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ψ1 of a new symmetry on Nψ . By Theorem 3.11, there exists a covering,
where both ψ and ψ1 are realized as nonlocal symmetries. Thus we can
continue the procedure applying ω to ψ1 and eventually arrive to a covering
in which the whole series {ψk } is realized.
Thus, we can state that classical recursion operators are nonlocal de-
formations of the equation structure. Algorithmically, computation of such
deformations ¬ts the following scheme:
(1)
1. Take an equation E and solve the linear equation E ω = 0, where ω
is an arbitrary Cartan form.
2. If solutions are trivial, take a covering • : N ’ E ∞ and try to ¬nd
shadows of recursion operators. Usually, such a covering is given by
conservation laws of the equation E.
3. If necessary, add another nonlocal variable (perhaps, de¬ned by a
nonlocal conservation law), etc.
4. If you succeeded to ¬nd a nontrivial solution „¦, then the correspond-
ing recursion operator acts by the rule R„¦ : ψ ’ ψ „¦, where ψ is
the generating function of a symmetry.
In the examples below, we shall see how this algorithm works.
Remark 5.12. Let us establish relation between recursion operators in-
troduced in this chapter with their interpretation as B¨cklund transforma-
a
tions given in Section 8 of Chapter 3.
Let „¦ be a shadow of a recursion operator in come covering • : N ’ E ∞ .
Then we can consider the following commutative diagram:
V
„N
N← VN
V„¦




• V•
“ “
V
„E






E← VE VE
5. DEFORMATIONS OF THE BURGERS EQUATION 221

V V
where „E and „N are the Cartan coverings of the equation and its covering
respectively, V• is naturally constructed by •, while the mapping V„¦ is
de¬ned by V„¦ (v) = v „¦, v ∈ V N . The pair (V• , V„¦ ) is the B¨cklund
a
transformation corresponding to the recursion operator de¬ned by „¦.
This interpretation is another way to understand why shadows of recur-
sion operators take symmetries to shadows of symmetries (see Section 8 of
Chapter 3).

5. Deformations of the Burgers equation
Deformations of the Burgers equation
ut = uu1 + u2 (5.83)
will be discussed from the point of view of the theory of deformations in
coverings. We start with the following theorem (see Theorem 5.19 above):
Theorem 5.26. The only solution of the deformation equation
(1)
E („¦) =0
for the Burgers equation (5.83) is ω = ±ω0 where ± is a constant and
ω0 = du ’ u1 dx ’ (uu1 + u2 ) dt
i.e., Cartan form associated to u. This leads to the trivial deformation of
UE for (5.83).
In order to ¬nd nontrivial deformations for the Burgers equation, we
have to discuss them in the nonlocal setting. So in order to arrive at an
augmented system, a situation similar to that one for the construction of
nonlocal symmetries (see Section 4 of Chapter 3), we ¬rst have to construct
conservation laws for the Burgers equation and from this we have to intro-
duce nonlocal variables.
The only conservation law for the Burgers equation is given by
1
Dt (u) = Dx u2 + u1 , (5.84)
2
which is just the Burgers equation itself.
In (5.84), the total derivative operators Dx and Dt are given in local
coordinates on E, x, t, u, u1 , u2 , . . . , by
‚ ‚ ‚
Dx = + u1 + u2 + ...,
‚x ‚u ‚u1
‚ ‚ ‚
Dt = + ut + u1t + ... (5.85)
‚t ‚u ‚u1
The conservation law (5.84) to the introduction of the new nonlocal variable
y, which satis¬es formally the additional partial di¬erential equations
yx = u,
1
yt = u 2 + u 1 . (5.86)
2
222 5. DEFORMATIONS AND RECURSION OPERATORS

We now start from the covering E 1 = E ∞ —R, where the Cartan distribution,
or equivalently the total derivative operators Dx , Dt , is given by

Dx = D x + u ,
‚y
12 ‚
Dt = D t + u + u1 , (5.87)
2 ‚y
x
where y is the (formal) nonlocal variable y = u dx with the associated
Cartan form ω’1 de¬ned by
12
ω’1 = dy ’ u dx ’ u + u1 dt. (5.88)
2
Local coordinates in E 1 are given by
(x, t, y, u, u1 , . . . ).
We now demonstrate the calculations involved in the computations of defor-
mations of a partial di¬erential equation or a system of di¬erential equations.
In order to construct deformations of the Burgers equation (5.83)

i
Dx („¦) —
U= , (5.89)
‚ui
we start at the generating form
„¦ = F 0 ω0 + F 1 ω1 + F 2 ω2 + F 3 ω3 + F ’1 ω’1 , (5.90)
where F i , i = ’1, . . . , 3, are functions dependent on u, u1 , u2 , u3 , u4 , u5 , y.
The Cartan forms ω’1 , . . . , ω3 are given by
ω0 = du ’ u1 dx ’ (uu1 + u2 ) dt,
ω1 = du1 ’ u2 dx ’ (u2 + uu2 + u3 ) dt,
1
ω2 = du2 ’ u3 dx ’ (uu3 + 3u1 u2 + u4 ) dt,
ω3 = du3 ’ u4 dx ’ (uu4 + 4u1 u3 + 3u2 + u5 ) dt,
2
12
ω’1 = dy ’ u dx ’ u + u1 dt, (5.91)
2
and it is a straightforward computation to show that
Dx (ωi ) = ωi+1 ,
Dt (ω’1 ) = uω0 + ω1 ,
Dt (ω0 ) = u1 ω0 + uω1 + ω2 ,
Dt (ω1 ) = u2 ω0 + 2u1 ω1 + uω2 + ω3 ,
Dt (ω2 ) = u3 ω0 + 3u2 ω1 + 3u1 ω2 + uω3 + ω4 ,
Dt (ω3 ) = u4 ω0 + 4u3 ω1 + 6u2 ω2 + 4u1 ω3 + uω4 + ω5 , (5.92)
where i = ’1, 0, . . .
5. DEFORMATIONS OF THE BURGERS EQUATION 223

Now the equation for nonlocal deformations is (4.65), see p. 185,
1
E 1 („¦) = 0.
Since this one amounts to
2
Dt („¦) ’ u1 „¦ ’ uDx („¦) ’ Dx („¦) = 0, (5.93)
we are led to an overdetermined system of partial di¬erential equations for
the functions F ’1 , . . . , F 3 , by equating coe¬cients of ω’1 , . . . , ω4 to zero,
i.e.,
0 = ’2Dx (F 3 ),
ω4 :
0 = ’2Dx (F 2 ) + (Dt ’ uDx ’ u1 ’ Dx )(F 3 ) + 4u1 F 3 ,
2
ω3 :
0 = ’2Dx (F 1 ) + (Dt ’ uDx ’ u1 ’ Dx )(F 2 ) + 6u2 F 3 + 3u1 F 2 ,
2
ω2 :
0 = ’2Dx (F 0 ) + (Dt ’ uDx ’ u1 ’ Dx )(F 1 ) + 4u3 F 3 + 3u2 F 2
2
ω1 :
+ 2u1 F 1 ,
0 = ’2Dx (F ’1 ) + (Dt ’ uDx ’ u1 ’ Dx )(F 0 ) + u4 F 3 + u3 F 2
2
ω0 :
+ u2 F 1 + u1 F 0 ,
0 = (Dt ’ uDx ’ u1 ’ Dx )(F ’1 ).
2
ω’1 : (5.94)
Note that in each coe¬cient related to ω’1 , . . . , ω3 there is always a number
of terms which together are just
Dt ’ uDx ’ u1 ’ Dx (F i ),
2
i = ’1, 0, 1, 2, 3, (5.95)
which arise by action of 1 1 on the coe¬cient F i of the term F i ωi in „¦,
E
(5.90). From these equations we obtain the solution by solving the system
in the order as given by the equations in (5.94).
This leads to the following solutions
F 3 = c1 ,
3
F 2 = c1 u + c 2 ,
2
3
F 1 = c1 u2 + 3u1 + c2 u + c3 ,
4
1 9 1 3 1
F 0 = c1 u3 + uu1 + 2u2 + c2 u2 + u1 + c3 u + c4 ,
8 4 4 2 2
3 3 3 1 1 1 1
F ’1 = c1 u2 u1 + uu2 + u2 + u3 + c2 uu1 + u2 + c3 u1 + c5 .
41 2
8 4 2 2 2
(5.96)
Combination of (5.90) and (5.96)) leads to the following independent solu-
tions
W 1 = ω0 ,
W 2 = u1 ω’1 + u0 ω0 + 2ω1 ,
224 5. DEFORMATIONS AND RECURSION OPERATORS

W 4 = 2(uu1 + u2 )ω’1 + (u2 + 6u1 )ω0 + 4uω1 + 4ω2 ,
W 7 = (3u2 u1 + 6uu2 + 6u2 + 4u3 )ω’1 + (u3 + 18uu1 + 16u2 )ω0
1
+ 6(u2 + 4u1 )ω1 + 12uω2 + 8ω3 . (5.97)
1

In case we start from functions F i , i = ’1, . . . , 2, in (5.90), dependent on
x, t, u, u1 , u2 , u3 , u4 , u5 , y, and taking F 3 = 0, and solving the system of
equations (5.96) in a straightforward way, we arrive to
F 2 = c1 (t),
1
F 1 = c1 (t)x + c1 (t)u + c2 (t),
2
1
F 0 = (c1 (t)x2 + 2c1 (t)xu + 2c1 (t)u2 + 12c1 (t)u1 + 4c2 (t)x
8
+ 4c2 (t)u + c3 (t)),
1
F ’1 = (c1 (t)x3 ’ 6c1 (t)x + 12c1 (t)u + 12c1 (t)xu1
48
+ 24c1 (t)(uu1 + u2 ) + 6c2 (t)x2 + 24c2 (t)u1 + 24c3 (t)x + 48c4 (t)).
(5.98)
Finally, from the last equation in (5.94) we arrive at
c1 (t) = ±1 + ±2 t + ±3 t2 ,
c2 (t) = ±4 + ±5 t,
3
c3 (t) = ±6 + t,
2
1
c4 (t) = ’ c5 , (5.99)
2
which leading to the six independent solutions
W 1 = ω0 ,
±6 :
W 2 = u1 ω’1 + u0 ω0 + 2ω1 ,
±4 :
W 3 = (tu1 + 1)ω’1 + (tu + x)ω0 + 2tω1 ,
±5 :
W 4 = 2(uu1 + u2 )ω’1 + (u2 + 6u1 )ω0 + 4uω1 + 4ω2 ,
±1 :
W 5 = (2tuu1 + 2tu2 + xu1 + u)ω’1 ,
±2 :
+ (tu2 + 6tu1 + xu)ω0 + (4tu + 2x)ω1 + 4tω2 ,
W 6 = (2t2 (uu1 + u2 ) + 2txu1 + 2tu + 2x)ω’1 ,
±3 :
+ (t2 (uu2 + 6u1 ) + 2txu + 6t + x2 )ω0 ,
+ (4t2 u + 4tx)ω1 + 4t2 ω2 . (5.100)
If we choose the term F 3 in (5.90) to be dependent of x, t, u, u1 , u2 ,
u3 , u4 , u5 , y too, the general solution of the deformation equation (5.93),
or equivalently the resulting overdetermined system of partial di¬erential
5. DEFORMATIONS OF THE BURGERS EQUATION 225

equations (5.94) for the coe¬cients F i , i = ’1, . . . , 3, is a linear combination
of the following ten solutions

W 1 = ω0 ,
W 2 = u1 ω’1 + u0 ω0 + 2ω1 ,
W 3 = (tu1 + 1)ω’1 + (tu + x)ω0 + 2tω1 ,
W 4 = 2(uu1 + u2 )ω’1 + (u2 + 6u1 )ω0 + 4uω1 + 4ω2 ,
W 5 = (2tuu1 + 2tu2 + xu1 + u)ω’1
+ (tu2 + 6tu1 + xu)ω0 + (4tu + 2x)ω1 + 4tω2 ,
W 6 = (2t2 (uu1 + u2 ) + 2txu1 + 2tu + 2x)ω’1
+ (t2 (u2 + 6u1 ) + 2txu + 6t + x2 )ω0
+ (4t2 u + 4tx)ω1 + 4t2 ω2 ,
W 7 = (3u2 u1 + 6uu2 + 6u2 + 4u3 )ω’1 + (u3 + 18uu1 + 16u2 )ω0
1
+ 6(u2 + 4u1 )ω1 + 12uω2 + 8ω3 ,
W 8 = (t(3u2 u1 + 6uu2 + 6u2 + 4u3 ) + x(2uu1 + 2u2 ) + u2 )ω’1
1
+ (t(u3 + 18uu1 + 16u2 ) + x(u2 + 6u1 ) + 2u)ω0
+ (t(6u2 + 24u1 ) + x(4u))ω1
+ (12tu + 4x)ω2
+ 8tω3 ,
W 9 = (t2 (3u2 u1 + 6uu2 + 6u2 + 4u3 ) + tx(4uu1 + 4u2 ) + x2 (u1 )
1
+ 2tu2 + 2xu ’ 6)ω’1 + (t2 (u3 + 18uu1 + 16u2 ) + tx(2u2 + 12u1 )
+ x2 u + 4tu ’ 2x)ω0 + (t2 (6u2 + 24u1 ) + 8txu + 2x2 )ω1
+ (12t2 u + 8tx)ω2 + (8t2 )ω3 ,
W 10 = (t3 (3u2 u1 + 6uu2 + 6u2 + 4u3 ) + t2 x(6uu1 + 6u2 ) + 3tx2 u1
1
+ t2 (3u2 + 12u1 ) + 6txu + 3x2 + 6t)ω’1 + (t3 (u3 + 18uu1 + 16u2 )
+ t2 x(3u2 + 18u1 ) + 3tx2 u + x3 + 18t2 u + 18tx)ω0 + (t3 (6u2 + 24u1 )
+ 12t2 xu + 6tx2 + 24t2 )ω1 + (12t3 u + 12t2 x)ω2 + (8t3 )ω3 . (5.101)

In order to compute the classical recursion operators for symmetries
resulting from the deformations constructed in (5.100) induced by the char-
acteristic functions W1 , W2 , . . . , we use Proposition 4.29. Suppose we start
at a (nonlocal) symmetry X of the Burgers equation; its presentation is

‚ ‚
i
= X’1 + Dx (X) . (5.102)
X
‚y ‚ui
i
226 5. DEFORMATIONS AND RECURSION OPERATORS

The nonlocal component X’1 is obtained from the invariance of the equa-
tions, cf. (5.87)
yx = u,
1
yt = u 2 + u 1 ,
2
i.e.,
Dx (X’1 ) = X, (5.103)
from which we have
’1
X’1 = Dx (X). (5.104)
Theorem 4.30, stating that U1 is a symmetry, yields for the component
X
‚/‚u,
’1
W 2 = u1 X’1 + uX + 2Dx X = u1 Dx + u + 2Dx X (5.105)
X

and similar for W 3

W 3 = (tu1 + 1)X’1 + (tu + x)X + 2tDx X
X
’1
= (tu1 + 1)Dx + (tu + x) + 2tDx X. (5.106)
From formulas (5.105) and (5.106) together with similar results with respect
to W4 , . . . , W7 we arrive in a straightforward way at the recursion operators
R1 = id,
’1
R2 = u1 Dx + u + 2Dx ,
’1 ’1
R3 = t(u1 Dx + u + 2Dx ) + x + Dx ,
’1
R4 = 2(uu1 + u2 )Dx + (u2 + 6u1 ) + 4uDx + 4Dx ,
2

’1
R5 = t (2uu1 + 2u2 )Dx + (u2 + 6u1 ) + 4uDx + 4Dx
2

’1 ’1
+ x u1 Dx + u + 2Dx + uDx ,
’1
R6 = t2 (2uu1 + 2u2 )Dx + (u2 + 6u1 ) + 4uDx + 4Dx
2

’1
+ 2tx u1 Dx + u + 2Dx + x2
’1 ’1
+ t(2uDx + 6) + 2xDx ,
’1
R7 = (3u2 u1 + 6uu2 + 6u2 + 4u3 )Dx + (u3 + 18uu1 + 16u2 )
1
+ 6(u2 + 4u1 )Dx + 12uDx + 8Dx .
2 3
(5.107)
The operator R1 is just the identity operator while R2 is the ¬rst classical
recursion operator for the Burgers equation.
This application shows that from the deformations of the Burgers equa-
tion one arrives in a straightforward way at the recursion operators for
symmetries. It will be shown in forthcoming sections that the representa-
tion of recursion operators for symmetries in terms of deformations of the
di¬erential equation is more favorable, while it is in e¬ect a more condensed
6. DEFORMATIONS OF THE KDV EQUATION 227

presentation of this recursion operator. Moreover the appearance of formal
integrals in these operators is clari¬ed by their derivation.
The deformation of an equation is a geometrical object, as is enlightened
in Chapter 6: it is a symmetry in a new type of covering.

6. Deformations of the KdV equation
Motivated by the results obtained for the Burgers equation, we search
for deformations in coverings of the KdV equation. In order to do this, we
¬rst have to construct conservation laws for the KdV equation
ut = uu1 + u3 , (5.108)
i.e., we have to ¬nd functions F x , F t , depending on x, t, u, u1 , . . . such that
on E ∞ one has
Dt (F x ) = Dx (F t ), (5.109)
where Dx , Dt are total derivative operators, which in local coordinates x, t,
u, u1 , u2 , u3 , . . . on E ∞ have the following presentation
‚ ‚ ‚ ‚
+ ··· ,
Dx = + u1 + u2 + u3
‚x ‚u ‚u1 ‚u2
‚ ‚ ‚ ‚
+ ···
Dt = + ut + ut1 + ut2 (5.110)
‚t ‚u ‚u1 ‚u2
Since the KdV equation is graded,
deg(x) = ’1, deg(t) = ’3
deg(u) = 2, deg(u1 ) = 3, . . . , (5.111)
F x , F t will be graded too being of degree k and k + 2 respectively.
In order to avoid trivialities in the construction of these conservation
laws, we start at a function F triv which is of degree k ’ 1 and remove in the
expression F x ’ Dx (F triv ) special terms by choosing coe¬cients in F triv in
an appropriate way, since the pair (Dx (F triv ), Dt (F triv )) leads to a trivial
conservation law.
After this, we restrict ourselves to conservation laws of the type (F x ’
Dx (F triv ), F t ’ Dt (F triv )). Searching for conservation laws satisfying the
condition deg(F x ) ¤ 6, we ¬nd the following three conservation laws
12
x t
F1 = u, F1 = u + u2 ,
2
1 13 12
F2 = u2 ,
x t
u ’ u1 + uu2 ,
F2 =
2 3 2
34
F3 = u3 ’ 3u2 , F3 =
x t
u + 3u2 u2 ’ 6uu2 ’ 6u1 u3 + 3u2 . (5.112)
1 1 2
4
We now introduce the new nonlocal variables y1 , y2 , y3 by the following
system of partial di¬erential equations
(y1 )x = u,
228 5. DEFORMATIONS AND RECURSION OPERATORS

1
(y1 )t = u2 + u2 ,
2
1
(y2 )x = u2 ,
2
1 1
(y2 )t = u3 ’ u2 + uu2
21
3
(y3 )x = u3 ’ 3u2 ,
1
3
(y3 )t = u4 + 3u2 u2 ’ 6uu2 ’ 6u1 u3 + 3u2 . (5.113)
1 2
4
The compatibility conditions for these equations (5.113) are satis¬ed because
of (5.109).
If we now repeat the construction of ¬nding conservation laws on E ∞ —
R3 , where local variables are given by x, t, u, y1 , y2 , y3 , u1 , u2 , . . . and
where the system of partial di¬erential equations is given for u, y1 , y2 , y3
by (5.113), we ¬nd yet another conservation law
x
F4 = y 1 ,
t
F4 = u 1 + y 2 (5.114)
leading to the nonlocal variable y4 , satisfying the partial di¬erential equa-
tions
(y4 )x = y1 ,
(y4 )t = u1 + y2 . (5.115)
x t
The conservation law (F4 , F4 ) is in e¬ect equivalent to the well-known clas-
sical (x, t)-dependent conservation law for the KdV equation, i.e.,
1
¯x
F4 = xu + tu2 ,
2
12 1 1
¯t
F4 = x u + u2 + t u3 + uu2 ’ u2 ’ u1 . (5.116)
21
2 3
We now start at the four-dimensional covering E ∞ —R4 of the KdV equation
E∞
ut = uu1 + u3 , (5.117)

where the prolongation of the Cartan distribution to E ∞ — R4 is given by
‚ 1 ‚ ‚ ‚
+ u2 + (u3 ’ 3u2 )
Dx = D x + u + y1 ,
1
‚y1 2 ‚y2 ‚y3 ‚y4
12 ‚ 13 12 ‚
u ’ u1 + uu2
Dt = D t + u + u2 +
2 ‚y1 3 2 ‚y2
34 ‚ ‚
u + 3u2 u2 ’ 6uu2 ’ 6u1 u3 + 3u2
+ + (u1 + y2 ) , (5.118)
1 2
4 ‚y3 ‚y4
6. DEFORMATIONS OF THE KDV EQUATION 229

where Dx , Dt are the total derivative operators on E ∞ , (5.110). In fact y1 ,
y2 , y3 are just potentials for the KdV equation, i.e.,
x
y1 = u dx,
x
12
y2 = u dx,
2
x
u3 ’ 3u2 dx,
y3 = (5.119)
1


while y4 is the nonlocal potential
x
y4 = y1 dx. (5.120)

The Cartan forms associated to y1 , . . . , y4 are denoted by ω’1 , . . . , ω’4 ,
while ω0 ,ω1 , . . . are the Cartan forms associated to u0 , u1 , . . . The generating
function for the deformation U1 is de¬ned by
6
F i ωi + F ’1 ω’1 + F ’2 ω’2 + F ’3 ω’3 + F ’4 ω’4 ,
„¦= (5.121)
i=0

where F i , i = ’4, . . . , 6, are dependent on the variables
x, t, u, . . . , u7 , y1 , . . . , y4 .
The overdetermined system of partial di¬erential equations resulting from
the deformation equation (4.65) on p. 185
(1)
(„¦) = 0,
E1
i.e.,
3
Dt („¦) ’ u1 „¦ ’ uDx („¦) ’ Dx („¦) = 0, (5.122)
can be solved in a straightforward way which yields the following character-
istic functions
W0 = ω 0 ,
2 1
W1 = uω0 + ω2 + u1 ω’1 ,
3 3
42 4 4
W2 = u + u2 ω0 + 2u1 ω1 + uω2 + ω4
9 3 3
1 1
+ (uu1 + u3 )ω’1 + u1 ω’2 ,
3 9
83 8
u + uu2 + 2u2 + 2u4 ω0 + (4uu1 + 5u3 )ω1
W3 = 1
27 3
4 2 20
+ u + u2 ω2 + 5u1 ω3 + 2uω4 + ω6
3 3
1 1
+ (5u2 u1 + 10uu3 + 20u1 u2 + 6u5 )ω’1 + (uu1 + u3 )ω’2
18 9
230 5. DEFORMATIONS AND RECURSION OPERATORS

1
+ u1 ω’3 . (5.123)
54
Note that the coe¬cients of ω’1 , ω’2 , ω’3 in (5.123) are just higher sym-
metries in a agreement with the remark made in the case of the Burgers
equation.
From these results it is straightforward to obtain recursion operators for
the KdV equation, i.e.,
2 1 ’1
2
R1 = u + Dx + u1 Dx ,
3 3
42 4 4 2 4
R2 = u + u2 + 2u1 Dx + uDx + Dx
9 3 3
1 1
’1 ’1
+ (uu1 + u3 )Dx + u1 Dx u, (5.124)
3 9
while
83 8 4 20
u + uu2 + 2u2 + 2u4 + 4uu1 + 5u3 )Dx + ( u2 + u2 Dx 2
R3 = 1
27 3 3 3
1 ’1
+ 5u1 Dx + 2uDx + Dx + (5u2 u1 + 10uu3 + 20u1 u2 + 6u5 )Dx
3 4 6
18
1 1
’1 ’1
+ (uu1 + u3 )Dx u + u1 Dx (3u2 ’ 6u1 Dx ). (5.125)
9 54
The last term in R2 and the last two terms in R3 arise due to the invariance
of
’1 1 2
y2 = D x u,
2
’1
y3 = Dx (u3 ’ 3u2 ). (5.126)
1

The operators R1 , R2 , R3 are just classical recursion operators for the KdV
equations (5.119). From (5.125) one observes the complexity of the recursion
operators in the last two terms of this expression, due to the complexity of
the conservation laws. The complexity of these operators increases more if
higher nonlocalities are involved.
Remark 5.13 (Linear coverings for the KdV equation). We also con-
sidered deformations of the KdV equations in the linear covering and the
prolongation coverings, performing computations related to these coverings.
1. Linear covering E ∞ — R2 . Local coordinates are x, t, u, u1 , . . . , s1 ,
s2 while the Cartan distribution is given by
1 ‚ 1 1 1 ‚
+ ’ s2 u1 + s2 u ’ »s2
Dx = D x + s 2 ,
6 ‚s1 6 18 9 ‚s2
‚ 1 1 1
+ ’ s1 u2 + s2 u1 ’ s1 u2 + »s1 u
Dt = Dt ’ (» + u)
‚s1 6 3 3
2 ‚
+ »2 s 1 . (5.127)
3 ‚s2
¨
7. DEFORMATIONS OF THE NONLINEAR SCHRODINGER EQUATION 231

The only deformation admitted here is the trivial one. There is how-
ever a yet unknown symmetry in this case, i.e.,

V = s 1 s2 . (5.128)
‚u
2. Prolongation covering E ∞ — R1 . In this case the Cartan distribution
is given by
1 ‚
Dx = Dx + (u + q 2 + ±) ,
6 ‚q
1 1 1 1
Dt = Dt + u2 + qu1 + u2 + u q 2 ’ ±
3 3 8 3
21 ‚
’ ± q2 + ± . (5.129)
36 ‚q
But here no nontrivial results were obtained.
In e¬ect these special coverings did not lead to new interesting deformation
structures.


7. Deformations of the nonlinear Schr¨dinger equation
o
In this section deformations and recursion operators of the nonlinear
Schr¨dinger (NLS) equation
o

ut = ’v2 + kv(u2 + v 2 ),
vt = u2 ’ ku(u2 + v 2 ) (5.130)
will be discussed in the nonlocal setting.
In previous sections we explained how to compute conservation laws for
partial di¬erential equations and how to construct from them the nonlocal
variables, thus “killing” the conservation laws, i.e., in the coverings the
conservation laws associated to the nonlocal variables become trivial.
We introduce the nonlocal variables y1 , y2 , y3 associated to the conser-
vation laws of the NLS equation and given by
y1x = u2 + v 2 ,
y1t = 2(’uv1 + vu1 ),
y2x = uv1 ,
3 1 1 1 12
y2t = ’ ku4 ’ ku2 v 2 + kv 4 + uu2 ’ u2 ’ v1 (5.131)
21 2
4 2 4
and
y3x = k(u2 + v 2 )2 + 2u2 + 2v1 ,
2
1
y3t = 4 (’kuv1 + kvu1 )(u2 + v 2 ) ’ u1 v2 + v1 u2 . (5.132)
232 5. DEFORMATIONS AND RECURSION OPERATORS

In the three-dimensional covering E ∞ — R3 of the NLS equation the Cartan
distribution is given by
3

Dx = D x + yix ,
‚yi
i=1
3

Dt = D t + yit , (5.133)
‚yi
i=1

while Dx , Dt are total derivative operators on E ∞ , which in internal coor-
dinates x, t, u, v, u1 , v1 , . . . have the representation
∞ ∞
‚ ‚ ‚
Dx = + ui+1 + vi+1 ,
‚x ‚ui ‚vi
i=0 i=0
∞ ∞
‚ ‚ ‚
Dt = + uit + vit . (5.134)
‚t ‚ui ‚vi
i=0 i=0

Now in order to construct a deformation of the NLS equation, we con-
struct a tuple of characteristic functions
3 3
iv ˜
u
(f i ωi
u
W= + f ωi ) + f i ωy i ,
i=0 i=1
3 3
Wv = (g i ωi + g i ωi ) +
u v
g i ωyi ,
˜ (5.135)
i=0 i=1
u v
where in (5.135) ωi , ωi , ωyi are the Cartan forms associated to ui , vi , yi

respectively; the coe¬cients f i , f , f i , g i , g i , g i are dependent on
˜
x, t, u, v, . . . , u4 , v4 , y1 , y2 , y3 .
The solution constructed from the deformation equation (4.65) leads to the
following nontrivial results.
1v
u
ω ’ vωy1 ,
W1 =
k1
1u
v
W1 = ’ ω1 + uωy1 ,
k
1u 1
W2 = (u2 + v 2 )ω0 + uvω0 ’
u u v
ω + u1 ωy1 ’ vωy2 ,
2k 2 2
1v 1
W 2 = v 2 ω0 ’
v v
ω + v1 ωy1 + uωy2 ,
2k 2 2
2v
W3 = 8uv1 ω0 + 12vv1 ω0 + 4uvω1 + (4u2 + 8v 2 )ω1 ’ ω3
u u v u v
k
2 2
+ 2(’k(u + v )v + v2 )ωy1 + 4u1 ωy2 ’ vωy3 ,
v u v
W3 = (’12uu1 ’ 4vv1 )ω0 + (’4uv1 ’ 8vu1 )ω0
8. DEFORMATIONS OF THE CLASSICAL BOUSSINESQ EQUATION 233

2u
+ (’8u2 ’ 4v 2 )ω1 ’ 4uvω1 + ω3
u v
k
+ 2(k(u2 + v 2 )u ’ u2 )ωy1 + 4v1 ωy2 + uωy3 . (5.136)
Suppose we have a shadow of a nonlocal symmetry
‚ ‚ ‚ ‚ ‚
X = Xu + . . . + Xv + . . . + X’1 + X’2 + X’3 . (5.137)
‚u ‚v ‚y1 ‚y2 ‚y3
Then the nonlocal component X’1 associated to y1 is obtained from the
invariance of the equations
y1x = u2 + v 2 ,
y1t = 2(’uv1 + vu1 ). (5.138)
So from (5.138) we arrive at the following condition
Dx (X’1 ) = 2uX u + 2vX v ,
or formally
’1
X’1 = Dx (2uX u + 2vX v ). (5.139)
From the invariance of the partial di¬erential equations for y2 , y3 , (5.131),
(5.132) we obtain in a similar way
’1
X’2 = Dx uDx (X v ) + v1 X u ,
’1
X’3 = Dx 4k(u2 + v 2 )(uX u + vX v ) + 4u1 Dx (X u ) + 4v1 Dx (X v ) .
(5.140)
u v
Using these results, we arrive from W1 , W1 in a straightforward way at
the well-known recursion operator
1
’1 ’1
’vDx (2u) ’vDx (2v) + k Dx
R1 = (5.141)
1
’1 ’1
+uDx (2u) ’ k Dx +uDx (2v)
Recursion operators resulting from Wiu , Wiv , i = 2, 3, . . . , can be ob-
tained similarly, using constructed formulas for X’2 , X’3 , see (5.140).

8. Deformations of the classical Boussinesq equation
Let us discuss now deformations of Classical Boussinesq equation
vt = u1 + vv1 ,
ut = u1 v + uv1 + σv3 . (5.142)
To this end, we start at a four-dimensional covering E ∞ — R4 of the Boussi-
nesq equation, where local coordinates are given by
(x, t, v, u, . . . , y1 , . . . , y4 )
with the Cartan distribution de¬ned by
‚ ‚ ‚ ‚
+ (u2 + uv 2 + vv2 σ)
Dx = D x + v +u + uv ,
‚y1 ‚y2 ‚y3 ‚y4
234 5. DEFORMATIONS AND RECURSION OPERATORS

1 ‚ ‚
Dt = D t + u + v 2 + (uv + v2 σ)
2 ‚y1 ‚y2
12 12 ‚
u + uv 2 + vv2 σ ’ v1 σ
+
2 2 ‚y3

+ (2u2 v + uv 3 + 2σuv2 + 2σv 2 v2 + σvu2 ’ σv1 u1 ) . (5.143)
‚y4
The nonlocal variables y1 , y2 , y3 , y4 satisfy the equations
(y1 )x = v,
1
(y1 )t = u + v 2 ,
2
(y2 )x = u,
(y2 )t = uv + v2 σ,

(y3 )x = uv,
1 12
(y3 )t = u2 + uv 2 + vv2 σ ’ v1 σ,
2 2
(y4 )x = u2 + uv 2 + vv2 σ,
(y4 )t = 2u2 v + uv 3 + 2σuv2 + 2σv 2 v2 + σvu2 ’ σv1 u1 . (5.144)
We assume the characteristic functions W v , W u to be dependent on ω0 , v
u v u
ω0 , . . . , ω5 , ω5 , ωy1 , . . . , ωy4 , whereas the coe¬cients are required to be de-
pendent on x, t, v, u, . . . , v5 , u5 , y1 , . . . , y4 .
Solving the overdetermined system of partial di¬erential equations re-
sulting from the deformation condition (4.65), we arrive at the following
nontrivial characteristic functions
v v u
W1 = vω0 + 2ω0 + v1 ωy1 ,
u v u v
W1 = 2uω0 + vω0 + 2σω2 + u1 ωy1 ,
W2 = (4u + v 2 )ω0 + 4vω0 + 4σω2 + (2vv1 + 2u1 )ωy1 + 2v1 ωy2 ,
v v u v

W2 = (4uv + 6σv2 )ω0 + (4u + v 2 )ω0 + 6v1 σω1 + 4σvω2 + 4σω2
u v u v v u

+ (2uv1 + 2vu1 + 2σv3 )ωy1 + 2u1 ωy2 (5.145)
and two more deformations.
As in the preceding section we use the invariance of the equations
(y1 )x = v,
(y2 )x = u (5.146)
to arrive at the associated recursion operators
’1
v + v 1 Dx 2
R1 = (5.147)
’1
2
2u + 2σDx + u1 Dx v
9. SYMMETRIES AND RECURSION FOR THE SYM EQUATION 235

and
« 
’1 4v + 2v D ’1
(4u + v 2 ) + 4σDx + (2vv1 + 2u1 )Dx
2
1x
¬ ·
R2 = ¬ (4uv + 6σ2 v2 ) + 6σv1 Dx + 4σvDx (4u + v 2 ) + 4σDx ·
2 2
 
’1 ’1
+(2uv1 + 2vu1 + 2σv3 )Dx +2u1 Dx
(5.148)
Note that R2 is just equivalent to double action of the operator R1 , i.e.,
R2 = R1 —¦ R1 = (R1 )2 . (5.149)

9. Symmetries and recursion for the Sym equation
The following system of partial di¬erential equations plays an interesting
role in some speci¬c areas of geometry [16]:
‚u ‚w
+ (u ’ v) = 0,
‚x ‚x
‚v ‚w
’ (u ’ v) = 0,
‚y ‚y
‚2w ‚2w
2w
uve + + = 0. (5.150)
‚x2 ‚y 2
The underlying geometry is de¬ned as the manifold of local surfaces which
admit nontrivial isometries conserving principal curvatures, the so-called
isothermic surfaces.
In this section we shall prove that this system (5.150) admits an in¬nite
hierarchy of commuting symmetries and conservation laws, [7]. Results will
be computed not for system (5.150), but for a simpli¬ed system obtained by
the transformation u ’ ue’w , v ’ ve’w , i.e.,
‚u ‚w
’v = 0,
‚x ‚x
‚v ‚w
’u = 0,
‚y ‚y
‚2w ‚2w
uv + + = 0. (5.151)
‚x2 ‚y 2
9.1. Symmetries. In this subsection we discuss higher symmetries for
system (5.151):
ux ’ vwx = 0, vy ’ uwy = 0, wyy + uv + wxx = 0. (5.152)
This system is a graded system of di¬erential equations, i.e.,
deg(x) = deg(y) = ’1,
deg(u) = deg(v) = 1,
deg(w) = 0. (5.153)
236 5. DEFORMATIONS AND RECURSION OPERATORS

All objects of interest for system (5.152), like symmetries and conservation
laws, turn out to be homogeneous with respect to this grading, e.g.,
deg(ui,j ) = deg(u) ’ i deg(x) ’ j deg(y) = 1 + i + j,

= deg(u) + deg(v) ’ deg(wxy ) = 0,
deg uv (5.154)
‚wxy
whereas in (5.154) ui,j = ux . . . x y . . . y .
i times j times

For computation of higher symmetries we have to introduce vertical
vector ¬elds ¦ with generating function ¦ = (¦u , ¦v , ¦w ), which has to
satisfy the symmetry condition
F (¦) = 0, (5.155)
where is the universal linearization operator for system (5.152), i.e.,
F
« 
Dx ’wx ’vDx
’wy Dy ’uDy 
F= (5.156)
2 + D2
v u Dx y
The system F (¦) = 0 is homogeneous with respect to the degree, so
the symmetry with the generating function ¦ = (¦u , ¦v , ¦w ) is homo-
geneous with respect to the degree, i.e., deg(¦u ‚/‚u) = deg(¦v ‚/‚v) =
deg(¦w ‚/‚w), leading to the required degree of ¦:
deg(¦u ) = deg(¦v ) = deg(¦w ) + 1. (5.157)
Internal coordinates of E ∞ , where E is system (5.152), are chosen to be
x, y, u, v, w, uy , vx , wx , wy , uyy , vxx , wxx , wxy , uyyy , vxxx , wxxx , wxxy , . . . .
(5.158)
Thus E ∞ is solved for ux , vy , wyy and their di¬erential consequences ux...x ,
vy...y , wx...xy...yy . With this choice of internal coordinates, the symmetry
equation (5.155) reads
Dx (¦u ) ’ wx ¦v ’ vDx (¦w ) = 0,
’wy ¦u + Dy (¦v ) ’ uDy (¦w ) = 0,
2 2
v¦u + u¦v + Dx (¦w ) + Dy (¦w ) = 0. (5.159)
The generating function ¦ = (¦u , ¦v , ¦w ) depends on a ¬nite number of in-
ternal coordinates, ¦ being de¬ned on E ∞ . Dependencies for the generating
function are selected with respect to degree, i.e., ¦ depends on the internal
coordinates of degree n or less. According to (5.157), this means that ¦ w
depends on internal coordinates of degree n ’ 1 or less.
The results for the generating function ¦ depending on the internal
coordinates of degree 6 or less are as follows. There are two symmetries of
degree 0:
X 0 = (0, 0, 1),
Y 0 = (u + xvwx + yuy , v + xvx + yuwy , xwx + ywy ). (5.160)
9. SYMMETRIES AND RECURSION FOR THE SYM EQUATION 237

The second symmetry in (5.160) corresponds to the scaling or grading of
systems (5.152), (5.153). Other symmetries appear in pairs of degrees 1, 3
and 5. The symmetries of degree 1 are
X 1 = (uy , uwy , wy ),
Y 1 = (vwx , vx , wx ). (5.161)
They are equivalent to the vector ¬elds of ‚/‚y and ‚/‚x respectively.
The symmetries of degree 3 are
Xu = 6u2 vwy + 3u2 uy + 6uwy wxx + 3uy wx ’ 3uy wy + 2uyyy ,
3 2 2

Xv = u3 wy + 3uwx wy ’ 3uwy ’ 2uwxxy + 2uy wxx + 2wy uyy ,
3 2 3

Xw = u2 wy ’ 2vuy + 3wx wy ’ wy ’ 2wxxy
3 2 3
(5.162)
and
Yu = 3v 3 wx ’ 3vwx + 3vwx wy + 2vwxxx + 2wx vxx ’ 2wxx vx ,
3 3 2

Yv3 = 3v 2 vx ’ 6vwx wxx ’ 3wx vx + 3wy vx + 2vxxx ,
2 2

Yw = 3v 2 wx ’ wx + 3wx wy + 2wxxx .
3 3 2
(5.163)
Finally the components of the generating functions ¦ = (¦u , ¦v , ¦w ) of the
two symmetries of degree 5 are given by
Xu = ’ 60u4 vwy + 15u4 uy ’ 60u3 wy wxx ’ 140u2 v 2 uy + 60u2 vwx wy
5 2

’ 60u2 vwy ’ 80u2 vwxxy + 30u2 uy wx + 110u2 uy wy ’ 40u2 wy vxx
3 2 2

+ 20u2 uyyy ’ 40uv 2 wy wxx ’ 200uvuy wxx ’ 120uvwx wy vx
2 3
’ 40uwx wy wxy ’ 60uwy wxx + 120uvwy uyy ’ 40uuy wx vx
2
+ 80uuy uyy + 60uwx wy wxx ’ 40uwy wxxxx ’ 80uwxx wxxy
’ 40v 2 uy wx + 80vu2 wy + 20u3 + 15uy wx ’ 50uy wx wy ’ 40uy wx wxxx
2 4 22
y y
4 2 2 2
+ 15uy wy + 80uy wy wxxy ’ 60uy wxx + 20uy wxy + 20wx uyyy
2
+ 40wx uyy wxy ’ 20wy uyyy + 80wy wxx uyy + 8uyyyyy ,

Xv = + 3u5 wy ’ 20u3 v 2 wy + 10u3 wx wy + 10u3 wy ’ 4u3 wxxy
5 2 3

’ 16u2 vwy wxx + 12u2 uy wxx ’ 8u2 wx wy vx + 20u2 wy uyy ’ 16uv 2 wx wy
2

+ 8uv 2 wxxy + 80uvuy wy + 24uvvx wxy + 20uu2 wy + 15uwx wy
2 4
y
23 2 5
’ 50uwx wy ’ 20uwx wxxy ’ 40uwx wy wxxx ’ 40uwx wxx wxy + 15uwy
+ 60uwy wxxy ’ 20uwy wxx + 20uwy wxy + 8uwxxxxy ’ 8v 2 uy wxx
2 2 2

2 2 2
’ 24vuy wx vx + 20uy wx wxx + 20uy wy wxx ’ 8uy wxxxx + 20wx wy uyy
3
’ 20wy uyy + 8wy uyyyy + 8wxx uyyy ’ 8wxxy uyy ,

Xw = 3u4 wy ’ 20u2 v 2 wy ’ 12u2 vuy + 10u2 wx wy ’ 10u2 wy ’ 4u2 wxxy
5 2 3

’ 16uvwy wxx ’ 8uwx wy vx + 8uwy uyy ’ 16v 2 wx wy + 8v 2 wxxy
2
238 5. DEFORMATIONS AND RECURSION OPERATORS

’ 20vuy wx + 20vuy wy ’ 8vuyyy + 24vvx wxy ’ 4u2 wy
2 2
y
4 23 2
+ 8uy vxx + 15wx wy ’ 30wx wy ’ 20wx wxxy ’ 40wx wy wxxx
5 2 2 2
’ 40wx wxx wxy + 3wy + 20wy wxxy ’ 20wy wxx + 20wy wxy + 8wxxxxy
(5.164)

and

Yu = + 15v 5 wx ’ 50v 3 wx + 30v 3 wx wy + 20v 3 wxxx + 60v 2 wx vxx
5 3 2

+ 20v 2 wxx vx + 15vwx ’ 50vwx wy ’ 60vwx wxxx
5 32 2

4 2 2 2
+ 15vwx wy + 40vwx wy wxxy ’ 20vwx wxx + 20vwx vx + 20vwx wxy
2 3 2
+ 20vwy wxxx + 40vwy wxx wxy + 8vwxxxxx ’ 20wx vxx ’ 20wx wxx vx
2 2
+ 20wx wy vxx + 8wx vxxxx ’ 20wy wxx vx ’ 8wxx vxxx + 8wxxx vxx
’ 8vx wxxxx ,

Yv5 = + 15v 4 vx ’ 60v 3 wx wxx ’ 90v 2 wx vx + 30v 2 wy vx + 20v 2 vxxx
2 2

3 2 2
+ 60vwx wxx ’ 40vwx wy wxy ’ 60vwx wy wxx ’ 40vwx wxxxx
4 22 2
’ 80vwxx wxxx + 80vvxx vx + 15wx vx ’ 50wx wy vx ’ 20wx vxxx
4 2
’ 80wx wxx vxx ’ 80wx wxxx vx + 15wy vx + 20wy vxxx + 40wy vxx wxy
2 3 2
+ 40wy vx wxxy ’ 60wxx vx + 20vx + 20vx wxy + 8vxxxxx ,

Yw = + 15v 4 wx ’ 30v 2 wx + 30v 2 wx wy + 20v 2 wxxx + 40vwx vxx
5 3 2

5 32 2 4
+ 40vwxx vx + 3wx ’ 30wx wy ’ 20wx wxxx + 15wx wy + 40wx wy wxxy
2 2 2 2
’ 20wx wxx + 20wx vx + 20wx wxy + 20wy wxxx + 40wy wxx wxy
+ 8wxxxxx . (5.165)

Apart from the second symmetry in (5.160), these symmetries commute,
i.e., [ ¦ , ¦ ] = 0. The Lie bracket with the second symmetry in (5.160)
acts as multiplication by the degree of the symmetry.

Remark 5.14. One should note that for system (5.152) there exists a
discrete symmetry

T : x ’ y, y ’ x, u ’ v, v ’ u, w ’ w, (5.166)

from which we have

T (X 0 ) = X 0 , T (Y 0 ) = Y 0 ,
T (X 1 ) = Y 1 , T (Y 1 ) = X 1 ,
T (X 3 ) = Y 3 , T (Y 3 ) = X 3 ,
T (X 5 ) = Y 5 , T (Y 5 ) = X 5 . (5.167)
9. SYMMETRIES AND RECURSION FOR THE SYM EQUATION 239

9.2. Conservation laws and nonlocal symmetries. As in previous
applications, we ¬rst construct conservation laws in order to arrive at non-
local variables and the augmented system of partial di¬erential equations
governing them.
To construct conservation laws, we start at functions F x and F y , such
that
Dy (F x ) = Dx (F y )
We construct conservation laws for functions F x and F y of degree 0 until 4.
For degree 2 we obtained two solutions,
’v 2 + wx ’ wy
2 2
x
F y = w x wy ,
F= ,
2
u2 + w x ’ w y
2 2
x y
F = ’wx wy , F= . (5.168)
2
Degree 4 yields two conservation laws, which are
F x = ’ (u2 wx wy ’ u2 wxy ’ 2uvwx wy + 2uvwxy + wx wy ’ wx wy
3 3

+ 2wxx wxy ),
F y =(u4 ’ 4u3 v + 4u2 v 2 + 2u2 wx ’ 6u2 wy ’ 4u2 wxx + 8uvwxx
2 2

+ 8uwy uy + wx ’ 6wx wy + wy + 4wxx ’ 4u2 ’ 4wxy )/4,
4 22 4 2 2
y

F x = ’ (v 4 ’ 6v 2 wx + 2v 2 wy + 4v 2 wxx + 8vwx vx + wx ’ 6wx wy + wy
2 2 4 22 4

2 2 2
+ 4wxx ’ 4wxy ’ 4vx )/4,
F y = ’ 2uvwx wy + v 2 wx wy ’ v 2 wxy ’ wx wy + wx wy ’ 2wxx wxy . (5.169)
3 3


Associated to the conservation laws given in (5.168), (5.169), we introduce
nonlocal variables.
The conservation laws (5.168) give rise to two nonlocal variables, p and
q of degree 1,
’v 2 + wx ’ wy
2 2
px = , p y = w x wy ,
2
u2 + w x ’ w y
2 2
qx = ’wx wy , qy = . (5.170)
2
To the conservation laws (5.169) there correspond two nonlocal variables r
and s of degree 3:
rx = ’ u2 wx wy + u2 wxy + 2uvwx wy ’ 2uvwxy ’ wx wy + wx wy ’ 2wxx wxy ,
3 3

ry =(u4 ’ 4u3 v + 4u2 v 2 + 2u2 wx ’ 6u2 wy ’ 4u2 wxx + 8uvwxx + 8uwy uy
2 2

+ wx ’ 6wx wy + wy + 4wxx ’ 4u2 ’ 4wxy )/4,
4 22 4 2 2
y

sx =(’v 4 + 6v 2 wx ’ 2v 2 wy ’ 4v 2 wxx ’ 8vwx vx ’ wx + 6wx wy ’ wy
2 2 4 22 4

2 2 2
’ 4wxx + 4wxy + 4vx )/4,
240 5. DEFORMATIONS AND RECURSION OPERATORS

sy = ’ 2uvwx wy + v 2 wx wy ’ v 2 wxy ’ wx wy + wx wy ’ 2wxx wxy .
3 3
(5.171)
We now discuss the existence of symmetries in the covering of (5.152) by
nonlocal variables p, q, r, s, i.e., in E ∞ —R4 . The system of partial di¬erential
equations in this covering is constituted by (5.152), (5.170) and (5.171).
Total derivative operators Dx , Dy are de¬ned on E ∞ — R4 , and are given by
’v 2 + wx ’ wy ‚
2 2
‚ ‚ ‚
’ w x wy
Dx = D x + + rx + sx ,
2 ‚p ‚q ‚r ‚s
u2 + w x ’ w y ‚
2 2
‚ ‚ ‚
D y = D y + w x wy + + ry + sy , (5.172)
‚p 2 ‚q ‚r ‚s
where rx , ry , sx , sy are given by (5.171).
Symmetries ¦ in this nonlocal setting, where the generating function
¦ = (¦u , ¦v , ¦w ) is dependent on the internal coordinates (5.158) as well as
on the nonlocal variables p, q, r, s, have to satisfy the symmetry condition

F (¦) = 0, (5.173)
where F is the universal linearization operator for the augmented system
(5.152) together with (5.170), (5.171), i.e.,
« 
Dx ’wx ’v Dx
¬ ·
F = ’wy (5.174)
’uDy 
Dy
2 2
v u Dx + Dy
This does lead to the following nonlocal symmetry of degree 2, where
Z

Zu = ’ 2pvwx ’ 2quy
+ x 3v 3 wx ’ 3vwx + 3vwx wy + 2vwxxx + 2wx vxx ’ 2wxx vx
3 2

+ y ’6u2 vwy ’ 3u2 uy ’ 6uwy wxx ’ 3uy wx + 3uy wy ’ 2uyyy
2 2

’ 2u3 ’ 2uv 2 ’ 4uwx + 6uwy ’ 2vwxx + 4wx vx ’ 6uyy ,
2 2


Zv = ’ 2pvx ’ 2quwy
+ x 3v 2 vx ’ 6vwx wxx ’ 3wx vx + 3wy vx + 2vxxx
2 2

+ y ’u3 wy ’ 3uwx wy + 3uwy + 2uwxxy ’ 2uy wxx ’ 2wy uyy
2 3

’ 2uwxx + 2v 3 ’ 6vwx + 4vwy ’ 4uy wy + 6vxx ,
2 2


Zw = ’ 2pwx ’ 2qwy
+ x 3v 2 wx ’ wx + 3wx wy + 2wxxx
3 2

+ y ’u2 wy + 2vuy ’ 3wx wy + wy + 2wxxy
2 3

+ 2uv + 4wxx . (5.175)
One should note that the coe¬cients at p, q, i.e., (’2vwx , ’2vx , ’2wx ) and
(’2uy , ’2uwy , ’2wy ), are just the generating functions of the symmetries
9. SYMMETRIES AND RECURSION FOR THE SYM EQUATION 241

(5.161). This nonlocal symmetry is just the recursion symmetry, acting by
the extended Jacobi brackets on generating functions on E ∞ — R4 .
There is another symmetry of degree 4, dependent on p, q, r, s. For an
explicit formula of this symmetry we refer to [10]. Finally we mention that
starting from E ∞ — R4 , there is an additional nonlocal conservation law
Dy (p) = Dx (’q).
The nonlocal variable associated to this conservation law did not play an
essential role in the construction of the nonlocal symmetry (5.175).
9.3. Recursion operator for symmetries. We now arrive at the
construction of the classical recursion operator for symmetries of the Sym
equation [7]
‚u ‚w
’v = 0,
‚x ‚x
‚v ‚w
’u = 0,
‚y ‚y
‚2w ‚2w
uv + + = 0. (5.176)
‚x2 ‚y 2
We could arrive at this recursion operator by the construction of deforma-
tions of system (5.176), but we decided not to do so. We shall demonstrate
how we can, from the knowledge we have of the nonlocal structure of defor-
mations, arrive at the formal classical recursion operator, which, by means of
its presentation as integral di¬erential operator is of a more complex struc-
ture. Due to the structure of conservation laws, we can make an ansatz for
the recursion operator.
We expect that as in the previous problems, in the deformation structure
of our system (5.176) the Cartan forms associated to the nonlocal variables
p, q, i.e.,
’v 2 + wx ’ wy
2 2
ωp = dp ’ dx ’ wx wy dy,
2
u2 + w x ’ w y
2 2
ωq = dq + wx wy dx ’ dy (5.177)
2
play an essential role. According to this, the associated nonlocal components
of the symmetries play a signi¬cant role too. These components have to be
constructed from the invariance of the associated di¬erential equations for
p and q. Since the system at hand is not of evolutionary type, we have a
choice to compute these components from the invariance of either px or py
and similar for the qx and qy .
Due to the discrete symmetry (5.166), we choose the invariance of the
following equations
py =wx wy ,
q x = ’ w x wy .
242 5. DEFORMATIONS AND RECURSION OPERATORS

From these invariances, we obtain for the generating function of a symmetry
¦ = (¦u , ¦v , ¦w ), terms like
’1
¦p = Dy (wx Dy (¦w ) + wy Dx (¦w )),
’1
¦q = Dx (wx Dy (¦w ) + wy Dx (¦w )). (5.178)
From the above considerations we expect the recursion operator to contain
’1 ’1
terms like Dy (wx Dy (·) + wy Dx (·)), Dx (wx Dy (·) + wy Dx (·)).
Moreover from the expected degree of the operator, which probably will
be equal to 2, due to the degrees of the symmetries of the previous sub-
section, we arrive at the ansatz for the recursion operator for symmetries.
From this ansatz we arrive at the following expression for R:
«2 
Dy + u2 + wx ’ wy ’wx Dx + uv + wxx uwx Dx ’ 2uwy Dy
2 2

R= ’Dx ’ v 2 + wx ’ wy 2vwx Dx ’ vwy Dy 
2 2 2
wy Dy + wxx
2
0 u Dy
« 
’1 ’1
0 0 vwx Dy (wy Dx + wx Dy ) ’ uy Dx (wy Dx + wx Dy )
+ 0 0 vx Dy (wy Dx + wx Dy ) ’ uwy Dx (wy Dx + wx Dy ) (5.179)
’1 ’1
’1 ’1
0 0 wx Dy (wy Dx + wx Dy ) ’ wy Dx (wy Dx + wx Dy )
It is a straightforward check that the operator R is a recursion operator for
higher symmetries since
—¦R=S —¦ F, (5.180)
F
where the matrix operator S is given by
«2 
Dy + u2 + wx ’ wy ’Dx ’ v 2 + wx ’ wy ’uDy ’ vwy
2 2 2 2 2

S =  ’wy Dx ’ 2wxy  (5.181)
wx Dy + 2wxy uwx
S31 S32 S33
where S31 , S32 , S33 are given by
’1 ’1
S31 = 2(uv + wxx )Dx u ’ wy Dx Dy u,
’1 ’1
S32 = 2wxx Dy v + wx Dy Dx v,
’1 ’1 ’1
S33 = 2wxx Dy wy + wx Dy Dx wy + 2(uv + wxx )Dx wx
’1 2 2 2
’ w y D x D y wx + D y + w x ’ w y . (5.182)
It would have been possible not to start from the invariance of py , qx , but
from the invariance of for instance px , qx , but in that case we had to in-
’1 ’1 ’1
corporate terms like Dx v, Dx wx Dx , Dx wy Dy into the matrix recursion
operator R.
CHAPTER 6


Super and graded theories

We shall now generalize the material of the previous chapters to the case
of super (or graded ) partial di¬erential equations. We con¬ne ourselves to
the case when only dependent variables admit odd gradings and develop a
theory closely parallel to that exposed in Chapters 1“5.
We also show here that the cohomological theory of recursion operators
may be considered as a particular case of the symmetry theory for graded
equations, which, in a sense, explains the main result of Chapter 5, i.e.,
(p)
p,0
HC (E) = ker E . It is interesting to note that this reduction is accom-
plished using an odd analog of the Cartan covering introduced in Example
3.3 of Chapter 3.
Our main computational object is a graded extension of a classical partial
di¬erential equation. We discuss the principles of constructing nontrivial
extensions of such a kind and illustrate them in a series of examples. Other
applications are considered in Chapter 7.

1. Graded calculus
Here we rede¬ne the Fr¨licher“Nijenhuis bracket for the case of n-graded
o
commutative algebras. All de¬nitions below are obvious generalizations of
those from 4. Proofs also follow the same lines and are usually omitted.

1.1. Graded polyderivations and forms. Let R be a commutative
ring with a unit 1 ∈ R and A be a commutative n-graded unitary algebra
over R, i.e.,
A= Ai , Ai Aj ‚ Ai+j
i∈Zn

and
ab = (’1)a·b ba
for any homogeneous elements a, b ∈ A. Here and below the notation (’1) a·b
means (’1)i1 j1 +...in jn , where i = (i1 , . . . , in ), j = (j1 , . . . , jn ) ∈ Zn are the
gradings of the elements a and b respectively. We also use the notation a · b
for the scalar product of the gradings of elements a and b. In what follows,
one can consider Zn -graded objects as well. We consider the category of n-
2
graded (left) A-modules Mod = Mod(A) and introduce the functors
Di : Mod(A) ’ Mod(A)
243
244 6. SUPER AND GRADED THEORIES

as follows (cf. [54, 58]):
D0 (P ) = P
for any P ∈ Ob(Mod), P = Pi , and
i∈Zn

D1,j (P ) = {∆ ∈ homR (A, P ) | ∆(Ai ) ‚ Pi+j , ∆(ab)
= ∆(a)b + (’1)∆·a a∆(b)},
where j = (j1 , . . . , jn ) = gr(∆) ∈ Zn is the grading of ∆; we set

D1 (P ) = D1,j (P ).
i∈Zn

Remark 6.1. We can also consider objects of Mod(A) as right A-mod-
ules by setting pa = (’1)a·p ap for any homogeneous a ∈ A, p ∈ P . In
a similar way, for any graded homomorphism • ∈ homR (P, Q), the right
action of • can be introduced by (p)• = (’1)p• •(p).
Further, if D0 , . . . , Ds are de¬ned, we set

Ds+1,j (P ) = {∆ ∈ homR (A, Ds (P )) | ∆(Ai ) ‚ Ds,i+j (P ),
∆(ab) = ∆(a)b + (’1)∆·a a∆(b), ∆(a, b) + (’1)a·b ∆(b, a) = 0}
and
Ds+1 (P ) = Ds+1,j (P ).
j∈Zn

Elements of Ds (P ) a called graded P -valued s-derivations of A and elements
of D— (P ) = s≥0 Ds (P ) are called graded P -valued polyderivations of A.
Proposition 6.1. The functors Ds , s = 0, 1, 2, . . . , are representable in
the category Mod(A), i.e., there exist n-graded modules Λ0 , Λ1 , . . . , Λs , . . . ,
such that
Ds (P ) = homA (Λs , P )
for all P ∈ Ob(Mod).
Elements of the module Λs = Λs (A) are called graded di¬erential forms
of degree s.
Our local target is the construction of graded calculus in the limits needed
for what follows. By calculus we mean the set of basic operations related to
the functors Ds and to modules Λs as well as most important identities con-
necting these operations. In further applications, we shall need the following
particular case:
(i) A0 = C ∞ (M ) for some smooth manifold M , where 0 = (0, . . . , 0);
(ii) All homogeneous components Pi of the modules under consideration
are projective A0 -modules of ¬nite type.
1. GRADED CALCULUS 245

Remark 6.2. In fact, the entire scheme of calculus over commutative
algebras is carried over to the graded case. For example, to de¬ne graded
linear di¬erential operators, we introduce the action δa : homR (P, Q) ’
homR (P, Q), a ∈ A, by setting δa • = a• ’ (’1)a• • · a, • ∈ homR (P, Q),
and say that • is an operator of order ¤ k, if
(δa0 —¦ · · · —¦ δak )• = 0
for all a0 , . . . , ak ∈ A, etc. A detailed exposition of graded calculus can be
found in [106, 52].
1.2. Wedge products. Let us now consider some essential algebraic
structures in the above introduced objects.
Proposition 6.2. Let A be an n-graded commutative algebra. Then:
(i) There exists a derivation d : A ’ Λ1 of grading 0 such that for any
A-module P and any graded derivation ∆ : A ’ P there exists a
uniquely de¬ned morphism f∆ : Λ1 ’ P such that f∆ —¦ d = ∆.
(ii) The module Λ1 is generated over A by the elements da = d(a), a ∈ A,
with the relations
a, b ∈ A.
d(±a + βb) = ±da + βdb, d(ab) = (da)b + adb,
The j-th homogeneous component of Λ1 is of the form
Λ1 = { adb | a, b ∈ A, gr(a) + gr(b) = j},
j

(iii) The modules Λs are generated over A by the elements of the form
ω1 , . . . , ω s ∈ Λ 1 ,
ω1 § · · · § ω s ,
with the relations
ω § θ + (’1)ωθ θ § ω = 0, ω, θ ∈ Λ1 , a ∈ A.
ω § aθ = ωa § θ,
The j-th homogeneous component of Λs is of the form
Λs = { ω1 § · · · § ωs | ωi ∈ Λ1 , gr(ω1 ) + · · · + gr(ωs ) = j}.
j

(iv) Let ω ∈ Λs , j = (j1 , . . . , jn ). Set gr1 (ω) = (j1 , . . . , jn , s). Then
j

Λ— = Λs = Λs
j
s≥0 j∈Zn
s≥0

is an (n + 1)-graded commutative algebra with respect to the wedge
product
ω ∈ Λs , θ ∈ Λr , ω± , θ β ∈ Λ 1 ,
ω § θ = ω § · · · § ω s § θ1 § · · · § θ r ,
i.e.,
ω § θ = (’1)ω·θ+sr θ § ω,
where ω · θ in the power of (’1) denotes scalar product of gradings
inherited by ω and θ from A.
246 6. SUPER AND GRADED THEORIES

Remark 6.3. When working with the algebraic de¬nition of di¬erential
forms in the graded situation, one encounters the same problems as in a pure
commutative setting, i.e., the problem of ghost elements. To kill ghosts, the
same procedures as in Chapter 4 (see Remark 4.4) are to be used.
A similar wedge product can be de¬ned in D— (A). Namely for a, b ∈
D0 (A) = A we set
a § b = ab
and then by induction de¬ne
def ·a+r
(∆ § )(a) = ∆ § ∆(a) §
(a) + (’1) , (6.1)
where a ∈ A, ∆ ∈ Ds (A), ∈ Dr (A) and in the power of (’1) denotes
the grading of in the sense of the previous subsection.
Proposition 6.3. For any n-graded commutative algebra A the follow-
ing statements are valid :
(i) De¬nition (6.1) determines a mapping
§ : Ds (A) —A Dr (A) ’ Ds+r (A),
which is in agreement with the graded structure of polyderivations:
Ds,i (A) § Dr,j (A) ‚ Ds+r,i+j (A).
(ii) The module D— (A) = s≥0 j∈Zn Ds,j is an (n + 1)-graded commu-
tative algebra with respect to the wedge product:
= (’1)∆· +rs
∆§ §∆
for any ∆ ∈ Ds (A), ∈ Dr (A).1
(iii) If A satis¬es conditions (i), (ii) on page 244, then the module D— (A)
is generated by D0 (A) = A and D1 (A), i.e., any ∆ ∈ Ds (A) is a sum
of the elements of the form
a∆1 § · · · § ∆s , ∆i ∈ D1 (A), a ∈ A.
Remark 6.4. One can de¬ne a wedge product § : Di (A) —A Dj (P ) ’
Di+j (P ) with respect to which D— (P ) acquires the structure of an (n + 1)-
graded D— (A)-module (see [54]), but it will not be needed below.
1.3. Contractions and graded Richardson“Nijenhuis bracket.

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