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 
’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)

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