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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

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