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V = f u [vn ] + f v [vn ] + f v1 [vn+1 ] + f v2 [vn+2 ] + ..., (2.91)
‚u ‚v ‚v1 ‚v2
we derive the following symmetry conditions from (2.84)
D(n) f u [vn ] ’ 2v2 f v2 [vn+2 ] = 0,
D(n) f v [vn ] ’ f v1 [vn+1 ] = 0,
D(n+1) f v1 [vn+1 ] ’ f v2 [vn+2 ] = 0. (2.92)
4. THE HILBERT“CARTAN EQUATION 89

In e¬ect, the second and third equation of (2.92) are just the de¬nitions
of f v1 [vn+1 ] and f v2 [vn+2 ], due to the evolutionary property of ¦ . We now
want to construct the general solution of system (2.92). In order to do so,
we ¬rst solve the third equation in (2.92) for f v2 [vn+2 ],
f v2 [vn+2 ] = D(n+1) f v1 [vn+1 ], (2.93)
and the system reduces to
D(n) f u [vn ] ’ 2v2 D(n+1) f v1 [vn+1 ] = 0,
D(n) f v [vn ] ’ f v1 [vn+1 ] = 0. (2.94)
Remark 2.11. At this stage it would be possible to solve the last equa-
tion for f v1 [vn+1 ], but we prefer not to do so.
Now (2.94) is a polynomial in vn+2 of degree 1 and (2.94) reduces to
‚f v1 [vn+1 ]
’2v2
vn+2 : = 0,
‚vn+1
D(n) f u [vn ] ’ 2v2 D(n) f v1 [vn ] = 0,
1:
D(n) f v [vn ] ’ f v1 [vn ] = 0.
: (2.95)
In (2.95) and below, “vn+2 :” refers to the coe¬cient at vn+2 in a particular
equation. From (2.95) we arrive, due to the fact that second and third
equation are polynomial in vn+1 , at
‚f u [vn ] ‚f u1 [vn ]
’ 2v2
vn+1 : = 0,
‚vn ‚vn
D(n’1) f u [vn ] ’ 2v2 D(n’1) f v1 [vn ] = 0,
1:
‚f v [vn ]
vn+1 : = 0,
‚vn
D(n’1) f v [vn ] ’ f v1 [vn ] = 0.
1: (2.96)
To solve system (2.96), we ¬rst note that
f v [vn ] = f v [vn’1 ]. (2.97)
By di¬erentiation of the fourth equation in (2.96) twice with respect to v n ,
we obtain
‚ 2 f v1 [vn ]
= 0. (2.98)
2
‚vn
By consequence, f v1 is linear with respect to vn , i.e.,
f v1 [vn ] = H 1 [vn’1 ] + vn H 2 [vn’1 ]. (2.99)
Now, substitution of (2.97) and (2.99) into (2.96) yields the following
system of equations
‚f u [vn ]
’ 2v2 H 2 [vn’1 ] = 0,
‚vn
90 2. HIGHER SYMMETRIES AND CONSERVATION LAWS


D(n’1) f u [vn ] ’ 2v2 D(n’1) H 1 [vn’1 ] ’ 2v2 vn D(n’1) H 2 [vn’1 ] = 0,
D(n’1) f v [vn’1 ] ’ H 1 [vn’1 ] ’ vn H 2 [vn’1 ] = 0. (2.100)
We solve the ¬rst equation in (2.100) for f u [vn ], i.e.,
f u [vn ] = 2v2 vn H 2 [vn’1 ] + H 3 [vn’1 ], (2.101)
and from the second and third equation in (2.100) we arrive at
2v3 vn H 2 [vn’1 ] + 2v2 vn D(n’1) H 2 [vn’1 ] + D(n’1) H 3 [vn’1 ]
’ 2v2 D(n’1) H 1 [vn’1 ] ’ 2v2 vn D(n’1) H 2 [vn’1 ] = 0,
D(n’1) f v [vn’1 ] ’ H 1 [vn’1 ] ’ vn H 2 [vn’1 ] = 0. (2.102)
Due to cancellation of second and ¬fth term in the ¬rst equation of (2.102)
and its polynomial structure with respect to vn , we obtain a resulting system
of four equations:
‚H 3 [vn’1 ] ‚H 1 [vn’1 ]
2
’ 2v2
vn : 2v3 H [vn’1 ] + = 0,
‚vn’1 ‚vn’1
D(n’2) H 3 [vn’1 ] ’ 2v2 D(n’2) H 1 [vn’1 ] = 0,
1:
‚f v [vn’1 ]
’ H 2 [vn’1 ] = 0,
vn :
‚vn’1
D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0.
1: (2.103)
From (2.103) we solve the third equation for H 2 [vn’1 ],
‚f v [vn’1 ]
2
H [vn’1 ] = , (2.104)
‚vn’1
and integrate the ¬rst one in (2.103):
‚f v [vn’1 ] ‚H 3 [vn’1 ] ‚H 1 [vn’1 ]
’ 2v2
2v3 + = 0, (2.105)
‚vn’1 ‚vn’1 ‚vn’1
which leads to
H 3 [vn’1 ] = 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ]. (2.106)
By obtaining (2.106), we have to put in the requirement n ’ 1 > 3 and we
shall return to this case in the next subsection.
We now proceed by substituting the results (2.104) and (2.106) into
(2.103), which leads to
2v3 H 1 [vn’1 ] + 2v2 D(n’2) H 1 [vn’1 ] ’ 2v4 f v [vn’1 ] ’ 2v3 D(n’2) f v [vn’1 ]
+ D(n’2) H 4 [vn’2 ] ’ 2v2 D(n’2) H 1 [vn’1 ] = 0,
D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0. (2.107)
By cancellation of the second and sixth term in the ¬rst equation of (2.107),
we ¬nally arrive at
D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0,
4. THE HILBERT“CARTAN EQUATION 91


D(n’2) H 4 [vn’2 ] ’ 2v4 f v [vn’1 ] = 0, (2.108)
where the ¬rst equation in (2.108) can be considered as de¬ning relation
for H 1 [vn’1 ], while the second equation determines f v [vn’1 ] in terms of an
arbitrary function H 4 [vn’2 ]. The ¬nal result can now be obtained by (2.104)
and (2.106):
‚f v [vn’1 ]
2
H [vn’1 ] = ,
‚vn’1
H 3 [vn’1 ] = 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ], (2.109)
together with (2.108) and (2.101):
‚f v [vn’1 ]
u
+ 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ],
f [vn ] = 2v2 vn
‚vn’1
f v [vn ] = f v [vn’1 ], (2.110)

whereas in (2.110) f v [vn’1 ], H 1 [vn’1 ] are de¬ned by (2.108) in terms of an
arbitrary function H 4 [vn’2 ]! The general result of this section can now be
formulated in the following
Theorem 2.27. Let H be an arbitrary function of the variables x, u,
v, . . . , vn’2 , i.e.,
H = H[vn’2 ], (2.111)
and let us de¬ne
1 (n’2)
f v [vn’1 ] = D H[vn’2 ],
2v4
f u [vn ] = 2v2 D(n’1) f v [vn’1 ] ’ 2v3 f v [vn’1 ] + H[vn’2 ]. (2.112)
Then the vector ¬eld
‚ ‚
V = f u [vn ] + f v [vn’1 ] (2.113)
‚u ‚v
is a higher symmetry of (2.85).
Conversely, given a higher symmetry of (2.85), then there exists a func-
tion H, such that the components f u , f v of V are de¬ned by (2.112).
4.3. Special cases. Due to the restriction n > 4 the result (2.109) and
(2.110) holds for
n = 5, . . . (2.114)

meaning that H 4 [vn’2 ] is a free function of x, u, v, . . . , vn’2 and f v [vn’1 ] is
obtained by (2.109)
1 (n’2) 4
f v [vn’1 ] = D H [vn’2 ]. (2.115)
2v4
92 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

From (2.115) and (2.109) it is clear that f v [vn’1 ] is linear with respect
to the variable vn’1 and
vn’1 ‚H 4 [vn’2 ]
v
+ f v [vn’2 ].
f [vn’1 ] = (2.116)
2v4 ‚vn’2
Moreover, the requirement that f v [vn’1 ] is independent of vn’1 reduces to
H 4 [vn’2 ] to be independent of vn’2 , i.e.,
‚f v [vn’1 ]
= 0 ’ H 4 [vn’2 ] = H 4 [vn’3 ]. (2.117)
‚vn’1
The result (2.117) holds for all n > 5.
The results for higher symmetries, or Lie“B¨cklund transformations, for
a
n < 6 are obtained by imposing additional conditions on the coe¬cient f v
of the evolutionary vector ¬eld.
The case n = 5.
1 ‚H 4 [v3 ] 4 ‚H 4 [v3 ]
2 ‚H [v3 ]
v
f [v4 ] = ( + v2 + v1
2v4 ‚x ‚u ‚v
‚H 4 [v3 ] ‚H 4 [v3 ] ‚H 4 [v3 ]
+ v2 + v3 + v4 ).
‚v1 ‚v2 ‚v3
The requirement that f v [v4 ] is independent of v4 now leads to a genuine ¬rst
order partial di¬erential equation, i.e.,
‚H 4 4 ‚H 4 ‚H 4 ‚H 4
2 ‚H
+ v2 + v1 + v2 + v3 = 0, (2.118)
‚x ‚u ‚v ‚v1 ‚v2
and the general solution is given in terms of the invariants of the correspond-
ing vector ¬eld
‚ 2‚ ‚ ‚ ‚
U= + v2 + v1 + v2 + v3 , (2.119)
‚x ‚u ‚v ‚v1 ‚v2
where the set of invariants is given by
z1 = v 3 ,
z2 = v2 ’ v3 x,
z3 = 2v1 ’ 2v2 x + v3 x2 ,
z4 = 6v ’ 6v1 x + 3v2 x2 ’ v3 x3 ,
z5 = 3u ’ 3v2 x + 3v2 v3 x2 ’ v3 x3 .
2 2
(2.120)
So H 4 is given by
H 4 = H 4 (z1 , z2 , z3 , z4 , z5 ), (2.121)
whereas the formulas for f v and f u reduce to
‚H 4 ‚H 4 ‚2H 4
u 4
f = H ’ v2 ’ v3 + v 2 v4 2,
‚v2 ‚v3 ‚v3
1 ‚H 4
v
f= . (2.122)
2 ‚v3
5. THE CLASSICAL BOUSSINESQ EQUATION 93

The requirement the function f v is independent of
The case n = 4.
v3 reduces to
‚2H 4
2 = 0, (2.123)
‚v3
and (2.118)
‚H 4 4 ‚H 4 ‚H 4 ‚H 4
2 ‚H
+ v2 + v1 + v2 + v3 = 0. (2.124)
‚x ‚u ‚v ‚v1 ‚v2
Substitution of (2.123) into (2.124) immediately leads to the condition
‚‚
H 4 = 0, (2.125)
‚v2 ‚v3
i.e.,
f v = f v (x, u, v, v1 ), (2.126)
and the result completely reduces to the second order higher symmetries
obtained by Anderson [3] and [2] leading to the 14-dimensional Lie algebra
G2 .

5. The classical Boussinesq equation
The classical Boussinesq equation is written as the following system
of partial di¬erential equations in J 3 (π), where π : R2 — R2 ’ R2 with
independent variables x, t and u, v for dependent ones:
ut = (uv + ±vxx )x = ux v + uvx + ±vxxx ,
1
vt = (u + v 2 )x = ux + vvx . (2.127)
2
So in this application u = (u, v) and (x1 , x2 ) = (x, t). In order to construct
higher symmetries of (2.127), we have to construct solutions of the symmetry
condition which are discussed in Section 2. For evolution equations it is
custom to choose internal coordinates as x, t, u, v, u1 , v1 , u2 , v2 , . . . , where
‚iu ‚iv
ui = , vi = . (2.128)
‚xi ‚xi
The partial derivative operators Dx and Dt are de¬ned on E ∞ by
‚ ‚ ‚
Dx = + ui+1 + vi+1 ,
‚x ‚ui ‚vi
i>0 i>0
‚ ‚ ‚
Dt = + uit + vit , (2.129)
‚t ‚ui ‚vi
i>0 i>0

while expressions for uit and vit are derived from (2.127) by
i i
uit = Dx (ut ), vit = Dx (vt ). (2.130)
94 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

From (2.127) we derive the universal linearization operator as a 2 — 2 matrix
operator by of the form
3
vDx + v1 ±Dx + uDx + u1
= . (2.131)

Dx vDx + v1
To construct higher symmetries for equations (2.127), we start from a
vertical vector ¬eld of evolutionary type, i.e.,
∞ ∞
‚ ‚
Dx (Y u )
i
Dx (Y v )
i
Y’ = + . (2.132)
Y
‚ui ‚vi
i=0 i=0
From this and the presentation of the universal linearization operator we
derive the condition for Y = (Y u , Y v ) to be a higher symmetry of (2.127),
i.e.,
vDx Y u + v1 Y u + (±Dx + uDx + u1 )Y v = 0,
3

Dx Y u + (vDx + v1 )Y v = 0. (2.133)
It is quite of interest to make some remarks here on the construction
of solutions of this overdetermined system of partial di¬erential equations
for Y u , Y v . Recall that we require Y u and Y v to be dependent of a ¬nite
number variables. Equations (2.127) are graded, i.e., they admit a scaling
symmetry,
‚ ‚ ‚ ‚
’x ’ 2t + 2u +v ,
‚x ‚t ‚u ‚v
from where we have

deg(x) = ’1, = ’2,
deg(u) = 2, deg
‚u

deg(t) = ’2, = ’1.
deg(v) = 1, deg
‚v
Due to the grading of (2.127), equations (2.132) and (2.133) are graded too
and we require
Y u to dependent on x, t, v, u, v1 , . . . , u4 , v5 , u5 , v6 ,
Y v to dependent on x, t, v, u, v1 , . . . , u4 , v5 .
The general solution of (2.133) is then given by the following eight vector
¬elds
= (Yiu ,Yiv ) , i = 1, . . . , 8,
Yi

where
Y1u = ±v3 + u1 v + v1 u,
Y1v = u1 + v1 v;

Y2u = u1 ,
Y2v = v1 ;
5. THE CLASSICAL BOUSSINESQ EQUATION 95

Y3u = tu1 ,
Y3v = tv1 + 1;
1
Y4u = xu1 + t(±v3 + u1 v + v1 u) + u,
2
1 1
Y4v = xv1 + t(u1 + v1 v) + v;
2 2
1 3 3
Y5u = u1 (v 2 + 2u)
x(±v3 + u1 v + v1 u) + t u3 + v3 v + 3v2 v1 +
2± 2 4±
3 3 1
v1 vu + v 2 + vu,
+
2± 2 ±
1 3 3 12 1
Y5v = v1 (v 2 + 2u) +
x(u1 + v1 v) + t v3 + u1 v + v + u;
2± 2± 4± 4± ±
Y6u = 2±v5 + 4u3 v + v3 (3v 2 + 5u) + 9u2 v1 + 10v2 u1 + 12v2 v1 v
1 3
+ u1 v(v 2 + 6u) + 3v1 + v1 u(v 2 + u),
3
± ±
3 1
Y6v = 2u3 + 4v3 v + 7v2 v1 + u1 (v 2 + u) + v1 v(v 2 + 6u);
± ±
5 15 5 25
Y7u = ±u5 + v5 v + ±v4 v1 + u3 (v 2 + u) + ±v3 v2
2 2 2 2
5 45 25 5
+ v3 v(v 2 + 5u) + 5u2 u1 + u2 v1 v + v2 u1 v + v2 v1 (3v 2 + 5u)
4 4 2 2
75 5 15 3 5
2
u1 (v 4 + 12v 2 u + 6u2 ) + v1 v + v1 vu(v 2 + 3u),
+ u 1 v1 +
8 16± 4 4
5 5 35
Y7v = ±v5 + u3 v + v3 (v 2 + u) + 5u2 v1 + 5v2 u1 + v2 v1 v
2 2 4
5 15 3 5
+ u1 v(v 2 + 3u) + v1 + v1 (v 4 + 12v 2 u + 6u2 );
4 8 16
3 3 3
Y8u = u3 + v3 v + 3v2 v1 + u1 (v 2 + 2u) + v1 vu,
2 4± 2±
3 3
Y8v = v3 + v1 (v 2 + 2u).
u1 v + (2.134)
2± 4±
The Lie algebra structure of these symmetries is constructed by comput-
ing the Jacobi brackets of the respective generating functions Yi = (Yiu , Yiv ).
The commutators of the associated vector ¬elds are given then in Fig. 2.2.
The generating function Y9 is de¬ned here by
5 15 5 105
Y9u = ±v7 + u5 v + v5 (15v 2 + 14u) + 25u4 v1 + v4 u 1
2 2 8 4
225 175 25 175 375
u3 v2 + u3 v(v 2 + 3u) +
+ v4 v1 v + v3 u 2 + v3 v2 v
4 4 4 4 4
1125 25 75
2
v3 (3v 4 + 30v 2 u + 14u2 ) +
+ v3 v1 + u2 u1 v
16 32± 2±
96 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

[Yi , Yj ] Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8
0 ’Y2 ’Y1 ’Y8
Y1 0 0 0
1 1
’ 2 Y2 ’ 2± Y4
Y2 0 0 0 0
1 1 5 3
Y3 2 Y3 ± Y4 4Y8 4 Y6 2± Y1
1 5 3
Y4 2 Y5 2Y6 2 Y7 2 Y8
4 3
Y5 ± Y7 Y9 4± Y6
Y6 0 0
Y7 0
Y8

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