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or equivalently, the generating functions associated to them, given by
V1u = ux ,
V2u = uux + uxxx ,
V3u = 1 ’ tux ,
V4u = xux + 3t(uux + uxxx ) + 2u.
Associated to (3.28), we can construct conservation laws Ax , At such that
Dt (Ax ) = Dx (At ), (3.29)
which leads to
A1 = u,
x
1
A1 = u2 + uxx ,
t
2
Ax = u 2 ,
2
112 3. NONLOCAL THEORY

2
A2 = u3 ’ u2 + 2uuxx . (3.30)
t x
3
A few higher conservation laws are given by
A3 = u3 ’ 3u2 ,
x x
3
A3 = (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ),
t x xx
4
36
A4 = u4 ’ 12uu2 + u2 ,
x x
5 xx
4
A4 = u5 + 4u3 uxx ’ 18u2 u2 ’ 24uux uxxx + 12u2 uxx ,
t x x
5
96 72 36
+ uu2 + uxx uxxxx ’ u2 . (3.31)
xx
5 xxx
5 5
We now introduce nonlocal variables associated to two of conservation
laws (3.30) in the form

p1 = u dx,

(u2 ) dx.
p3 = (3.32)

We also introduce the grading to the polynomial functions on the KdV
equation by setting
[x] = ’1, [t] = ’3, [u] = 2, [ux ] = 3, [ut ] = 5, . . . (3.33)
Then the nonlocal variables p1 and p3 are of degree
[p1 ] = 1, [p3 ] = 3.
In order to study nonlocal symmetries of the KdV equation, we consider the
augmented system
ut = uux + uxxx ,
(p1 )x = u,
1
(p1 )t = u2 + uxx ,
2
(p3 )x = u2 ,
2
(p3 )t = u3 ’ u2 + 2uuxx . (3.34)
x
3
We note here that system (3.34) is in e¬ect a system of partial di¬eren-
tial equations in three dependent variables u, p1 , p3 and two independent
variables x, t. We choose internal coordinates on E ∞ — R2 as
x, t, u, p1 , p3 , ux , uxx , uxxx , uxxxx , uxxxxx , . . . , (3.35)
while the total derivative operators D x , Dt are given as
‚ ‚
+ u2
D x = Dx + u ,
‚p1 ‚p3
5. NONLOCAL SYMMETRIES OF THE KDV EQUATION 113

12 ‚ 23 ‚
u ’ u2 + 2uuxx
D t = Dt + u + uxx + . (3.36)
x
2 ‚p1 3 ‚p3
A vertical vector ¬eld V on E ∞ — R2 has as its generating functions V u ,
V p1 , V p3 . The symmetry conditions resulting from (3.34) are
3
Dt V u = V u ux + uDx V u + Dx V u ,
D x V p1 = V u ,
Dx V p3 = 2uV u . (3.37)
For the vertical vector ¬elds V1 , . . . , V4 we derive from this after a short
computation
V1u = ux , V2u = uux + uxxx ,
1
V1p1 = u, V2p1 = u2 + uxx ,
2
2
V1p3 = u2 , V2p3 = u3 + 2uuxx ’ u2 ,
x
3
V3u = 1 ’ tux , V4u = xux + 3t(uux + uxxx ) + 2u,
12
V3p1 = x ’ tu, V4p1 = xu + 3t u + uxx + p1 ,
2
23
V3p3 = 2p1 ’ tu2 , V4p3 = xu2 + 3t u + 2uuxx ’ u2 + 3p3 . (3.38)
x
3
It is a well-known fact [80] that the KdV equation (3.28) admits the Lenard
recursion operator for higher symmetries, i.e.,
2 1 ’1
2
L = D x + u + u x Dx . (3.39)
3 3
From this we have
L(V1u ) = V2u ,
5 10 5
L(V2u ) = V5u = uxxxxx + uxxx u + uxx ux + ux u2 ,
3 3 6
2 1 1
L(V3u ) = u + xux + t(uux + uxxx ) = V4u . (3.40)
3 3 3
We now compute the action of the Lenard recursion operator L on the
generating function V4u of the symmetry V4 . The result is
V5u = L(V4u ) = x(uxxx + uux )
5 10 5 4 1
+ 3t uxxxxx + uxxx u + uxx ux + ux u2 + 4uxx + u2 + ux p1 .
3 3 6 3 3
(3.41)
It is a straightforward check that V5u satis¬es the ¬rst condition of (3.37),
i.e.,
3
Dt (V5u ) = V5u ux + uDx V5u + Dx V5u . (3.42)
114 3. NONLOCAL THEORY

The component V5p1 can be computed directly from the second condition
in (3.37), i.e.,
Dx (V5p1 ) = V5u , (3.43)
which readily leads to
1
V5p1 = x uxx + u2
2
5 5 5 1 1
+ 3t uxxxx + uxx u + u2 + u3 + 3ux + up1 + p3 . (3.44)
x
3 6 18 3 2
The construction of the component V5p3 , which should result from the third
condition in (3.37), i.e.,
Dx (V5p3 ) = 2uV5u , (3.45)
causes a problem:
It is impossible to derive a formula for V5p3 in this setting.
The way out of this problem is to augment system (3.34) once more with
the nonlocal variable p5 resulting from
(p5 )x = u3 ’ 3u2 ,
x
3
(p5 )t = (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ), (3.46)
x xx
4
or equivalently

(u3 ’ 3u2 ) dx,
p5 = (3.47)
x

and extending total derivative operators D x , Dt to

Dx = Dx + (u3 ’ 3u2 ) ,
x
‚p5
3 ‚
Dt = Dt + (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ) . (3.48)
x xx
4 ‚p5
Within this once more augmented setting, i.e., having a system of par-
tial di¬erential equations for u, p1 , p3 , and p5 , it is posssible to solve the
symmetry condition for p3 , (3.34):
Dx (V5p3 ) = 2uV5u , (3.49)
the result being the vertical vector ¬eld V5 whose generating functions are
given by (3.41), (3.44), and from (3.49) we obtain
5 10 5
V5u = x(uxxx + uux ) + 3t uxxxxx + uxxx u + uxx ux + ux u2
3 3 6
4 1
+ 4uxx + u2 + ux p1 ,
3 3
1 5 5 5
V5p1 = x uxx + u2 + 3t uxxxx + uxx u + u2 + u3
6 x 18
2 3
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 115

1 1
+ 3ux + up1 + p3 ,
3 2
1 1
V5p3 = 2x uuxx ’ u2 + u3
x
2 3
1 5 5
+ 6t uuxxxx ’ ux uxxx + u2 + u2 uxx + u5
2 xx 3 24
1 5
+ 6uux + u2 p1 + p5 . (3.50)
3 3
The outline above indicates that we are working in e¬ect in an aug-
mented system of partial di¬erential equations in which all nonlocal vari-
ables associated to all conservation laws for the KdV equation are incorpo-
rated (cf. Theorem 3.8).
The computation of Lie brackets of vertical vector ¬elds, or equivalently,
the computation of the Jacobi brackets for the associated generating func-
tions, is to be carried out in this augmented setting. To demonstrate this,
we want to compute the Lie bracket of the symmetry V1 and the nonlocal
symmetry V5 with the generating functions
V1u = ux ,
5 10 5
V5u = x(uxxx + uux ) + 3t uxxxxx + uxxx u + uxx ux + ux u2
3 3 6
4 1
+ 4uxx + u2 + ux p1 . (3.51)
3 3
The associated Jacobi bracket {V5u , V1u } is de¬ned as
V u = {V5u , V1u } = u u

V5 (V1 ) V1 (V5 ), (3.52)
which, using in this computation the equality V1p1 = u, results in
V u = uxxx + uux = V2u .
In a similar way the Jacobi bracket {V5u , V2u } equals
5 10 5
{V5u , V2u } = 3 uxxxxx + uxxx u + uxx ux + ux u2 ,
3 3 6
which is just the generating function of the classical ¬rst higher symmetry
of the KdV equation.
Remark 3.2. The functions Viu , i = 1, . . . , 5, are just the so-called shad-
ows (see the previous section) of the symmetries Vi , i = 1, . . . , 5, in the
augmented setting, including all nonlocal variables.

6. Symmetries of the massive Thirring model
We shall establish higher and nonlocal symmetries of the so-called mas-
sive Thirring model [32], which is de¬ned as the following system E 0 of
partial di¬erential equations de¬ned on J 1 (π), where π : R4 — R2 ’ R2 is
116 3. NONLOCAL THEORY

the trivial bundle with the coordinates u1 , v1 , u2 , v2 in the ¬ber (unknown
functions) and x, t in the base (independent variables):
‚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
‚v2 ‚v2
= mu1 ’ (u2 + v1 )u2 .
2
’ ’ (3.53)
1
‚x ‚t
For this system of equations we choose internal coordinates on E 1 as x, t, u1 ,
v1 , u2 , v2 , u1,1 , v1,1 , u2,1 , v2,1 , while internal coordinates on E 4 are chosen as
x, t, u1 , v1 , u2 , v2 , . . . , u1,4 , v1,4 , u2,4 , v2,4 , where ui,j , vi,j refer to ‚ j ui /‚xj ,
‚ j vi /‚xj , i = 1, 2, j = 1, . . . , 4. In a similar way coordinates can be choosen
on E ∞ .

6.1. Higher symmetries. According to Theorem 2.15 on p. 72, we
construct higher symmetries (symmetries of order 2) by constructing vertical
vector ¬elds • , where the generating functions •u1 , •v1 , •u2 , •v2 depend
on the local variables x, t, u1 , v1 , u2 , v2 , u1,1 , v1,1 , u2,1 , v2,1 , u1,2 , v1,2 , u2,2 ,
v2,2 [41]. The symmetry condition then is
’Dx •u1 + Dt •u1 = m•v2 ’ 2(u2 •u2 + v2 •v2 )v1 + (u2 + v2 )•v1 ,
2
2
Dx •u2 + Dt •u2 = m•v1 ’ 2(u1 •u2 + v1 •v2 )v2 + (u2 + v1 )•v2 ,
2
1
Dx •v1 ’ Dt •v1 = m•u2 ’ 2(u2 •u2 + v2 •v2 )u1 + (u2 + v2 )•u1 ,
2
2
’Dx •v2 ’ Dt •v2 = m•u1 ’ 2(u1 •u2 + v1 •v2 )u2 + (u2 + v1 )•u2 .
2
(3.54)
1

The result then is the existence of four symmetries X1 , . . . , X4 of order 1
the generating functions of which, •u1 , •v1 , •u2 , •v2 , i = 1, . . . , 4, are given
i i i i
as
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
•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
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 117

1
•v2 = (’mu1 + u2 (u2 + v1 )),
2
1
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
•u 1 = v 1 ,
4
•v1 = ’u1 ,
4
•u 2 = v 2 ,
4
•v2 = ’u2 . (3.55)
4

Thus in e¬ect, the ¬elds X1 , X2 , X3 are of the ¬rst order, while X4 is of
order zero.
In order to ¬nd symmetries of higher order, we take great advantage of
the fact that the massive Thirring model is a graded system, as is the case
with all equations possessing a scaling symmetry, i.e.,
deg(x) = deg(t) = ’2,
deg(u1 ) = deg(v1 ) = deg(u2 ) = deg(v2 ) = 1,
‚u1
deg(m) = 2, deg = 3, . . . (3.56)
‚x
Due to this grading, all equations in (3.53) are of degree three; the total
derivative operators Dx , Dt are graded too as is the symmetry condition
0
) = 0 mod E 3 .
• (E (3.57)

The solutions of (3.57) are graded too. Note that the ¬elds X1 , . . . , X4 are
of degrees 2, 2, 0, 0 respectively.
We now introduce the following notation:
[u] refers to u1 , v1 , u2 , v2 ,
[u]x refers to u1,1 , v1,1 , u2,1 , v2,1 ,
..............................
In our search for higher symmetries we are not constructing the general
solution of the overdetermined system of partial di¬erential equations for
the generating functions •u1 , •v1 , •u2 , •v2 , resulting from (3.57).
We are just looking for those (x, t)-independent functions which are of
degree ¬ve; so the presentation of these functions is as follows:
•— = [u]xx + ([u]2 + [m])[u]x + ([u]5 + [m][u]3 + [m]2 [u]). (3.58)
118 3. NONLOCAL THEORY

Using the presentation above, we derive two higher symmetries, X5 and X6
of degree 4 and order 2, whose generating functions are given as
1

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