vector ¬elds, solutions of overdetermined systems of partial di¬erential equa-

tions, covering conditions, etc.

Since then, quite a number of programs has been constructed and it

seems that nowadays each individual researcher in this ¬eld of mathematical

physics uses his or her own developed software to do the required computa-

tions in more or less the most or almost most e¬cient way. An overview of

existing programs in all distinct related areas was recently given by Hereman

in his extensive paper [30].

In the following sections, we shall discuss in some detail a number of

types of computations which can be carried through on a computer sys-

tem. The basis of these programs has been constructed by Gragert [22],

Kersten [37], Gragert, Kersten and Martini [24, 25], Roelofs [85, 86], van

Bemmelen [9, 8] at the University of Twente, starting in 1979 with exterior

di¬erential forms, construction and solution of overdetermined systems of

partial di¬erential equations arising from symmetry computations, exten-

sion of the software to work in a graded setting, meaning supercalculus,

349

350 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

required for the interesting ¬eld of super and supersymmetric extensions of

classical di¬erential equations and at the end a completely new package,

being extremely suitable for classical as well as supersymmetrical systems,

together with packages for computation of covering structures of completely

integrable systems, and a package to handling the computations with to-

tal derivative operators. We should mention here too (super) Lie algebra

computations for covering structures by Gragert and Roelofs [23, 26].

We prefer to start in Section 1 with setting down the basic notions of the

graded or supercalculus, since classical di¬erential geometric computations

can be embedded in a very e¬ective way in this more general setting, which

will be done in Section 2.

In Section 3 we shall give an idea how the software concerning construc-

tion of solutions of overdetermined systems of partial di¬erential equations

works, and what the facilities are.

Finally we shall present in Subsection 3.2 a computer session concerning

the construction of higher symmetries of third order of the Burgers equation,

i.e., de¬ning functions involving derivatives (with respect to x up to order 3),

cf. Chapter 2.

1. Super (graded) calculus

We give here a concise exposition of super (or graded) calculus needed

for symbolic computations.

At ¬rst sight the introduction of graded calculus requires a completely

new set of de¬nitions and objects. It has been shown that locally a graded

manifold, or equivalently the algebra of functions de¬ned on it, is given as

C ∞ (U ) — Λ(n), where Λ(n) is the exterior algebra of n (odd) variables, [50].

Below we shall set down the notions involved in the graded calculus and

graded di¬erential geometry.

Thus we give a short review of the notions of graded di¬erential ge-

ometry as far as they are needed for implementation by means of software

procedures, i.e., graded commutative algebra, graded Lie algebra, graded man-

ifold, graded derivation, graded vector ¬eld, graded di¬erential form, exterior

di¬erentiation, inner di¬erentiation or contraction by a vector ¬eld, Lie de-

rivative along a vector ¬eld, etc.

The notions and notations have been taken from Kostant [50] and the

reader is referred to this paper for more details, compare with Chapter 6.

Throughout this section, the basic ¬eld is R or C and the grading will be

with respect to Z2 = {0, 1}.

1. A vector space V over R is a graded vector space if one has V0 and V1

subspaces of V , such that

V = V0 • V1 (8.1)

is a direct sum. Elements of V0 are called even, elements of V1 are

called odd. Elements of V0 or V1 are called homogeneous elements.

1. SUPER (GRADED) CALCULUS 351

If v ∈ Vi , i = 0, 1, then i is called the degree of v, i.e.,

|v| = i, i = 0, 1, or i ∈ Z2 . (8.2)

The notation |v| is used for homogeneous elements only.

2. A graded algebra B is a graded vector space B = B0 • B1 with a

multiplication such that

Bi · Bj ‚ Bi+j , i, j ∈ Z2 . (8.3)

3. A graded algebra B is called graded commutative if for any two ho-

mogeneous elements x, y ∈ B we have

xy = (’1)|x||y| yx. (8.4)

4. A graded space V is a left module over the graded algebra B, if V is

a left module in the usual sense and

Bi · Vj ‚ Vi+j , i, j ∈ Z2 ; (8.5)

right modules are de¬ned similarly.

5. If V is a left module over the graded commutative algebra B, then V

inherits a right module structure, where we de¬ne

def

v · b = (’1)|v||b| b · v, v ∈ V, b ∈ B. (8.6)

Similarly, a left module structure is determined by a right module

structure.

6. A graded vector space g = g0 • g1 , together with a bilinear operation

[·, ·] on g such that [x, y] ∈ g|x|+|y| is called a graded Lie algebra if

[x, y] = ’(’1)|x||y| [y, x],

(’1)|x||z| [x, [y, z]] + (’1)|z||y| [z, [x, y]] + (’1)|y||x| [y, [z, x]] = 0, (8.7)

where the last equality is called the graded Jacobi identity.

If V is a graded vector space, then End(V ) has the structure of a

graded Lie algebra de¬ned by

[±, β] = ±β ’ (’1)|±||β| β±, ±, β ∈ End(V ). (8.8)

7. If B is a graded algebra, an operator h ∈ Endi (B) is called a graded

derivation of B if

h(xy) = h(x)y + (’1)|i||x| xh(y). (8.9)

An operator h ∈ End(B) is a derivation if its homogeneous compo-

nents are so.

The graded vector space of derivations of B, denoted by Der(B),

is a graded Lie subalgebra of End(B). Equality (8.9) is called graded

Leibniz rule. If B is a graded commutative algebra then Der(B) is a

left B-module: if ζ ∈ Der(B), f, g ∈ B, then f ζ ∈ Der(B), where

(f ζ)g = f (ζg). (8.10)

352 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

8. The local picture of a graded manifold is an open neighborhood U ‚

Rm together with the graded commutative algebra

C ∞ (U ) — Λ(n), (8.11)

where Λ(n) is the antisymmetric (exterior) algebra on n elements

|si | = 1, si sj = ’sj si ,

s1 , . . . , sn , i, j = 1, . . . , n. (8.12)

The pair (m|n) is called the dimension of the graded manifold at

hand. A particular element f ∈ C ∞ (U ) — Λ(n) is represented as

f= fµ s µ , (8.13)

µ

where µ is a multi-index: µ ∈ Mn = {µ = (µ1 , . . . , µn ) | µi ∈ N, 1 ¤

µ1 ¤ µ2 · · · ¤ µk ¤ n},

fµ ∈ C ∞ (U ).

s µ = s µ1 · s µ2 · . . . · s µk , (8.14)

9. Graded vector ¬elds on a graded manifold (U, C ∞ (U ) — Λ(n)) are

introduced as graded derivations of the algebra C ∞ (U ) — Λ(n). They

constitute a left C ∞ (U ) — Λ(n)-module. Locally, a graded vector ¬eld

V is represented as

m n

‚ ‚

V= fi + gj , (8.15)

‚ri ‚sj

i=1 j=1

where fi , gj ∈ C ∞ (U ) — Λ(n), and ri , i = 1, . . . , m, are local coordi-

nates in U ‚ Rm .

The derivations ‚/‚ri , i = 1, . . . , m, are even, while the deriva-

tions ‚/‚sj , j = 1, . . . , n, are odd. They satisfy the relations

‚sj

‚rk ‚rk ‚sl

= δik , = 0, = 0, = δjl (8.16)

‚ri ‚ri ‚sj ‚sj

for all i, k = 1, . . . , m, j, l = 1, . . . , n.

10. A graded di¬erential k-form is introduced as k-linear mapping β on

Der(C ∞ (U ) — Λ(n)) which has to satisfy the identities

Pl’1

ζ1 , . . . , f ζl , . . . , ζk | β = (’1)|f | |ζi |

f ζ 1 , . . . , ζk | β (8.17)

i=1

and

ζ1 , . . . , ζj , ζj+1 . . . , ζk | β

= (’1)1+|ζj ||ζj+1 | ζ1 , . . . , ζj , ζj+1 , . . . , ζk | β , (8.18)

for all ζi ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ C ∞ (U ) — Λ(n). The set of

k-forms is denoted by „¦k (U ).

Remark 8.1. Actually we have to write „¦k (U, C ∞ (U ) — Λ(n)),

but we made our choice for the abbreviated „¦k (U ).

1. SUPER (GRADED) CALCULUS 353

The set „¦k (U ) has the structure of a right C ∞ (U ) — Λ(n)-module

by

ζ1 , . . . , ζk | βf = ζ1 , . . . , ζk | β f. (8.19)

We also set „¦0 (U ) = C ∞ (U ) — Λ(n) and „¦(U ) = •∞ „¦k (U ).

k=0

Moreover „¦(U ) can be given a structure of a bigraded (Z+ , Z2 )-com-

mutative algebra, that is, if βi ∈ „¦ki (U )ji , i = 1, 2, then

β1 β2 ∈ „¦k1 +k2 (U )j1 +j2 (8.20)

and

β1 β2 = (’1)k1 k2 +j1 j2 β2 β1 . (8.21)

For the general de¬nition of β1 β2 see [50].

11. One de¬nes the exterior derivative (or de Rham di¬erential )

d : „¦0 (U ) ’ „¦1 (U ), f ’ df, (8.22)

by the condition

ζ | df = ζf (8.23)

for ζ ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ „¦0 (U ) = C ∞ (U ) — Λ(n). By

[50] and the de¬nition of β1 β2 ,

dri , i = 1, . . . , m, dsj , j = 1, . . . , n, (8.24)

de¬ned by

‚ ‚

| dri = δik , | dri = 0,

‚rk ‚sj

‚ ‚

| dsl = 0, | dsl = δjl , (8.25)

‚rk ‚sj

generate „¦(U ) and any β ∈ „¦(U ) can be uniquely written as

drµ dsν fµ,ν ,

β= (8.26)

µ,ν

where

1 ¤ µ1 ¤ . . . ¤ µk ¤ n,

µ = (µ1 , . . . , µk ), l(µ) = k,

νi ∈ N = Z+ \ {0},

ν = (ν1 , . . . , νn ),

n

fµ,ν ∈ C ∞ (U ) — Λ(n).

|ν| = νi , (8.27)

i=1

Note in particular that by (8.21),

dri drj = ’drd ri , dri dsj = ’dsj dri , dsj dsk = dsk dsj , (8.28)

and by consequence

dsj . . . dsj = 0. (8.29)

k times

354 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

By means of (8.22) and (8.23), the operator d : „¦0 (U ) ’ „¦1 (U ) has

the following explicit representation

m n

‚f ‚f

df = dri + dsj . (8.30)

‚ri ‚sj

i=1 j=1

12. Since „¦(U ) is a (Z+ , Z2 )-bigraded commutative algebra, the algebra

End(„¦(U )) is bigraded too and if u ∈ End(„¦(U )) is of bidegree (b, j) ∈

(Z+ , Z2 ), then

u(„¦a (U )i ) ∈ „¦a+b (U )i+j . (8.31)

Now, an element u ∈ End(„¦(U )) of bidegree (b, j) is a bigraded deriva-

tion of „¦(U ), if for any ± ∈ „¦a (U )i , β ∈ „¦(U ) one has the Leibniz

rule

u(±β) = u(±)β + (’1)ab+ij ±u(β). (8.32)

There exists a unique derivation, the exterior di¬erentiation,

d : „¦(U ) ’ „¦(U ) (8.33)

of bidegree (1, 0), such that d is de¬ned by (8.22), (8.30), and

„¦0 (U )

d2 = 0. (8.34)

If β ∈ „¦(U ),

drµ dsν fµ,ν ,

β= (8.35)

µ,ν

then

(’1)l(µ)+|ν| drµ dsν dfµ,ν .

dβ = (8.36)

µ,ν

Other familiar operations on ordinary manifolds have their counter-

parts in the graded case too.

13. If ζ ∈ Der(C ∞ (U ) — Λ(n)), inner di¬erentiation by ζ, or contraction

by ζ, iζ is de¬ned by

Pb

|ζ| |ζi |

ζ1 , . . . , ζb | iζ β = (’1) ζ, ζ1 , . . . , ζb | β (8.37)

i=1

for ζ, ζ1 , . . . , ζb ∈ Der(C ∞ (U ) — Λ(n)) and β ∈ „¦b+1 (U ). Moreover

iζ : „¦(U ) ’ „¦(U ), β ∈ „¦b+1 (U ), iζ β ∈ „¦b (U ), is a derivation of

bidegree (’1, |ζ|).

Bigraded derivations on „¦(U ) can be shown to constitute a bi-

graded Lie algebra Der „¦(U ) by the following Lie bracket. If u1 , u2 ∈

Der „¦(U ) of bidegree (bi , bj ), i = 1, 2, then

[u1 , u2 ] = u1 u2 ’ (’1)b1 b2 +j1 j2 u2 u1 ∈ Der „¦(U ). (8.38)

2. CLASSICAL DIFFERENTIAL GEOMETRY 355

14. From (8.38) we have that Lie derivative by the vector ¬eld ζ de¬ned

by

Lζ = diζ + iζ d (8.39)

is a derivation of „¦(U ) of bidegree (0, |ζ|).

The fact that exterior di¬erentiation d, inner di¬erentiation by ζ,

iζ , and Lie derivative by ζ, Lζ are derivations, has been used to imple-

ment them on the computer system starting from the representation

of vector ¬elds and di¬erential forms (8.15) and (8.35).

15. If one has a graded manifold (U, C ∞ (U ) — Λ(n)), the exterior deriv-

ative is easy to be represented as an odd vector ¬eld in the following

way

m n

‚ ‚

dri § dsj §

d= + , (8.40)

‚ri ‚sj

i=1 j=1

where now the initial system has been augmented by n even variables

ds1 , . . . , dsn and m odd variables dr1 , . . . , drm . The implementation

of the supercalculus package is based on the theorem proved in [50]

that locally a supermanifold, or a graded manifold, is represented as

U, C ∞ (U ) — Λ(n), U ‚ Rn , from which it is now easy to construct

the di¬erential geometric operations.

Suppose we have a supermanifold of dimension (m|n). Local

variables are given by (r, s) = (ri , sj ), i = 1, . . . , m, j = 1, . . . , n.

Associated to these coordinates, we have (dri , dsj ), i = 1, . . . , m,

j = 1, . . . , n. We have to note that dri , i = 1, . . . , m, are odd while

dsj , j = 1, . . . , n, are even.

So the exterior algebra is

C ∞ (Rm ) — R[ds] — Λ(n) — Λ(m), (8.41)

where in (8.41) a speci¬c element is given by

dsk1 . . . dskm drµ1 . . . drµr fk,µ

f= (8.42)

m

1

while in (8.42) ki ≥ 0, i = 1, . . . , m, 1 ¤ µ1 < · · · < µr ¤ n, while

fk,µ ∈ C ∞ (Rm ) — Λ(n).

2. Classical di¬erential geometry

We shall describe here how classical di¬erential geometric objects are

realised in the graded setting of the previous section. We start at a super-

algebra A on n even elements, r1 , . . . , rn , and n odd elements s1 , . . . , sn ,

i.e.,

A = C ∞ (Rn ) — Λ(n), (8.43)

356 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

where Λ(n) is the exterior algebra on n elements, s1 , . . . , sn . A particular

element f ∈ A = C ∞ (Rn ) — Λ(n) is represented as

f= fµ s µ , (8.44)

µ

where µ is a multi-index µ ∈ Mn = {µ = (µ1 , . . . , µk ) | µi ∈ N, 1 ¤ µ1 ¤

· · · ¤ µk ¤ n} and

fµ ∈ C ∞ (Rn ),

s µ = s µ1 s µ2 . . . s µk , (8.45)

where we in e¬ect formally assume:

si = dri , i = 1, . . . , n. (8.46)

1. Functions are represented as elements of the algebra A0 = C ∞ (Rn ).

2. Derivations of A0 can be identi¬ed with vector ¬elds

‚ ‚

+ · · · + Vn

V = V1 , (8.47)

‚r1 ‚rn

where Vi ∈ C ∞ (Rn ), i = 1, . . . , n.

3. Di¬erential forms are just speci¬c elements of A.

4. Exterior derivative is a derivation of A which is odd and can be

represented as the vector ¬eld

‚ ‚

+ · · · + drn

d = dr1 , dri = si . (8.48)

‚r1 ‚rn

5. Contraction by a V , where V is given by (8.47), can be represented

as an odd derivation of A by

‚ ‚

+ · · · + Vn

V ±= V1 (±). (8.49)

‚s1 ‚sn

6. The Lie derivative by V can be easily implemented by the formula

LV (±) = V d(±) + d(V ±). (8.50)

3. Overdetermined systems of PDE

In construction of classical and higher symmetries, nonlocal symmetries

and deformations or recursion operators, one is always left with an overde-

termined system of partial di¬erential equations for a number of so-called

generating functions (or sections). The ¬nal result is obtained as the general

solution to this resulting system.

In Section 3.1 we shall describe how by the procedure which is called

here solve equation, written in the symbolic language LISP, one is able

to solve the major part of the construction of the general solution of the

overdetermined system of partial di¬erential equations resulting from the

symmetry condition (2.29) on p. 72 or the deformation condition (6.42) on

p. 266.

It should be noted that each speci¬c equation or system of equations

arising from mathematical physics has its own speci¬cs, e.g., the sine-Gordon

3. OVERDETERMINED SYSTEMS OF PDE 357

equation is not polynomial but involves the sine function, similar to the

Harry Dym equations, where radicals are involved.

In Subsection 3.2 we discuss, as an application, symmmetries of the

Burgers equation, while ¬nally in Subection 3.3 we shall devote some words

to the polynomial and graded cases.

3.1. General case. Starting at the symmetry condition (2.29), one

arrives at an overdetermined system of homogeneous linear partial di¬eren-

tial equations for the generating functions Fi , i = 1, . . . , m. First of all,

one notes that in case one deals with a di¬erential equation1 E k ‚ J k (x, u),

x = (x1 , . . . , xn ), u = (u1 , . . . , um ), then the r-th prolongation E k+r is always

polynomial with respect to the higher jet variables in the ¬bre E k+r ’ E k .

The symmetry condition (2.29) is also polynomial with respect to these

variables, cf. Subsection 3.2. So the overdetermined system of partial di¬er-

ential equations can always be splitted with respect to the highest variables

leading to a new system of equations.

These equations are stored in the computer system memory as right-

hand sides of operators equ(1), . . . , equ(te), where the variable te stands

for the Total Number of Equations involved.

If at a certain stage, the computer system constructs new expressions

which have to vanish in order to generate the general solution to the system

of equations (for instance, the derivative of an equation is a consequence,

which might be easier to solve). These new equations are added to the

system as equ(te + 1), . . . and the value of te is adjusted automatically to

the new situation.

In the construction of solutions to the system of equations we distinguish

between a number of di¬erent cases:

1. CASE A: A partial di¬erential equation is of a polynomial type in

one (or more) of the variables, the functions F— appearing in this

equation are independent of this (or these) variable(s). By conse-

quence, each of the coe¬cients of the polynomial has to be zero, and

the partial di¬erential equation decomposes into some new additional

and smaller equations.

Example 8.1. The partal di¬erential equation is

equ(.) :=x2 (F1 )x2 + x1 F2 , (8.51)

1

where in (8.51) the functions F1 , F2 are independent of x1 .

By consequence, the coe¬cients of the polynomial in x1 have to

be zero, i.e., (F1 )x2 and F2 . So equation (8.51) is equivalent to the

system

equ(.):=(F1 )x2 ,

(8.52)

equ(.):=F2

1

We use the notation J k (x, u) as a synonim for J k (π), where π : (x, u) ’ (x).

358 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

2. CASE B: The partial di¬erential equations equ(.) represents a de-

rivative of a function F— . In general