tmR ’ p2 ,
3
2
¦ u1 = v 1 ,
4
¦v1 = ’u1 ,
4
¦ u2 = v 2 ,
4
¦v2 = ’u2 ,
4
126 3. NONLOCAL THEORY

¦p1 = 0,
4
¦ p2 = 0 (3.76)
4


and similar for ¦5 , ¦6

1
¦u1 = (2u2,1 (’m + 2v1 v2 ) ’ 4v2,1 u2 v1 ’ mv2 (R1 + R2 )
5
4
2
’ 2mv1 R + v1 (R2 + 2R1 R2 )),
1
¦v1 = (2v2,1 (’m + 2u1 u2 ) ’ 4u2,1 v2 u1 + mu2 (R1 + R2 )
5
4
2
+ 2mu1 R ’ u1 (R2 + 2R1 R2 )),
1
¦u2 = (’4v2,2 + 2u1,1 (’m + 2u1 u2 ) + 4u2,1 (R1 + R2 )
5
4
2
+ 4v1,1 u2 v1 ’ mv1 (R1 + R2 ) ’ 2mv2 R + v2 (R1 + 2R1 R2 )),
1
¦v2 = (’4u2,2 + 2v1,1 (’m + 2v1 v2 ) + 4v2,1 (R1 + R2 ) + 4u1,1 v2 u1
5
4
2
+ mu1 (R1 + R2 ) + 2mu2 R ’ u2 (R1 + 2R1 R2 )),
1 1 1 1
¦p1 = ’ mv1 u2,1 + mu1 v2,1 ’ mR(R1 + R2 ) + m2 (R1 + R2 ),
5
2 2 4 4
1
¦p2 = u2,2 u2 + v2,2 v2 ’ u2 ’ v2,1 ’ mu2 v1,1 + mv1 u2,1
2
2,1
5
2
1
+ mv2 u1,1 ’ mu1 v2,1 ’ u2,1 v2 (R2 + 2R1 ) + v2,1 u2 (R2 + 2R1 )
2
1 3 1
’ m2 (R1 + R2 ) + mR(R1 + R2 ) + R1 R2 (R1 + R2 ),
4 4 2
1
¦u1 = (4v1,2 + 2u2,1 (’m + 2u1 u2 ) + 4u1,1 (R1 + R2 )
6
4
2
+ 4v2,1 u1 v2 + mv2 (R1 + R2 ) + 2mv1 R + v1 (R2 + 2R1 R2 )),
1
¦v1 = (’4u1,2 + 2v2,1 (’m + 2v1 v2 ) + 4v1,1 (R1 + R2 )
6
4
2
+ 4u2,1 u2 uv1 ’ mu2 (R1 + R2 ) ’ 2mu1 R + u1 (R2 + 2R1 R2 )),
1
¦u2 = (+2u1,1 (’m + 2v1 v2 ) ’ 4v1,1 u1 v2 + mv1 (R1 + R2 ) + 2mv2 R
6
4
2
’ v2 (R1 + 2R1 R2 )),
1
¦v2 = (+2v1,1 (’m + 2u1 u2 ) ’ 4u1,1 u2 v1 ’ mu1 (R1 + R2 )
6
4
2
’ 2mu2 R + u2 (R1 + 2R1 R2 )),
1
¦p1 = ’u1,2 u1 ’ v1,2 v1 + v1,1 + u2 ’ mu1 v2,1 + mv2 u1,1
2
1,1
6
2
1
+ mv1 u2,1 ’ mu2 v1,1 ’ u1,1 v1 (R1 + 2R2 ) + v1,1 u1 (R1 + 2R2 )
2
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 127

1 3 1
+ m2 (R1 + R2 ) ’ mR(R1 + R2 ) ’ R1 R2 (R1 + R2 ),
4 4 2
1 1 1 1
¦p2 = ’ mv2 u1,1 + mu2 v1,1 + mR(R1 + R2 ) ’ m2 (R1 + R2 ). (3.77)
6
2 2 4 4
The ‚/‚p1 -component of Z1 and the ‚/‚p2 -component of Z2 are given by
1
p
Z1 1 = (x ’ t)(’2mu1 v2,1 + 2mv1 u2,1 ’ (’m2 + mR)(R1 + R2 )
2
1
’ m(u1 v2 ’ u2 v1 ),
2
1
p
Z2 2 = (x + t)(’2mu2 v1,1 + 2mv2 u1,1 + (+m2 ’ mR)(R1 + R2 )
2
1
+ m(u1 v2 ’ u2 v1 ). (3.78)
2
Computation of the Jacobi brackets (3.74) then leads to the following com-
mutators for the evolutionary vector ¬elds:
1
] = ’ m2 ¦4 ’ 2 ¦5 ,
[ Z1 , ¦1
2
12
[ Z2 , ¦1 ] = m ¦4 ,
2
12
¦2 ] = ’ m
[ Z1 , ¦4 ,
2
12
¦4 ’ 2 ¦6 ,
[ Z2 , ¦2 ] = m
2
[ Z1 , ¦ 3 ] = Z1 ,
[ Z2 , ¦3 ] = Z2 ,
[ Z1 , ¦4 ] = 0,
[ Z2 , ¦4 ] = 0,
’ 2m2 ’ m2
[ Z1 , ¦5 ] =4 ¦2 ,
¦7 ¦1

= m2
[ Z2 , ¦5 ] ¦1 ,

= m2
[ Z1 , ¦6 ] ¦2 ,

’ m2 ’ 2m2
[ Z2 , ¦6 ] =4 ¦2 ,
¦8 ¦1

= ’2m2
[ Z1 , Z2 ] ¦3 . (3.79)
Transformation of the vector ¬elds by
Y1 = ¦1 ,
Y2 = ¦2 ,
Y3 = ¦3 ,
Y4 = ¦4 ,

m2
Y5 = + ¦4 ,
¦5
4
128 3. NONLOCAL THEORY

[—, —] Z1 Z2 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8
2
’2m2 Y3 m2 Y 2
’2Y5 ’ m Y4 — —
Z1 0 Z1 0 4Y7
2
m2
m2 Y 1
’2Y6 ’Z2 — —
Z2 0 2 Y4 0 4Y8
Y1 0 0 Y1 0 0 0 0 0
’Y2
Y2 0 0 0 0 0 0
’2Y5 ’3Y7
Y3 0 0 2Y6 3Y8
Y4 0 0 0 0 0
Y5 0 0 0 0
Y6 0 0 0
Y7 0 0
Y8 0

Figure 3.2. Commutator table for nolocal symmetries of
the massive Thirring model



m2

Y6 = ¦4 ,
¦6
4
m2 m2
’ ’
Y7 = ¦2 ,
¦7 ¦1
2 4
m2 m2
’ ’
Y8 = ¦2 , (3.80)
¦7 ¦1
4 2

leads us to the following commutator table presented on Fig. 3.2.
From the commutator table we conclude that Z1 acts as a generating
recursion operator on the hierarchy Y = (Y1 , Y5 , . . . ) while Z2 acts as a
ˆ
generating recursion operator on the hierarchy Y = (Y2 , Y6 , . . . ). The
action of Z1 on Y2 , Y6 is of a decreasing nature just as Z2 acts on Y1 , Y5 .
We expect that the vector ¬elds Z1 , Z2 generate a hierarchy of commuting
higher symmetries.

Remark 3.3. In (3.78), only those components of Z1 and Z2 are given
that are necessary to compute the Jacobi bracket of the generating functions,
i.e., for Z1 the ‚/‚p1 - and for Z2 the ‚/‚p2 -component


{Z1 , Z2 } = ’2m2 Y3 . (3.81)


We should mention here that Z1 does not admit a ‚/‚p2 -component,
while Z2 does not admit a ‚/‚p1 -component in this formulation. The asso-
ciated components can be obtained after introduction of nonlocal variables
arising from higher conservation laws, a situation similar to the nonlocal
symmetries of the KdV equation, Section 5.
7. SYMMETRIES OF THE FEDERBUSH MODEL 129

7. Symmetries of the Federbush model
We present here results of symmetry computations for the Federbush
model. The Federbush model is described by the matrix system of equations
|Ψ’s,2 |2 Ψs,1
’m(s)
i(‚/‚t + ‚/‚x) Ψs,1
= 4πs» ,
|Ψ’s,1 |2 Ψs,2
’m(s) i(‚/‚t ’ ‚/‚x) Ψs,2
(3.82)
where in (3.82) s = ±1 and Ψs (x, t) are two component complex-valued
functions R2 ’ C.
Suppressing the factor 4π from now on (we set » = 4π») and introducing
the eight variables u1 , v1 , u2 , v2 , u3 , v3 , u4 , v4 by
Ψ+1,1 = u1 + iv1 , Ψ+1,2 = u2 + iv2 , m(+1) = m1 ,
Ψ’1,1 = u3 + iv3 , Ψ’1,2 = u4 + iv4 , m(’1) = m2 , (3.83)
equation (3.82) is rewritten as a system of eight nonlinear partial di¬erential
equations for the component functions u1 , . . . , v4 , i.e.,
u1,t + u1,x ’ m1 v2 = »(u2 + v4 )v1 ,
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
(3.84)
2
The contents of this section is strongly related to a number of papers [42,
36, 92] and references therein.
7.1. Classical symmetries. The symmetry condition (2.29) on p. 72
leads to the following ¬ve classical symmetries

V1 = ,
‚x

V2 = ,
‚t
‚ ‚ 1 ‚ ‚ ‚ ‚
’ u2 ’ v2
V3 = t + x + (u1 + v1
‚x ‚t 2 ‚u1 ‚v1 ‚u2 ‚v2
‚ ‚ ‚ ‚
’ u4 ’ v4
+ u3 + v3 ),
‚u3 ‚v3 ‚u4 ‚v4
‚ ‚ ‚ ‚
V4 = ’v1 ’ v2
+ u1 + u2 ,
‚u1 ‚v1 ‚u2 ‚v2
‚ ‚ ‚ ‚
V5 = ’v3 ’ v4
+ u3 + u4 . (3.85)
‚u3 ‚v3 ‚u4 ‚v4
130 3. NONLOCAL THEORY

Associated to these classical symmetries, we construct in a straightforward
i i
way the conservation laws (Cx , Ct ), satisfying
i i
Dx (Ct ) ’ Dt (Cx ) = 0, (3.86)
i.e.,
1
Cx = u1x v1 ’ u1 v1x + u2x v2 ’ u2 v2x + u3x v3 ’ u3 v3x + u4x v4 ’ u4 v4x ,
1
Ct = ’u1x v1 + u1 v1x + u2x v2 ’ u2 v2x ’ u3x v3 + u3 v3x + u4x v4 ’ u4 v4x
+ »(R1 R4 ’ R2 R3 ),
2
Cx = ’u1x v1 + u1 v1x + u2x v2 ’ u2 v2x ’ u3x v3 + u3 v3x + u4x v4 ’ u4 v4x
+ 2m1 (u1 u2 + v1 v2 ) + 2m2 (u3 u4 + v3 v4 ) + »(R1 R4 ’ R2 R3 ),
2
Ct = u1x v1 ’ u1 v1x + u2x v2 ’ u2 v2x + u3x v3 ’ u3 v3x + u4x v4 ’ u4 v4x ,
3 2 1
Cx = xCx + tCx ,
3 2 1
Ct = xCt + tCt ,
4
Cx = R 1 + R 2 ,
4
Ct = ’R1 + R2 ,
5
Cx = R 3 + R 4 ,
5
Ct = ’R3 + R4 . (3.87)
In (3.87) we used the notations
R1 = u2 + v1 ,
2
R2 = u 2 + v 2 ,
2
R3 = u 2 + v 3 ,
2
R4 = u 2 + v 4 .
2
(3.88)
1 2 3 4

7.2. First and second order higher symmetries. We now con-
struct ¬rst and second order higher symmetries of the Federbush model.
In obtaining the results, we observe the remarkable fact of the existence of
¬rst order higher symmetries, which are not equivalent to classical symme-
tries.
The results for ¬rst order symmetries are
» ‚ » ‚ » ‚ » ‚
’ u1 R4 ’ u2 R4
X1 = v1 R4 + v2 R4
2 ‚u1 2 ‚v1 2 ‚u2 2 ‚v2
1 ‚ 1 ‚
’ m 2 u4
+ m 2 v4
2 ‚u3 2 ‚v3
1 ‚
+ (2u4x + m2 v3 + »v4 (R1 + R2 ))
2 ‚u4
1 ‚
+ (2v4x ’ m2 u3 ’ »u4 (R1 + R2 )) ,
2 ‚v4
» ‚ » ‚ » ‚ » ‚
’ u1 R3 ’ u2 R3
X2 = v1 R3 + v2 R3
2 ‚u1 2 ‚v1 2 ‚u2 2 ‚v2
1 ‚
+ (2u3x ’ m2 v4 + »v3 (R1 + R2 ))
2 ‚u3
7. SYMMETRIES OF THE FEDERBUSH MODEL 131

1 ‚
+ (2v3x + m2 u4 ’ »u3 (R1 + R2 ))
2 ‚v3
1 ‚ 1 ‚
’ m 2 v3 + m 2 u3 ,
2 ‚u4 2 ‚v4
1 ‚ 1 ‚
’ m 1 u2
X 3 = m 1 v2
2 ‚u1 2 ‚v1
1 ‚
+ (2u2x + m1 v1 ’ »v2 (R3 + R4 ))
2 ‚u2
1 ‚
+ (2v2x ’ m1 u1 + »u2 (R3 + R4 ))
2 ‚v2
» ‚ » ‚ » ‚ » ‚
’ v3 R2 ’ v4 R2
+ u3 R2 + u 4 R2 ,
2 ‚u3 2 ‚v3 2 ‚u4 2 ‚v4
1 ‚
X4 = (2u1x ’ m1 v2 ’ »v1 (R3 + R4 ))
2 ‚u1
1 ‚
+ (2v1x + m1 u2 + »u1 (R3 + R4 ))
2 ‚v1
1 ‚ 1 ‚
’ m 1 v1 + m 1 u1
2 ‚u2 2 ‚v2
» ‚ » ‚ » ‚ » ‚
’ v3 R1 ’ v4 R1
+ u3 R1 + u 4 R1 .
2 ‚u3 2 ‚v3 2 ‚u4 2 ‚v4
Recall that two symmetries, X and Y are equivalent (we use the notation
.
=), see Chapter 2, if their exist functions f, g ∈ F(E) such that
X = Y + f Dx + gDt , (3.89)
where Dx , Dt are the total derivative operators.
From this one notes that
.1‚ ‚
X2 + X 4 = ’ ’ ,
2 ‚x ‚t
.1‚ ‚
X1 + X 3 = ’ + . (3.90)
2 ‚x ‚t
We did ¬nd these ¬rst order higher symmetries of the Federbush model
using the following grading of the model:
deg(x) = deg(t) = ’2, nonumber (3.91)
‚ ‚
deg( ) = deg( ) = 2, nonumber (3.92)
‚x ‚t
deg(u1 ) = · · · = deg(v4 ) = 1, nonumber (3.93)
‚ ‚
) = · · · = deg( ) = ’1, nonumber
deg( (3.94)
‚u1 ‚v4
deg(m1 ) = deg(m2 ) = 2. (3.95)
In order to ¬nd ¬rst order higher symmetries which are equivalent to the
vector ¬eld V3 (3.85), we searched for a vertical vector ¬eld of the following
132 3. NONLOCAL THEORY

presentation:
V = xH1 + tH2 + C, (3.96)
where H1 , H2 are combinations of the vector ¬elds V4 , V5 , X1 , . . . , X4 , while
C is a correction of an appropriate degree.
From (3.96) and condition (3.91) we obtain two additional ¬rst order
higher symmetries X5 , X6 , i.e.,
1 ‚ ‚ ‚ ‚
X5 = x(X1 ’ X2 ) + t(X1 + X2 ) ’ ’ u4 ’ v4
u3 + v3 ,
2 ‚u3 ‚v3 ‚u4 ‚v4
1 ‚ ‚ ‚ ‚
X6 = x(X3 ’ X4 ) + t(X3 + X4 ) ’ ’ u2 ’ v2
u1 + v1 .
2 ‚u1 ‚v1 ‚u2 ‚v2
(3.97)
Note that
.
X5 + X6 = ’V3 . (3.98)
In order to construct second order higher symmmetries of the Feder-
bush model, we searched for a vector ¬eld V , whose de¬ning functions
V u1 , . . . , V v4 are dependent on the variables u1 , . . . , v4 , . . . , u1xx , . . . , v4xx .
Due to the above introduced grading (3.91) the presentation of the de¬ning
functions V u1 , . . . , V v4 is of the folowing structure:
V — = [u]xx + ([u]2 + [m])[u]x + ([u]5 + [m][u]3 + [m]2 [u]) (3.99)
whereas in (3.99)
[u] refers to u1 , . . . , v4 ,
[u]x refers to u1x , . . . , v4x ,
[u]xx refers to u1xx , . . . , v4xx ,
[m] refers to m1 , m2 .
From presentation (3.99) and the symmetry condition we derive an overde-
termined system of partial di¬erential equations. The solution of this sys-
tem leads to four second-order higher symmetries of the Federbush model,
X7 , . . . , X10 , i.e.:
» » » »
u v u v
X 7 1 = v1 K 7 , X 7 1 = ’ u 1 K 7 , X 7 2 = v2 K 7 , X 7 2 = ’ u 2 K 7 ,
2 2 2 2
1 1
u v
X7 3 = m2 2u4x + »v4 (R1 + R2 ) , X7 3 = m2 2v4x ’ »u4 (R1 + R2 ) ,
4 4
1
u
’ 4v4xx + 2»u4 (R1 + R2 )x + 4»u4x (R1 + R2 ) + 2m2 u3x
X7 4 =
4
+ »m2 v3 (R1 + R2 ) + »2 v4 (R1 + R2 )2 ,
1
v
X7 4 = 4u4xx + 2»v4 (R1 + R2 )x + 4»v4x (R1 + R2 ) + 2m2 v3x
4
’ »m2 u3 (R1 + R2 ) ’ »2 u4 (R1 + R2 )2 ,

» » » »
u v u v
v1 K 8 , X 8 1 = ’ u 1 K 8 , X 8 2 = v2 K 8 , X 8 2 = ’ u 2 K 8 ,
X8 1 =
2 2 2 2
7. SYMMETRIES OF THE FEDERBUSH MODEL 133

1
u
’ 4v3xx + 2»u3 (R1 + R2 )x + 4»u3x (R1 + R2 ) ’ 2m2 u4x
X8 3 =
4
’ »m2 v4 (R1 + R2 ) + »2 v3 (R1 + R2 )2 ,
1
v
= 4u3xx + 2»v3 (R1 + R2 )x + 4»v3x (R1 + R2 ) ’ 2m2 v4x
X8 3
4
+ »m2 u4 (R1 + R2 ) ’ »2 u3 (R1 + R2 )2 ,
1
u
= m2 ’ 2u3x ’ »v3 (R1 + R2 ) ,
X8 4
4
1
v
= m2 ’ 2v3x + »u3 (R1 + R2 ) ,
X8 4
4
1
u
X9 1 = m1 2u2x ’ »v2 (R3 + R4 ) ,
4
1
v
X9 1 = m1 2v2x + »u2 (R3 + R4 ) ,
4
1
u
’ 4v2xx ’ 2»u2 (R3 + R4 )x ’ 4»u2x (R3 + R4 ) + 2m1 u1x
X9 2 =
4
’ »m1 v1 (R3 + R4 ) + »2 v2 (R3 + R4 )2 ,
1
v
X9 2 = 4u2xx ’ 2»v2 (R3 + R4 )x ’ 4»v2x (R3 + R4 ) + 2m1 v1x
4
+ »m1 u1 (R3 + R4 ) ’ »2 u2 (R3 + R4 )2 ,
» » » »
u v u v
X 9 3 = v3 K 9 , X 9 3 = ’ u 3 K 9 , X 9 4 = v4 K 9 , X 9 4 = ’ u 4 K 9 ,
2 2 2 2
1
u1
’ 4v1xx ’ 2»u1 (R3 + R4 )x ’ 4»u1x (R3 + R4 ) ’ 2m1 u2x
X10 =
4
+ »m1 v2 (R3 + R4 ) + »2 v1 (R3 + R4 )2 ,
1
v1
= 4u1xx ’ 2»v1 (R3 + R4 )x ’ 4»v1x (R3 + R4 ) ’ 2m1 v2x
X10
4
’ »m1 u2 (R3 + R4 ) ’ »2 u1 (R3 + R4 )2 ,
1
u2
= m1 ’ 2u1x + »v1 (R3 + R4 ) ,
X10
4
1
v2
= m1 ’ 2v1x ’ »u1 (R3 + R4 ) ,
X10
4
» »
u3 v3
= v3 K10 , X10 = ’ u3 K10 ,
X10
2 2
» »
u4 v4
= v4 K10 , X10 = ’ u4 K10 ,
X10 (3.100)
2 2
whereas in (3.100)

K7 = 2u4x v4 ’ 2u4 v4x + m2 (u3 u4 + v3 v4 ) + »R4 (R1 + R2 ),
K8 = 2u3x v3 ’ 2u3 v3x ’ m2 (u3 u4 + v3 v4 ) + »R3 (R1 + R2 ),
K9 = ’2u2x v2 + 2u2 v2x ’ m1 (u1 u2 + v1 v2 ) + »R2 (R3 + R4 ),
134 3. NONLOCAL THEORY

K10 = ’2u1x v1 + 2u1 v1x + m1 (u1 u2 + v1 v2 ) + »R1 (R3 + R4 ). (3.101)

The Lie bracket for vertical vector ¬elds Vi , i ∈ N, de¬ned by
‚ ‚ ‚ ‚
Vi = Viu1 + Viv1 + · · · + Viu4 + Viv4 , (3.102)
‚u1 ‚v1 ‚u4 ‚v4
is given by

[Vi , Vj ]± = Vi (Vj± ) ’ Vj (Vi± ), ± = u 1 , . . . , v4 . (3.103)

The commutators of the associated vector ¬elds V4 , V5 , X1 , . . . , X4 , X5 , X6 ,
X7 , . . . , X10 are given by the following nonzero commutators:

[X1 , X5 ] = ’X1 ,
[X2 , X5 ] = X2 ,
[X3 , X6 ] = ’X3 ,
[X4 , X6 ] = X4 ,
1
[X5 , X7 ] = 2X7 ’ m2 V5 ,
22
1
[X5 , X8 ] = ’2X8 + m2 V5 ,
22
1
[X6 , X9 ] = 2X9 ’ m2 V4 ,
21
1
[X6 , X10 ] = ’2X10 + m2 V4 . (3.104)
21
We now transform the vector ¬elds by

Y0’ = V5 ,
Y0+ = V4 ,
Y1’ = X1 ,
Y1+ = X3 ,

+
Y’1 = X4 , Y’1 = X2 ,
1 1
Y2’ = X7 ’ m2 V5 ,
Y2+ = X9 ’ m2 V4 ,
41 42
1 1

+
Y’2 = X10 ’ m2 V4 , Y’2 = X8 ’ m2 V5 ,
41 42

+
Z0 = X 6 , Z0 = X 5 . (3.105)

From (3.103) and (3.105) we obtain a direct sum of two Lie algebras: each
“+”-denoted element commutes with any “’”-denoted element and

i, j = ’2, . . . , 2.
[Z0 , Yi ] = iYi , [Yi , Yj ] = 0, (3.106)

In (3.106) Z0 , Yi , where i = ’2, . . . , 2, are assumed to have the same upper
sign, + or ’.
7. SYMMETRIES OF THE FEDERBUSH MODEL 135

7.3. Recursion symmetries. We shall now construct four (x, t)-de-
pendent higher symmetries which act, by the Lie bracket for vertical vector
¬elds, as recursion operators on the above constructed (x, t)-independent
vector ¬elds X1 , . . . , X4 , X7 , . . . , X10 . We are motivated by the results for
the massive Thirring model, which were discussed in Subsections 6.1 and
6.2, and the results of Subsection 7.2, leading to the direct sum of two Lie
algebras, each of which having a similar structure to the Lie algebra for
the massive Thirring model. So we are forced to search for nonlocal higher
symmetries, including the nonlocal variables (3.87) associated to the vector
¬elds V1 , V2 in (3.85).
Surprisingly, carrying through the huge computations, the nonlocal vari-
ables dropped out automatically from intermediate results, ¬nally leading
to local (x, t)-dependent higher symmetries. So, for simplicity we shall dis-
cuss the search for creating and annihilating symmetries, assuming from the
beginning that they are local.
The formulation of creating and annihilating symmetries will follow from
the Lie brackets of these symmetries with Yi± , meaning going up or down in
the hierarchy. The symmetries Y0+ , Y0’ are of degree 0, Y1+ , Y’1 , Y1’ , Y’1 ’
+

are of degree 2, while the symmetries Y2+ , Y’2 , Y2’ , Y’2 are of degree 4, see

+

(3.105).
We now search for an (x, t)-dependent higher symmetry of second order,
linear with respect to x, t, and of degree 2, i.e., for a vector ¬eld V of the
form
V = xH1 + tH2 + C — , (3.107)
where H1 , H2 are higher symmetries of degree four and, due to the fact that
m1 , m2 are of degree two, H1 , H2 are assumed to be linear with respect
to Y0+ , Y0’ , . . . , Y2+ , Y’2 , Y2’ , Y’2 , while V in (3.107) has to satisfy the

+

symmetry condition. From these conditions we obtained the following result.
The symmetry condition is satis¬ed under the special assumption for V ,
(3.107), leading to the following four higher symmetries:
1 1
X11 = x ’Y’2 + m2 Y0+ + t Y’2 + m2 Y0+ + C11 ,
+ +
41 41
1 1
X12 = x Y2+ ’ m2 Y0+ + t Y2+ + m2 Y0+ + C12 ,
1
41
4
1 1
X13 = x ’Y’2 + m2 Y0’ + t Y’2 + m2 Y0’ + C13 ,
’ ’
42 42
1 1
X14 = x Y’2 ’ m2 Y0’ + t Y’2 + m2 Y0’ + C14 .
’ ’
(3.108)
2
42
4
where in (3.108) the functions C11 , . . . , C14 are given by the following ex-
pressions
1 ‚
C11 = 2v1x + m1 u2 + »u1 (R3 + R4 )
2 ‚u1
136 3. NONLOCAL THEORY

1 ‚
’ 2u1x + m1 v2 + »v1 (R3 + R4 )
+ ,
2 ‚v1
1 ‚
’ 2v2x + m1 u1 ’ »u2 (R3 + R4 )
C12 =
2 ‚u2
1 ‚
2u2x + m1 v1 ’ »v2 (R3 + R4 )
+ ,
2 ‚v2
1 ‚
2v3x + m2 u4 ’ »u3 (R1 + R2 )
C13 =
2 ‚u3
1 ‚
’ 2u3x + m2 v4 ’ »v3 (R1 + R2 )
+ ,
2 ‚v3
1 ‚
’ 2v4x + m2 u3 + »u4 (R1 + R2 )
C14 =
2 ‚u4
1 ‚
+ 2u4x + m2 v3 + »v4 (R1 + R2 ) . (3.109)
2 ‚v4
From (3.108) and (3.109) we de¬ne
’ ’
+ +
Z’1 = X11 , Z1 = X12 , Z’1 = X13 , Z1 = X14 . (3.110)

Computation of the commutators of Z’1 , Z1 , Z’1 , Z1 and Yi± , where
’ ’
+ +

i = ’2, . . . , 2, leads to the following result:
1
[Z’1 , Y2+ ] = ’ m2 Y1+ ,
+
[Z1 , Y2+ ] = Y3+ ,
+
21
1
[Z’1 , Y1+ ] = m2 Y0+ ,
+
[Z1 , Y1+ ] = Y2+ ,
+
41
[Z’1 , Y0+ ] = 0,
+
[Z1 , Y0+ ] = 0,
+

1
+ + +
[Z1 , Y’1 ] = ’ m2 Y0+ , nonumber
+ +
[Z’1 , Y’1 ] = ’Y’2 , (3.111)
41
1
+ + + + + +
[Z1 , Y’2 ] = m2 Y’1 , nonumber
[Z’1 , Y’2 ] = Y’3 , (3.112)
1
2
1
[Z’1 , Y2’ ] = ’ m2 Y1’ ,

[Z1 , Y2’ ] = Y3’ ,

22
1
[Z’1 , Y1’ ] = m2 Y0’ ,

[Z1 , Y1’ ] = Y2’ ,

42
[Z’1 , Y0’ ] = 0,

[Z1 , Y0’ ] = 0,


1
’ ’ ’
[Z1 , Y’1 ] = ’ m2 Y0’ , nonumber
’ ’
[Z’1 , Y’1 ] = ’Y’2 , (3.113)
42
1
’ ’ ’ ’ ’ ’
[Z1 , Y’2 ] = m2 Y’1 ,
[Z’1 , Y’2 ] = Y’3 , (3.114)
2
2
while
1 1
’ ’ ’
+ + +
[Z’1 , Z1 ] = ’ m2 Z0 , [Z’1 , Z1 ] = ’ m2 Z0 . (3.115)
1 2
2 2
7. SYMMETRIES OF THE FEDERBUSH MODEL 137

All other commutators are zero. The vector ¬eld Y3+ is given by

m1
Y3+,u1 = ’ 4v2xx ’ 4»R34 u2x + 2m1 u1x ’ 4»u2 (R34 )(1)
4
+ m2 v2 ’ »m1 R34 v1 + »2 R34 v2 ,
2
1
m1
Y3+,v1 = + 4u2xx ’ 4»R34 v2x + 2m1 v1x ’ 4»v2 (R34 )(1)
4
’ m2 u2 + »m1 R34 u1 ’ »2 R34 u2 ,
2
1
1
Y3+,u2 = ’ 8u2xxx ’ 4m1 v1xx + 12»R34 v2xx + 8»v2 (R34 )(2)
4
+ 24»v2x (R34 )(1) + 8»v2 (R34 )(1,1) + u2x (4m2 + 6»2 R34 )
2
1
’ 4»m1 R34 u1x + 12»2 u2 R34 (R34 )(1) ’ 4»m1 u1 (R34 )(1)
+ m3 v1 ’ 2»m2 R34 v2 + »2 m1 R34 v1 ’ »3 R34 v2 ,
2 3
1 1
1
Y3+,v2 = ’ 8v2xxx + 4m1 u1xx ’ 12»R34 u2xx + 8»u2 (R34 )(2)
4
’ 24»u2x (R34 )(1) ’ 8»u2 (R34 )(1,1) + v2x (4m2 + 6»2 R34 )
2
1
’ 4»m1 R34 v1x + 12»2 v2 R34 (R34 )(1) ’ 4»m1 v1 (R34 )(1)
’ m3 u1 + 2»m2 R34 u2 ’ »2 m1 R34 u1 + »3 R34 u2 ,
2 3
1 1

» »
Y3+,u3 = Y3+,v3 = ’ u3 L,
v3 L,
4 4
» »
Y3+,u4 Y3+,v4 = ’ u4 L,
= v4 L, (3.116)
4 4

where in (3.116)

R34 = R3 + R4 ,
(R34 )(1) = u3 u3x + v3 v3x + u4 u4x + v4 v4x ,
(R34 )(2) = u3 u3xx + v3 v3xx + u4 u4xx + v4 v4xx ,
(R34 )(1,1) = u2 + v3x + u2 + v4x ,
2 2
3x 4x
L = 8(u2 u2xx + v2 v2xx ) ’ 4(u2 + v2x ) + 12»R34 (u2x v2 ’ v2x u2 )
2
2x
+ 4m1 (u1 v2x ’ v1 u2x + u2 v1x ’ v2 u1x ) ’ m2 (2R2 + R1 )
1
+ 4m1 »R34 (u1 u2 + v1 v2 ) ’ 3»2 R2 R34 .
2



The results for the vector ¬elds Y’3 , Y3’ , Y’3 are similar to (3.116) and

+

are not given here, but are obtained from discrete symmetries σ and „ , to
be described in the next section.
’ ’
+ +
From the above it is clear now, why the vector ¬elds Z’1 , Z1 , Z’1 , Z1
are called creating and annihilating operators.
138 3. NONLOCAL THEORY

We thus have four in¬nite hierarchies of symmetries of the Federbush

+ ’
+
model, i.e., Y’n , Yn , Y’n , Yn , n ∈ N. A formal proof of the in¬niteness of
the hierarchies is given in Subsection 7.5.3.
7.4. Discrete symmetries. In deriving the speci¬c results for the
symmetry structure of the Federbush model, we realised that there are dis-
crete transformations which transform the Federbush model into itself and
by consequence transform symmetries into symmetries. Existence of these
disrete symmetries allow us to restrict to just one part of the Lie algebra of
symmetries, the discrete symmetries generating the remaining parts. These
discrete symmetries σ, „ are given by
σ : u1 ” u3 , v1 ” v3 , u2 ” u4 , v2 ” v4 , m1 ” m2 , » ” ’», t ” t;
„ : u1 ” u2 , v1 ” v2 , u3 ” u4 , v3 ” v4 , » ” ’», x ” ’x, t ” t. (3.117)
The transformations satisfy the following rules:
σ 2 = id,
„ 2 = id,
σ —¦ „ = „ —¦ σ.
Physically, the transformation σ denotes the exchange of two particles.
The action of the discrete smmetries on the Lie algebra of symmetries
is as follows:
σ(Yi+ ) = Yi’ ,
„ (Yi+ ) = Y’i ,
+

„ (Yi’ ) = Y’i ,



where i = 0, 1, 2,

+
σ(Z1 ) = Z1 ,
+ +
„ (Z1 ) = Z’1 ,
’ ’
„ (Z1 ) = Z’1 , (3.118)
while Y’3 , Y3’ , Y’3 , arising in the previous section, are de¬ned by

+


Y3’ = σ(Y3+ ), ’
Y’3 = „ (Y3+ ),
+
Y’3 = „ σ(Y3+ ). (3.119)
7.5. Towards in¬nite number of hierarchies of symmetries. In
this subsection, we demonstrate the existence of an in¬nite number of
hiearchies of higher symmetries of the Federbush model. We shall do this
by the construction of two (x, t)-dependent symmetries of degree 0 which
are polynomial with respect to x, t and of degree 2. This will be done in
Subsection 7.5.1.
Then, after writing the Federbush model as a Hamiltonian system, we
show that all higher symmetries obtained thusfar are Hamitonian vector
¬elds; this will be done in Subsection 7.5.2. Finally in Subsection 7.5.3
7. SYMMETRIES OF THE FEDERBUSH MODEL 139

we give a proof of a lemma from which the existence of in¬nite number of
hierarchies of Hamiltonians becomes evident, and from this we then obtain
the obvious result for the symmetry structure of the Federbush model.
7.5.1. Construction of Y + (2, 0) and Y + (2, 0). First, we start from the
presentation of these vector ¬elds, which is assumed to be of the following
structure
Y + (2, 0) = x2 (±1 Y2+ + ±2 m1 Y1+ + ±3 m2 Y0+ + ±4 m1 Y’1 + ±5 Y’2 )
+ +
1
+ 2xt(β1 Y2+ + β2 m1 Y1+ + β3 m2 Y0+ + β4 m1 Y’1 + β5 Y’2 )
+ +
1
+ t2 (γ1 Y2+ + γ2 m1 Y1+ + γ3 m2 Y0+ + γ4 m1 Y’1 + γ5 Y’2 )
+ +
1
+ + +
+ xC1 + tC2 + C0 , (3.120)

In (3.120), the ¬elds Yi+ , i = ’2, . . . , 2, are given in previous sections, ±1 ,
+ + +
βi , γi , i = 1, . . . , 5, are constant, while C1 , C2 , C0 , which are of degree 2,
2 and 1 respectively, have to be determined.
From the symmetry condition (2.29) on p. 72 we obtained the following
result: There does exist a symmetry of presentation (3.120), which is given
by
1
Y + (2, 0) = x2 (Y2+ ’ m2 Y0+ + Y’2 ) + 2xt(Y2+ ’ Y’2 )
+ +
1
2
1
+ t2 (Y2+ + m2 Y0+ + Y’2 ) + xC1 + tC2 ,
+ + +
(3.121)
1
2
whereas in (3.120) and (3.121),
‚ ‚
+
C1 = (’2v1x ’ m1 u2 ’ »R34 u1 ) + (2u1x ’ m1 v2 ’ »R34 v1 )
‚u1 ‚v1
‚ ‚
+ (’2v2x + m1 u1 ’ »R34 u2 ) + (2u2x + m1 v1 ’ »R34 v2 ) ,
‚u2 ‚v2
‚ ‚
+
C2 = (2v1x + m1 u2 + »R34 u1 ) + (’2u1x + m1 v2 + »R34 v1 )
‚u1 ‚v1
‚ ‚
+ (’2v2x + m1 u1 ’ »R34 u2 ) + (2u2x + m1 v1 ’ »R34 v2 ) ,
‚u2 ‚v2
+
C0 = 0. (3.122)
In a similar way, motivated by the structure of the Lie algebra obtained
thusfar, we get another higher symmetry of a similar structure, i.e.,
1
Y ’ (2, 0) = x2 (Y2’ ’ m2 Y0’ + Y’2 ) + 2xt(Y2’ ’ Y’2 )
’ ’
2
2
1
+ t2 (Y2’ ’ m2 Y0’ + Y’2 ) + xC1 + tC2 ,
’ ’ ’
(3.123)
2
2
whereas in (3.123),
‚ ‚

C1 = (’2v3x ’ m2 u4 + »R12 u3 ) + (2u3x ’ m2 v4 + »R12 v3 )
‚u3 ‚v3
140 3. NONLOCAL THEORY

‚ ‚
+ (’2v4x + m2 u3 + »R12 u4 ) + (2u4x + m2 v3 + »R12 v4 ) ,
‚u4 ‚v4
‚ ‚

C2 = (2v3x + m2 u4 ’ »R12 u3 ) + (’2u3x + m2 v4 ’ »R12 v3 )
‚u3 ‚v3
‚ ‚
+ (’2v4x + m2 u3 + »R12 u4 ) + (2u4x + m2 v3 + »R12 v4 ) ,
‚u4 ‚v4

C0 = 0. (3.124)

To give an idea of the action of the vector ¬elds Y + (2, 0), Y ’ (2, 0), we
compute their Lie brackets with the vector ¬elds Y1+ , Y0+ , Y’1 , Y1’ , Y0’ ,
+

Y’1 , yielding the following results

[Y ’ (2, 0), Y1’ ] = 2Z1 ,

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