<< . .

. 19
( : 58)



. . >>

•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 ) + 4v1,1 u2 v1
5
4
2
’ 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
•u1 = (4v1,2 + 2u2,1 (’m + 2u1 u2 ) + 4u1,1 (R1 + R2 ) + 4v2,1 u1 v2
6
4
2
+ mv2 (R1 + R2 ) + 2mv1 R + v1 (R2 + 2R1 R2 )),
1
•v1 = (’4u1,2 + 2v2,1 (’m + 2v1 v2 ) + 4v1,1 (R1 + R2 ) + 4u2,1 u2 v1
6
4
2
’ mu2 (R1 + R2 ) ’ 2mu1 R + u1 (R2 + 2R1 R2 )),
1
•u2 = (2u1,1 (’m + 2v1 v2 ) ’ 4v1,1 u1 v2 + mv1 (R1 + R2 )
6
4
2
+ 2mv2 R ’ 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 )), (3.59)
whereas in (3.59)
R1 = u2 + v1 ,
2
R2 = u 2 + v 2 ,
2
R = u 1 u 2 + v 1 v2 .
1 2
For third order higher symmetries the representation of the generating func-
tions, whose degree is seven, is
•— = [u]xxx + ([u]2 + [m])[u]xx + [u][u]2
x
+ ([u]4 + [m][u]2 + [m]2 )[u]x
+ ([u]7 + [m][u]5 + [m]2 [u]3 + [m]3 [u]).
After a massive computation, we arrive at the existence of higher symmetries
X7 and X8 of degree 6 and order 3, given by
1
•u1 = (8u2,2 u2 v1 + 4v2,2 (2v1 v2 ’ m) ’ 4u2 v1
2,1
7
8
2 2 2
+ 4u2,1 (m(R1 + R2 + v1 + v2 ) ’ 3v1 v2 (R1 + R2 )) ’ 4v2,1 v1
6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 119

+ 4v2,1 (’m(u1 v1 + u2 v2 ) + 3u2 v1 (R1 + R2 )) + 4u1,1 mR
’ 2m2 v1 (R1 + R2 ) ’ 4v2 m2 R + 4v1 mR(R1 + 2R2 )
2 2 3 2 2
+ v2 m(R1 + 4R1 R2 + R2 ) ’ v1 (R2 + 6R2 R1 + 3R2 R1 )),
1
• v1 2
= (’8v2,2 v2 u1 ’ 4u2,2 (2u1 u2 ’ m) + 4v2,1 u1
7
8
+ 4v2,1 (m(R1 + R2 + u2 + u2 ) ’ 3u1 u2 (R1 + R2 ))
1 2
+ 4u2 u1 + 4u2,1 (’m(u1 v1 + u2 v2 ) + 3v2 u1 (R1 + R2 )) + 4v1,1 mR
2,1
+ 2m2 u1 (R1 + R2 ) + 4u2 m2 R ’ 4u1 mR(R1 + 2R2 )
2 2 3 2 2
’ u2 m(R1 + 4R1 R2 + R2 ) + u1 (R2 + 6R2 R1 + 3R2 R1 )),
1
•u 2 = (8u2,3 + 12v2,2 (R1 + R2 ) + 8u1,2 u1 v2 + 4v1,2 (2v1 v2 ’ m)
7
8
’ 12u2 v2 + 24u2,1 v2,1 u2 + 2u2,1 (10mR ’ 3R1 ’ 12R1 R2 ’ 3R2 )
2 2
2,1
+ 12v2,1 v2 + 24v2,1 u1,1 u1 + 24v2,1 v1,1 v1 + 8u2 v2
2
1,1
+ 4u1,1 (m(R1 + R2 + u2 + u2 ) ’ 3u1 u2 (R1 + R2 )) + 8v1,1 v2
2
1 2
+ 4v1,1 (m(u1 v1 + u2 v2 ) ’ 3u2 v1 (R1 + R2 )) ’ 4m2 v1 R
’ 2m2 v2 (R1 + R2 ) + mv1 (R2 + 4R1 R2 + R1 ) + 4mv2 R(R2 + 2R1 )
2 2

3 2 2
’ v2 (R1 + 6R1 R2 + 3R1 R2 )),
1
• v2 = (8v2,3 ’ 12u2,2 (R1 + R2 ) + 8v1,2 u2 v1 ’ 4u1,2 (2u1 u2 ’ m)
7
8
2 2 2
’ 12v2,1 u2 ’ 24u2,1 v2,1 v2 + 2v2,1 (10mR ’ 3R1 ’ 12R1 R2 ’ 3R2 )
+ 12u2 u2 + 24u2,1 v1,1 v1 + 24u2,1 u1,1 u1 ’ 8v1,1 u2
2
2,1
+ 4v1,1 (m(R1 + R2 + v1 + v2 ) ’ 3v1 v2 (R1 + R2 )) ’ 8u2 u2
2 2
1,1
+ 4u1,1 (m(u1 v1 + u2 v2 )) ’ 3v2 u1 (R1 + R2 ) + 4m2 u1 R
+ 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

<< . .

. 19
( : 58)



. . >>