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+ (x2 un+2 + 2(n + 1)xun+1 + n(n + 1)un )(2xr un+1 + rxr’1 un ). (3.146)
By expanding the expressions in (3.146), we arrive, after a short calculation,
at
r+1
{H1 , Kn } = (2n + 1 ’ r)(4Hn+1 ’ r2 Hn ),
2 r r’1
(3.147)
which proves the fourth relation in (3.141).
We are now in a position to formulate and prove the main theorem of this
subsection.
Theorem 3.13. The conserved functionals F (Y ± (2, 0)) associated to
the symmetries Y ± (2, 0) generate in¬nite number of hierarchies of Hamilto-
nians, starting at the hierarchies F (Yi+ ), F (Yi’ ), where i ∈ Z, by repeated

+
action of the Poisson bracket (3.140). The hierarchies F (Zj ), F (Zj ),
j ∈ Z, are obtained by the ¬rst step of this procedure.
146 3. NONLOCAL THEORY

Moreover, the hierarchies F (Yj+ ), F (Yj’ ), j ∈ Z, are obtained from
± ±
F (Y±1 ) by repeated action of the conserved functionals F (Z±1 )
1
± ±
F (Z±1 ) = ± F [Y ± (2, 0), Y±1 ] . (3.148)
2
Proof. The proof of this theorem is a straightforward application of
the previous lemma, and the observation that the (», m1 , m2 )-independent
±
parts of the conserved densities Y±1 , Y + (2, 0), Y ’ (2, 0) are just given by
1
F(Y1+ ) ’’ ’ (u2x v2 ’ v2x u2 ),
2
1
+
F(Y’1 ) ’’ ’ (u1x v1 ’ v1x u1 ),
2
1
F(Y1’ ) ’’ ’ (u4x v4 ’ v4x u4 ),
2
1

F(Y’1 ) ’’ ’ (u3x v3 ’ v3x u3 ),
2
1 1
F(Y + (2, 0)) ’’ ’ (x + t)2 (u2 + v2x ) ’ (x ’ t)2 (u2 + v1x ),
2 2
2x 1x
2 2
1 1
F(Y ’ (2, 0)) ’’ ’ (x + t)2 (u2 + v4x ) ’ (x ’ t)2 (u2 + v3x ).
2 2
4x 3x
2 2
Note that in applying the lemma we have to choose (u, v) = (u1 , v1 ), etc.
7.6. Nonlocal symmetries. In this last subsection concerning the
Federbush model, we discuss existence of nonlocal symmetries. We start
from the conservation laws, conserved quantities and the associated nonlo-
cal variables p1 , p2 :
p1t = ’R1 + R2 ,
p1x = R1 + R2 ,
p2t = ’R3 + R4 .
p2x = R3 + R4 , (3.149)
Including these two nonlocal variables, we ¬nd two new nonlocal symmetries
‚ ‚ ‚ ‚ ‚ ‚
Z + (0, 0) = u1 ’ »p1 v3 ’ u3
+ v1 + u2 + v2
‚u1 ‚v1 ‚u2 ‚v2 ‚u3 ‚v3
‚ ‚ ‚
’ u4
+ v4 + 2p1 ,
‚u4 ‚v4 ‚p1
‚ ‚ ‚ ‚ ‚ ‚
Z ’ (0, 0) = u3 ’ u1
+ v3 + u4 + v4 + »p2 v1
‚u3 ‚v3 ‚u4 ‚v4 ‚u1 ‚v1
‚ ‚ ‚
’ u2
+ v2 + 2p2 . (3.150)
‚u2 ‚v2 ‚p2
Analogously to the construction of conservation laws and nonlocal variables
in previous sections, we obtained nonlocal variables p3 , p4 , p5 , p6 de¬ned by
1
p3x = »(R1 + R2 )R4 + m2 (u3 u4 + v3 v4 ) ’ u4 v4x + v4 u4x ,
2
1
p3t = »(R1 + R2 )R4 ’ u4 v4x + v4 u4x ,
2
7. SYMMETRIES OF THE FEDERBUSH MODEL 147

1
p4x = »(R1 + R2 )R3 + m2 (u3 u4 + v3 v4 ) + u3 v3x ’ v3 u3x ,
2
1
p4t = »(R1 + R2 )R3 ’ u3 v3x + v3 u3x ,
2
1
p5x = »(R3 + R4 )R2 ’ m1 (u1 u2 + v1 v1 ) + u2 v2x ’ v2 u2x ,
2
1
p5t = »(R3 + R4 )R2 + u2 v2x ’ v2 u2x ,
2
1
p6x = »(R3 + R4 )R1 + m1 (u1 u2 + v1 v1 ) + u1 v1x ’ v1 u1x ,
2
1
p6t = ’ »(R3 + R4 )R1 ’ u1 v1x + v1 u1x . (3.151)
2
Using these nonlocal variables we ¬nd four additional nonlocal symmetries
Z + (0, ’1), Z + (0, +1), Z ’ (0, ’1), Z ’ (0, +1):
1 ‚
Z + (0, ’1) = ’ »u1 (R3 + R4 ) ’ m1 u2 ’ 2v1x
2 ‚u1
1 ‚
’ »v1 (R3 + R4 ) ’ m1 v2 + 2u1x
+
2 ‚v1
1 ‚ 1 ‚
’ m 1 u1 ’ m 1 v1
2 ‚u2 2 ‚v2
‚ ‚ ‚ ‚
’ u3 ’ u4
+ »p6 v3 + v4 ,
‚u3 ‚v3 ‚u4 ‚v4
1 ‚ 1 ‚
Z + (0, +1) = m1 u2 + m 1 v2
2 ‚u1 2 ‚v1
1 ‚
’ »u2 (R3 + R4 ) + m1 u1 ’ 2v2x
+
2 ‚u2
1 ‚
’ »v2 (R3 + R4 ) + m1 v1 + 2u2x
+
2 ‚v2
‚ ‚ ‚ ‚
’ u3 ’ u4
+ »p5 v3 + v4 ,
‚u3 ‚v3 ‚u4 ‚v4
‚ ‚ ‚ ‚
Z ’ (0, ’1) = ’»p4 (v1 ’ u1 ’ u2
+ v2 )
‚u1 ‚v1 ‚u2 ‚v2
1 ‚
’ »u3 (R1 + R2 ) ’ m2 u4 ’ 2v3x
+
2 ‚u3
1 ‚
+ »v3 (R1 + R2 ) ’ m2 v4 + 2u3x
+
2 ‚v3
1 ‚ 1 ‚
’ m 2 u3 ’ m 2 v3 ,
2 ‚u4 2 ‚v4
‚ ‚ ‚ ‚
Z ’ (0, +1) = »p3 v1 ’ u1 ’ u2
+ v2
‚u1 ‚v1 ‚u2 ‚v2
1 ‚ 1 ‚
+ m 2 u4 + m 2 v4
2 ‚u3 2 ‚v3
148 3. NONLOCAL THEORY

1 ‚
»u4 (R1 + R2 ) + m2 u3 ’ 2v4x
+
2 ‚u4
1 ‚
+ »v4 (R1 + R2 ) + m2 v3 + 2u4x .
2 ‚v4
According to standard lines of computations, including prolongation towards
nonlocal variables as explained in previous sections, we arrive at the follow-
ing commutators:
[Y ± (1, ±1), Z ± (0, 0)] = 0,

[Y + (1, ’1), Z + (0, ’1)] = Z + (0, ’2),
1
[Y + (1, ’1), Z + (0, +1)] = ’ m2 Z + (0, 0),
41
1
[Y + (1, +1), Z + (0, ’1)] = m2 Z + (0, 0),
41
[Y + (1, +1), Z + (0, +1)] = Z + (0, +2)
and
[Y ’ (1, ’1), Z ’ (0, ’1)] = Z ’ (0, ’2),
1
[Y ’ (1, ’1), Z ’ (0, +1)] = ’ m2 Z ’ (0, 0),
42
1
[Y ’ (1, +1), Z ’ (0, ’1)] = m2 Z ’ (0, 0),
42
[Y ’ (1, +1), Z ’ (0, +1)] = Z ’ (0, +2), (3.152)
the vector ¬elds Z + (0, ’2), Z + (0, +2), Z ’ (0, ’2), Z ’ (0, +2) just being new
nonlocal symmetries.
Summarising these results, we conclude that the action of the symmetries
± (1, ±1) on Z ± (0, ±1) constitute hierarchies of nonlocal symmetries.
Y
Finally we compute the Lie brackets of Y + (2, 0), (3.121), and Z + (0, ±1)
which results in
[Y + (2, 0), Z + (0, ’1)] = Z + (1, ’1),
[Y + (2, 0), Z + (0, +1)] = Z + (1, +1), (3.153)
whereas in (3.153) Z + (0, ±1) are de¬ned by
1
Z + (1, ’1) = 2(’x + t)Z + (0, ’2) + m2 (x + t)Z + (0, 0)
21
‚ ‚
+ »v1 R34 + m1 v2 ’ 2u1x ’ »u1 R34 + m1 u2 + 2v1x
‚u1 ‚v1
» ‚ ‚ ‚ ‚ +
’ ’ u3 ’ u4
v3 + v4 K’1 ,
2 ‚u3 ‚v3 ‚u4 ‚v4
1
Z + (1, +1) = 2(x + t)Z + (0, ’2) + m2 (x ’ t)Z + (0, 0)
21

+ »v2 R34 ’ m1 v1 ’ 2u2x
‚u2
¨
8. BACKLUND TRANSFORMATIONS AND RECURSION OPERATORS 149


’ »u2 R34 + m1 u1 + 2v2x
‚v2
» ‚ ‚ ‚ ‚ +
’ ’ u3 ’ u4
v3 + v4 K+1 , (3.154)
2 ‚u3 ‚v3 ‚u4 ‚v4
+
while K±1 are given by
x x x x
+ +
m2 F(Y + (0, 0)),
F(Y (0, ’2)) ’
K’1 =8 1
’∞ ’∞ ’∞ ’∞
x x x x
+
F(Y + (0, +2)) ’ m2 F(Y + (0, 0)).
K+1 = 8 (3.155)
1
’∞ ’∞ ’∞ ’∞

The previous formulas re¬‚ect the fact that Y + (2, 0) constructs an (x, t)-
dependent hierarchy Z + (1, —) from Z + (0, —) by action of the Lie bracket. We
expect similar results for the action of Y + (2, 0) on the hierarchy Z + (1, —).
Results conserning the action of Y ’ (2, 0) on Z ’ (0, —) and from this, on
Z ’ (1, —) will be similar.

8. B¨cklund transformations and recursion operators
a
In this section, we mainly follow the results by M. Marvan exposed in
[73]. Our aim here is to show that recursion opeartors for higher symmetries
may be unberstood as B¨cklund transformations of a special type.
a
Let E1 and E2 be two di¬erential equations in unknown functions u1 and
u2 respectively. Informally speaking, a B¨cklund transformation between E1
a
and E2 is a third equation E containing both independent variables u1 and
u2 and possessing the following property:
1. If u1 is a solution of E1 , then solving the equation E[u1 ] with respect
0 0
to u2 , we obtain a family of solutions to E2 .
2. Vice versa, if u2 is a solution of E2 , then solving the equation E[u2 ]
0 0
with respect to u1 , we obtain a family of solutions to E1 .
Geometrically this construction is expressed in a quite simple manner.
Definition 3.10. Let N1 and N2 be objects of the category DM∞ . A
B¨klund transformation between N1 and N2 is a pair of coverings
a
N
•2
1











N1 N2
where N is a third object of DM∞ . A B¨cklund transformation is called a
a
B¨cklund auto-transformation, if N1 = N2 .
a
In fact, let Ni = Ei∞ , i = 1, 2, and s ‚ E1 be a solution. Then the set


•’1 s ‚ N is ¬bered by solutions of N and they are projected by •2 (at
1

nonsingular points) to a family of solutions of E2 .
150 3. NONLOCAL THEORY

We are now interested in B¨cklund auto-transformations of the total
a
v : V E ∞ ’ E ∞ (see Example 3.2). The reason
space of the Cartan covering „
to this is the following
Proposition 3.14. A section X : E ∞ ’ V E ∞ of the projection „ v is a
symmetry of the equation E if and only if it is a morphism in the category
DM∞ , i.e., if it preserves Cartan distributions.
The proof is straightforward and is based on the de¬nition of the Cartan
distribution on V E ∞ . The result is in full agreement with equalities (3.2)
on p. 101: the equations for V E ∞ are just linearization of E and symmetries
are solutions of the linearized equation.
Thus, we can hope that B¨cklund auto-transformations of V E ∞ will
a
relate symmetries of E to each other. This motivates the following
Definition 3.11. Let E ∞ be an in¬nitely prolonged equation. A recur-
sion operator for symmetries of E is a pair of coverings K, L : R ’ V E ∞
such that the diagram
R
K




L









V E∞ V E∞
v

v












E∞
is commutative. A recursion operator is called linear, if both K and L are
linear coverings.
Example 3.4. Consider the KdV equation E = {ut = uux + uxxx }.
Then V E ∞ is described by additional equation
vt = uvx + ux v + vxxx .
Let us take for R the system of equations
wx = v,
wt = vxx + uv,
vt = vxxx uvx + ux v,
ut = uxxx + uux ,
while the mappings K and L are given by
K : v = wx ,
2 1
L : v = vxx + uv + ux w.
3 3
¨
8. BACKLUND TRANSFORMATIONS AND RECURSION OPERATORS 151

Obviously, K and L determine covering structures over V E ∞ (the ¬rst being
one-dimensional and the second three-dimensional) while the triple (R, K, L)
corresponds to the classical Lenard operator Dx + 3 u + 1 ux Dx .
2 ’1
2
3
Let us now study action of recursion operators on symmetries in more
details. Let X be a symmetry of an equation E. Then, due to Proposition
3.14, it can be considered as a section X : E ∞ ’ V E ∞ which is a morphism
in DM∞ . Thus we obtain the following commutative diagram
X— L

’ V E∞
R ’R


P = X — (K) „v
K
“ “ “
„v
X
∞ ∞
’ E∞
E ’VE
where the composition of the arrows below is the identity while P = X — (K)
is the pull-back. As a consequence, we obtain the following morphism of
coverings
L —¦ X—

’ V E∞
R
v
P










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