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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

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