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45 : f(8) := f(8);
F8 := (4(F12 )t u1 x + (F7 )t2 u1 x2 + 4(F7 )t uu1 x + 8F16 u1 + 8F13
+ 8F12 uu1 + 6F7 u2 u1 + 12F7 u2 )/8$ (8.82)
1

while F1 and the remaining equation, equ(20), are given as:
46 : f(1) := f(1);
F1 := (4(F12 )t u1 x + (F7 )t2 u1 x2 + 4(F7 )t uu1 x + 4(F7 )t u2 x + 8F16 u1
+ 8F13 + 8F12 uu1 + 8F12 u2 + 6F7 u2 u1 + 12F7 uu2
+ 12F7 u2 + 8F7 u3 )/8$
1

47 : print equations(1, te);
equ(20) := (8(F16 )t u1 + 8(F13 )t ’ 16(F13 )u,x u1 ’ 8(F13 )u2 u2
1
’ 8(F13 )x2 ’ 8(F13 )x u + 4(F12 )t2 u1 x + 4(F12 )t uu1 + (F7 )t3 u1 x2
+ 2(F7 )t2 uu1 x ’ 2(F7 )t2 u1 + 2(F7 )t u2 u1 + 4(F7 )t u2 ’ 8F13 u1 )/4$
1
Functions occurring :
F7 (t)
F12 (t)
F13 (u, t, x)
F16 (t) (8.83)
The remaining equation can then be solved in a straightforward way leading
to the ¬nal result:
48 : solve equation(20);
equ(20) breaks into equ(23), . . . , equ(25) by u1 , u2 , u3 , u4 , u5 , . . .

49 : equ(23);
4(’2(F13 )u2 + (F7 )t )$

50 : solve equation(23);
equ(23) : Inhomogeneous integration of (F13 )u2

51 : f(13) := f(13);
F13 := ((F7 )t u2 + 4F18 u + 4F17 )/4$

52 : print equations(24, 24);
equ(24) := ’16(F18 )x + 8(F16 )t + 4(F12 )t2 x + 4(F12 )t u + (F7 )t3 x2
+ 2(F7 )t2 ux ’ 2(F7 )t2 ’ 8F18 u ’ 8F17 $
368 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

Functions occurring :
F17 (t, x)
F7 (t)
F12 (t)
F16 (t)
F18 (t, x)

53 : solve equation(24);
equ(24) breaks into equ(26), . . . , equ(27) by u, u1 , u2 , u3 , u4 , . . .

54 : print equations(26, 27);
equ(26) := 2(2(F12 )t + (F7 )t2 x ’ 4F18 )$
Functions occurring :
F18 (t, x)
F7 (t)
F12 (t)

equ(27) := ’16(F18 )x + 8(F16 )t + 4(F12 )t2 x + (F7 )t3 x2
+ 2(F7 )t2 ’ 8F17 $
Functions occurring :
F17 (t, x)
F7 (t)
F12 (t)
F16 (t)
F18 (t, x)

55 : solve equation(26);
equ(26) : Solved for F18

56 : solve equation(27);
equ(27) : Solved forF17

57 : print equations(1, te);
equ(25) := 8(F16 )t2 + 4(F12 )t3 x
+ (F7 )t4 x2 ’ 8(F7 )t3 $
Functions occurring :
F7 (t)
F12 (t)
F16 (t) (8.84)
3. OVERDETERMINED SYSTEMS OF PDE 369

and
58 : solve equation(25);
equ(25) breaks into equ(28), . . . , equ(30) by x, u, u1 , u2 , u3 , . . .

59 : print equations(28, 30);
equ(28) := (F7 )t4 $
Functions occurring :
F7 (t)

equ(29) := 4(F12 )t3 $
Functions occurring :
F12 (t)

equ(30) := 8((F16 )t2 ’ (F7 )t3 )$
Functions occurring :
F7 (t)
F16 (t)

60 : solve equation(28);
equ(28) : Homogeneous integration of (F7 )t4

61 : f(7) := f(7);
F7 := c(4)t3 + c(3)t2 + c(2)t + c(1)$

62 : solve equation(29);
equ(29) : Homogeneous integration of (F12 )t3

63 : f(12) := f(12);
F12 := c(7)t2 + c(6)t + c(5)$

64 : equ(30) := equ(30);
equ(30) := 8(’6c(4) + (F16 )t2 )$

65 : solve equation(30);
equ(30) : Inhomogeneous integration of (F16 )t2

66 : f(16) := f(16);
F16 := c(9)t + c(8) + 3c(4)t2 $

67 : factor t, x; (8.85)
and
68 : f(1) := f(1);
370 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

F1 := (t3 c(4)(3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 6t2 xc(4)(uu1 + u2 )
+ t2 (4c(7)uu1 + 4c(7)u2 + 3c(4)u2 + 12c(4)u1 + 3c(3)u2 u1
+ 6c(3)uu2 + 6c(3)u2 + 4c(3)u3 )
1
+ 3tx2 c(4)u1
+ 2tx(2c(7)u1 + 3c(4)u + 2c(3)uu1 + 2c(3)u2 )
+ t(4c(9)u1 + 4c(7)u + 4c(6)uu1 + 4c(6)u2 + 6c(4)
+ 2c(3)u2 + 3c(2)u2 u1 + 6c(2)uu2 + 6c(2)u2 + 4c(2)u3 )
1
+ x2 (3c(4) + c(3)u1 )
+ 2x(2c(7) + c(6)u1 + c(3)u + c(2)uu1 + c(2)u2 )
+ 4c(9) + 4c(8)u1 + 2c(6)u + 4c(5)uu1 + 4c(5)u2 ’ 6c(3)
+ c(2)u2 + 3c(1)u2 u1 + 6c(1)uu2 + 6c(1)u2 + 4c(1)u3 )/4$ (8.86)
1


and

69 : for i := 1 : 9 do write vec(i) := df(f(1), c(i));
vec(1) := (3u2 u1 + 6uu2 + 6u2 + 4u3 )/4$
1

vec(2) := (t(3u2 u1 + 6uu2 + 6u2 + 4u3 ) + 2x(uu1 + u2 ) + u2 )/4$
1

vec(3) := (t2 (3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 4tx(uu1 + u2 ) + 2tu2 + x2 u1 + 2xu ’ 6)/4$

vec(4) := (t3 (3u2 u1 + 6uu2 + 6u2 + 4u3 )
1
+ 6t2 x(uu1 + u2 ) + 3t2 (u2 + 4u1 ) + 3tx2 u1 + 6txu + 6t + 3x2 )/4$

vec(5) := uu1 + u2 $

vec(6) := (2t(uu1 + u2 ) + xu1 + u)/2$

vec(7) := t2 (uu1 + u2 ) + txu1 + tu + x$

vec(8) := u1 $

vec(9) := tu1 + 1$
70 : (8.87)

The previous application demonstrates in a nice way how calculations con-
cerning symmetries and other invariants of partial di¬erential equations are
performed.
3. OVERDETERMINED SYSTEMS OF PDE 371

We ¬nish this section with the remark that it is possible to run the
program automatically on this system (8.66). Doing this, the complete con-
struction does take 0.3 seconds. Most problems need however the researcher
as operator in the construction of the general solution.
3.3. Polynomial and graded cases. A very often arising situation
is the construction of symmetries and of conservation laws for equations
admitting scaling symmetry.
Let us take for example:
Example 8.6. The KdV equation is given by:
ut = uux + uxxx , (8.88)
which as we have seen in Section 5 of Chapter 3 admits a scaling symmetry
‚ ‚ ‚
S = ’x ’ 3t + 2u + ··· (8.89)
‚x ‚t ‚u
This means that in physical terms all variables are of appropriate dimen-
sions, whereas in mathematical terms it means that all variables are graded 2 ,
i.e.,
degree(x) ≡ [x] = ’1, [t] = ’3, [u] = 2, [ux ] = 3, [ut ] = 5, . . . . (8.90)
This grading means that all objects are graded too, and for the generat-
ing functions of symmetries and conservation laws only those functions are
of interest which are of a speci¬ed degree in the variables.
Example 8.7. Suppose that in the previous example we are interested
to have the most general functions F and G of degree 5 and 7 respectively,
with respect to the graded variables u, ux , uxx , uxxx , uxxxx , uxxxxx which
are of degree 2, 3, 4, 5, 6, 7 respectively. The result will be:
G = c3 uxxxxx + c4 uuxxx + c5 ux uxx + c6 u2 ux .
F = c1 uxxx + c2 uux ,
(8.91)
If, however, we are in the situation that F is of degree 5 with respect to the
graded variables p1 , u, ux , uxx , uxxx , uxxxx , uxxxxx which are of degree 1, 2,
3, 4, 5, 6, 7 respectively, then the result will be:
F = c1 uxxx + c2 p1 uxx + (c3 u + c4 p2 )ux + c5 p3 u + c6 p5 , (8.92)
1 1 1

while for G we have the general presentation
G = c1 uxxxxx + c2 p1 uxxxx + (c3 u + c4 p2 )uxxx
1
+ (c5 ux + c6 p1 u + c7 p3 )uxx + c8 p1 u2
1 x
+ (c9 u2 + c10 p2 u + c11 p4 )ux + c12 p1 u3
1 1
+ c13 p3 u2 + c14 p5 u + c15 p7 . (8.93)
1 1 1

2
The term graded here means that some weights can be assigned to all variables in
such a way that the equation becomes homogeneous with respect to these weights.
372 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

Procedures are availabe to construct the most general presentation of
a function of a speci¬ed degree, with respect to a speci¬ed list of graded
variables.
Once one knows that all objects are graded, the conditions (1.37) do lead
to polynomial equations with respect to the jet variables, the coe¬cients
of which have to vanish. This process does lead to just algebraic linear
equations for the constants in the original expressions (8.92) and (8.93).
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