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equ(.) :=(F— )xk1 ,...,xkr , (8.53)
i1 ir


is a mixed (k1 + · · · + kr )-th order derivative.
The general solution of (8.53) is
r ks ’1
Fis,t xts ,
F— := (8.54)
i
s=1 t=0

whereas in (8.54) Fis,t depends on the same variables as F— , except
for xis , t = 0, . . . , ks ’ 1, s = 1, . . . , r.
Example 8.2.
equ(.) :=(F1 )x1 ,x2 . (8.55)
The general solution to this equation is given by
F1 := F2 + F3 , (8.56)
where F2 depends on the same variables as F1 , except for x1 , while
F3 depends on the same variables as F1 , except for x2 .
3. CASE C: The partial di¬erential equation equ(.) contains a func-
tion F— , depending on all variables present as arguments of some
other function(s) F—— , occuring in this equation, whereas there is no
derivative of a function F— present in the equation.
The partial di¬erential equation can then be solved for the func-
tion F— .
Example 8.3.
equ(.) :=x1 F1 + x2 (F2 )x1 , (8.57)
where in (8.57) F1 , F2 are dependent on x1 , x2 , x3 . The solution is
F1 := (’x2 (F2 )x1 )/x1 (8.58)
We have to make a remark here. There is a switch in the system
that checks for the coe¬cient for the function F— to be a number. In
case the switch coefficient check is on, equ(.) will not be solved.
In case the switch coefficient check is o¬, the result is given as in
(8.58).
4. CASE D: In the partial di¬erential equation there is a derivative
of a function F— with respect to variables which are not present as
argument of any other function F—— , while the coe¬cient of F— is a
number. By the assumption that x1 , . . . , xn appear as polynomials,
the partial di¬erential equation can be integrated.
3. OVERDETERMINED SYSTEMS OF PDE 359

Example 8.4. Let the partial di¬erential equation be given by
equ(.) :=(F1 )x3 + x2 F2 , (8.59)
where F1 depends on x1 , x2 , x3 and F2 depends on x1 , x2 .
The solution to (8.59) is
F1 := ’x2 x3 F2 + F3 , (8.60)
whereas F3 depends on x1 , x2 and is independent of x3 .
5. CASE E: In the partial di¬erential equation a speci¬c variable xi is
present just once as argument of some function F— . By appropriate
di¬erentiation, one may arrive at a simple equation, which can be
solved.
Evaluation of the original equation can result in an equation which
can be solved too.
Example 8.5.
equ(.) :=x2 (F1 )x2 ,x3 + x3 F2 , (8.61)
where F1 depends on x1 , x2 , x3 and F2 depends on x1 , x2 .
Di¬erentiation with respect to x3 twice results in
equ(.) :=x2 (F1 )x2 ,x3 . (8.62)
3

The solution to (8.62) is CASE B:
F1 := F3 x2 + F4 x3 + F5 + F6 , (8.63)
3
where F1 , F4 , F5 are dependent on x1 , x2 , F6 depends on x1 , x3 .
Substitution of the result (8.63) into the original equation (8.61)
leads to
equ(.) :=2x2 x3 (F3 )x2 + x2 (F4 )x2 + x3 F2 . (8.64)
Due to CASE A, the procedure solve equation constructs two new
equations
equ(.):=2x2 (F3 )x2 + F2 ,
equ(.):=x2 (F4 )x2 (8.65)
The complete result of the procedure solve equation will in this case
be (8.63) and (8.65).
Now the procedure solve equation is then useful to solve the last
two equations (8.65) constructed before; this last step is not carried
through automatically.
For this case there is a switch “differentiation” too, similar to
the previous case.
In practical situations, one is able to solve the overdetermined system
of partial di¬erential equations, using the methods described in the CASES
A, B, C, D, E and some additional considerations, which are speci¬c for the
problem at hand.
360 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

3.2. The Burgers equation. We shall discuss the construction of
higher symmetries of order three of the Burgers equation in order to demon-
strate the facilities of the INTEGRATION package, in e¬ect the procedure
solve equation described in the previous subsection.
The Burgers equation is given by the following partial di¬erential equa-
tion
ut = uu1 + u2 , (8.66)
where partial derivatives with respect to x are given by indices 1, 2, . . . We
start this example by introduction of the vector ¬elds Dx , Dt in the jet
bundle where local coordinates are given by x, t, u, u1 , u2 , u3 , u4 , u5 , u6 ,
u7 , u8 and a generating function F1 , which is dependent on the jet variables
x, t, u, u1 , u2 , u3 .
So the representation of the vector ¬elds Dx , Dt is given by
‚ ‚ ‚ ‚ ‚ ‚ ‚
Dx = + u1 + u2 + u3 + u4 + u5 + u6
‚x ‚u ‚u1 ‚u2 ‚u3 ‚u4 ‚u5
‚ ‚
+ u7 + u8 ,
‚u6 ‚u7
‚ ‚ ‚ ‚ ‚ ‚
Dt = + (ut ) + (ut )1 + (ut )2 + (ut )3 + (ut )4
‚t ‚u ‚u1 ‚u2 ‚u3 ‚u4
‚ ‚
+ (ut )5 + (ut )6 , (8.67)
‚u5 ‚u6
where (ut ), . . . , (ut )6 are given by
(ut ) = uu1 + u2 ,
(ut )1 = uu2 + u2 + u3 ,
1
(ut )2 = uu3 + 3u1 u2 + u4 ,
(ut )3 = uu4 + 4u1 u3 + 3u2 + u5 ,
2
(ut )4 = uu5 + 5u1 u4 + 10u2 u3 + u6 ,
(ut )5 = uu6 + 6u1 u5 + 15u2 u4 + 10u2 + u7 ,
3
(ut )6 = uu7 + 7u1 u6 + 21u2 u5 + 35u3 u4 + u8 . (8.68)
In the remaining part of this section we shall present in e¬ect a computer
session and give some comments on the construction and use of the procedure
solve equation. We shall stick as close as possible to the real output of
the computer system. Boldtext will refer to real input to the system, while
the rest is just the output on screen.
Now from the symmetry condition (2.29) we obtain
2 : equ(1) = Dt (F1 ) ’ Dx (Dx F1 ) ’ uDx (F1 ) ’ u1 F1 ; (8.69)
where the solution is to be determined in such a way that the right-hand
side in (8.69) has to vanish.
3. OVERDETERMINED SYSTEMS OF PDE 361

The resulting equation is now given by
equ(1) = (F1 )t ’ 2(F1 )u,u1 u1 u2 ’ 2(F1 )u,u2 u1 u3 ’ 2(F1 )u,u3 u1 u4
’ 2(F1 )u,x u1 ’ (F1 )u2 u2 ’ 2(F1 )u1 ,u2 u2 u3 ’ 2(F1 )u1 ,u3 u2 u4
1
’ 2(F1 )u1 ,x u2 ’ (F1 )u2 u2 + (F1 )u1 u2 ’ 2(F1 )u2 ,u3 u3 u4
12 1

’ 2(F1 )u2 ,x u3 ’ (F1 )u2 u2 + 3(F1 )u2 u1 u2 ’ 2(F1 )u3 ,x u4 ’ (F1 )u2 u2
23 34

+ 4(F1 )u3 u1 u3 + 3(F1 )u3 u2 ’ (F1 )x2 ’ (F1 )x u ’ F1 u1 $ (8.70)
2

The dependency of the function F1 is stored on a depl!* (dependency list):
3 : lisp depl!—;
(((f 1) u3 u2 u1 u t x)) (8.71)

Equ(1) is an equation, which is a polynomial with respect to the variable u4 ,
so in order to be 0, its coe¬cients should be zero.
These coe¬cients will be detected by the procedure solve equation(*),
i.e., CASE A:
4 : solve equation(1);
equ(1) breaks into equ(2), . . . , equ(4) by u4 , u5 , u6 , u7 , u8

5 : print equations(2, 4);
equ(2) := ’ (F1 )u2 $
3
Functions occurring :
F1 (u3 , u2 , u1 , u, t, x)

equ(3) := ’ 2((F1 )u,u3 u1 + (F1 )u1 ,u3 u2 + (F1 )u2 ,u3 u + (F1 )u3 ,x )$
Functions occurring :
F1 (u3 , u2 , u1 , u, t, x)

equ(4) :=(F1 )t ’ 2(F1 )u,u1 u1 u2 ’ 2(F1 )u,u2 u1 u3 ’ 2(F1 )u,x u1
’ (F1 )u2 u2 ’ 2(F1 )u1 ,u2 u2 u3 ’ 2(F1 )u1 ,x u2 ’ (F1 )u2 u2 + (F1 )u1 u2
1 12 1

’ 2(F1 )u2 ,x u3 ’ (F1 )u2 u2 + 3(F1 )u2 u1 u2 + 4(F1 )u3 u1 u3 + 3(F1 )u3 u2
23 2
’ (F1 )x2 ’ (F1 )x u ’ F1 u1 $

Functions occurring :
F1 (u3 , u2 , u1 , u, t, x) (8.72)
We now are left with a system of three partial di¬erential equations for the
function F1 .
Equ(2) can now be solved, the result being a polynomial of degree 1 with
respect to the variable u3 , while coe¬cients are functions still dependent on
x, t, u, u1 , u2 .
362 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

The construction of this solution, as in CASE B, is performed by the
procedure solve equation(*) too, i.e.:
6 : solve equation(2);
equ(2) : Homogeneous integration of (F1 )u2
3

7 : F1 := F1 ;
F1 := F3 u3 + F2 $ (8.73)
Substitution of this result into the third equation leads to:
8 : equ(3) := equ(3);
equ(3) := ’2((F3 )u u1 + (F3 )u1 u2 + (F3 )u2 u3 + (F3 )x )$ (8.74)
and this equation splits up, since it is a polynomial of degree 1 with respect
to u3 , CASE A:
9 : solve equation(3);
equ(3) breaks into equ(5), ..., equ(6) by u3 , u4 , u5 , u6 , u7 , · · ·

10 : print equations(5, 6);
equ(5) := ’2(F3 )u2 $
Functions occurring :
F3 (u2 , u1 , u, t, x)

equ(6) := ’2((F3 )u u1 + (F3 )u1 u2 + (F3 )x )$
Functions occurring :
F3 (u2 , u1 , u, t, x) (8.75)
Now the procedure can be repeated, since equ(5) indicates that F3 is inde-
pendent of u2 , in e¬ect a polynomial of degree 0, and equ(6) can be splitted
with respect to u2 :
11 : solve equation(5);
equ(5) : Homogeneous integration of (F3 )u2

12 : solve equation(6);
equ(6) breaks into equ(7), . . . , equ(8) by u2 , u3 , u4 , u5 , u6 , . . .

13 : print equations(7, 8);
equ(7) := ’2(F4 )u1 $
Functions occurring :
F4 (u1 , u, t, x)

equ(8) := ’2((F4 )u u1 + (F4 )x )$
Functions occurring :
3. OVERDETERMINED SYSTEMS OF PDE 363

F4 (u1 , u, t, x) (8.76)
From equ(7) we have that F4 is independent of u1 and combination with
equ(8) then results in the fact that F4 is independent of u and x too:
14 : solve equation(7);
equ(7) : Homogeneous integration of (F4 )u1

15 : solve equation(8);
equ(8) breaks into equ(9), ..., equ(10) by u1 , u2 , u3 , u4 , u5 , . . .

16 : print equations(9, 10);
equ(9) := ’2(F5 )u $
Functions occurring :
F5 (u, t, x)

equ(10) := ’2(F5 )x $
Functions occurring :
F5 (u, t, x)

17 : solve equation(9);
equ(9) : Homogeneous integration of (F5 )u

18 : solve equation(10);
equ(10) : Homogeneous integration of (F6 )x (8.77)
Summarising the results obtained thusfar, we are left with an expression for
the function F1 in terms of F2 and F7 and one equation, equ(4), which is
polynomial with respect to u3 :
19 : f(1) := f(1);
F1 := F7 u3 + F2 $

20 : print equations(1, te);
equ(4) := (F7 )t u3 + (F2 )t ’ 2(F2 )u,u1 u1 u2 ’ 2(F2 )u,u2 u1 u3
’ 2(F2 )u,x u1 ’ (F2 )u2 u2 ’ 2(F2 )u1 ,u2 u2 u3 ’ 2(F2 )u1 ,x u2 ’ (F2 )u2 u2
1 12

+ (F2 )u1 u2 ’ 2(F2 )u2 ,x u3 ’ (F2 )u2 u2 + 3(F2 )u2 u1 u2 ’ (F2 )x2
1 23

’ (F2 )x u + 3F7 u1 u3 + 3F7 u2 ’ F2 u1 $
2
Functions occurring :
F2 (u2 , u1 , u, t, x)
F7 (t)

21 : solve equation(4);
equ(4) breaks into equ(11), . . . , equ(13) by u3 , u4 , u5 , u6 , u7 , . . .
364 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

22 : print equations(11, 13);
equ(11) := ’(F2 )u2 $
2
Functions occurring :
F2 (u2 , u1 , u, t, x)

equ(12) := (F7 )t ’ 2(F2 )u,u2 u1 ’ 2(F2 )u1 ,u2 u2 ’ 2(F2 )u2 ,x + 3F7 u1 $
Functions occurring :
F2 (u2 , u1 , u, t, x)
F7 (t)

equ(13) := (F2 )t ’ 2(F2 )u,u1 u1 u2 ’ 2(F2 )u,x u1 ’ (F2 )u2 u2
1
’ 2(F2 )u1 ,x u2 ’ (F2 )u2 u2 + (F2 )u1 u2 + 3(F2 )u2 u1 u2 ’ (F2 )x2
12 1

’ (F2 )x u + 3F7 u2 ’ F2 u1 $
2
Functions occurring :
F7 (t)
F2 (u2 , u1 , u, t, x) (8.78)

The remaining system, equ(11), equ(12), equ(13), can be handled in a sim-
ilar way as before, leading to an expression for the function F2 :

23 : solve equation(11);
equ(11) : Homogeneous integration of (F2 )u2
2


24 : equ(12) := equ(12);
equ(12) := ’2(F9 )u u1 ’ 2(F9 )u1 u2 ’ 2(F9 )x + (F7 )t + 3F7 u1 $

25 : solve equation(12);
equ(12) breaks into equ(14), . . . , equ(15) by u2 , u3 , u4 , u5 , u6 , . . .

26 : equ(14);
’ 2(F9 )u1 $

27 : solve equation(14);
equ(14) : Homogeneous integration of (F9 )u1

28 : equ(15);
’ 2(F10 )u u1 ’ 2(F10 )x + (F7 )t + 3F7 u1 $

29 : solve equation(15);
equ(15) breaks into equ(16), . . . , equ(17) by u1 , u2 , u3 , u4 , u5 , . . .

30 : print equations(16, 17);
3. OVERDETERMINED SYSTEMS OF PDE 365

equ(16) := ’2(F10 )u + 3F7 $
Functions occurring :
F7 (t)
F10 (u, t, x) (8.79)
and
equ(17) := ’2(F10 )x + (F7 )t $
Functions occurring :
F7 (t)
F10 (u, t, x)

31 : solve equation(16);
CASE C :
equ(16) : Inhomogeneous integration of (F10 )u

32 : solve equation(17);
equ(17) : Inhomogeneous integration of (F11 )x

33 : f(2) := f(2);
F2 := ((F7 )t u2 x + 2F12 u2 + 2F8 + 3F7 uu2 )/2$ (8.80)
while the original de¬ning function F1 , and the remaining equation, equ(13),
are given by:
34 : f(1) := f(1);
F1 := ((F7 )t u2 x + 2F12 u2 + 2F8 + 3F7 uu2 + 2F7 u3 )/2$

35 : print equations(1, te);
equ(13) := (2(F12 )t u2 + 2(F8 )t ’ 4(F8 )u,u1 u1 u2 ’ 4(F8 )u,x u1
’ 2(F8 )u2 u2 ’ 4(F8 )u1 ,x u2 ’ 2(F8 )u2 u2 + 2(F8 )u1 u2 ’ 2(F8 )x2
1 12 1
’ 2(F8 )x u + (F7 )t2 u2 x + 2(F7 )t uu2 + 2(F7 )t u1 u2 x + 4F12 u1 u2
’ 2F8 u1 + 6F7 uu1 u2 + 6F7 u2 )/2$
2
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)
F12 (t) (8.81)
Equ(13) is a polynomial with respect to the variable u2 , and the result is
again a system of three equations, the ¬rst two of them can be solved in
exactly the same way as before, leading to an expression for F8 :
36 : solve equation(13);
equ(13) breaks into equ(18), . . . , equ(20) by u2 , u3 , u4 , u5 , u6 , . . .
366 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

37 : print equations(18, 19);
equ(18) := 2(’(F8 )u1 ,2 + 3F7 )$
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)

equ(19) := 2(F12 )t ’ 4(F8 )u,u1 u1 ’ 4(F8 )u1 ,x + (F7 )t2 x + 2(F7 )t u
+ 2(F7 )t u1 x + 4F12 u1 + 6F7 uu1 $
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)
F12 (t)
38 : solve equation(18);
equ(18) : Inhomogeneous integration of (F8 )u2
1


39 : print equations(19, 19);
equ(19) := ’4(F14 )u u1 ’ 4(F14 )x + 2(F12 )t + (F7 )t2 x + 2(F7 )t u
+ 2(F7 )t u1 x + 4F12 u1 + 6F7 uu1 $
Functions occurring :
F7 (t)
F12 (t)
F14 (u, t, x)

40 : solve equation(19);
equ(19) breaks into equ(21), . . . , equ(22) by u1 , u2 , u3 , u4 , u5 , . . .

41 : equ(21);
2(’2(F14 )u + (F7 )t x + 2F12 + 3F7 u)$

42 : solve equation(21);
equ(21) : Inhomogeneous integration of (F14 )u

43 : print equations(22, 22);
equ(22) := ’4(F15 )x + 2(F12 )t + (F7 )t2 x$
Functions occurring :
F7 (t)
F12 (t)
F15 (t, x)
3. OVERDETERMINED SYSTEMS OF PDE 367

44 : solve equation(22);
equ(22) : Inhomogeneous integration of (F15 )x

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