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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

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