±

±=1

where »j are smooth functions, or

±

X(F j ) |E = 0, j = 1, . . . , r. (1.37)

These conditions can be rewritten in terms of generating sections and we

shall do it in Chapter 2 in a more general situation.

Let E ‚ J k (π) be a di¬erential equation and X be its symmetry. Then

for any solution • of this equation, the one-parameter group {At } corre-

sponding to X transforms • to some new solution •t almost everywhere. In

special local coordinates, evolution of • is governed by the following evolu-

tionary equation:

‚• ‚• ‚•

= f (x1 , . . . , xn , •, ,..., ), (1.38)

‚t ‚x1 ‚xn

if π is one-dimensional and f is the generating function of X, or by a system

of evolutionary equations of the form

‚•j 1 ‚•m

m ‚•

j 1

= f (x1 , . . . , xn , • , . . . , • , ,..., ), (1.39)

‚t ‚x1 ‚xn

where j = 1, . . . , m = dim π and f j are the components of the generating

section.

In particular, we say that a solution is invariant with respect to X, if

it is transformed by {At } to itself, which means that it has to satisfy the

28 1. CLASSICAL SYMMETRIES

equation

‚• ‚•

f (x1 , . . . , xn , •, ,..., )=0 (1.40)

‚x1 ‚xn

or a similar system of equations when dim π > 1. If g is a subalgebra in the

symmetry algebra of E, we can also de¬ne g-invariant solutions as solutions

invariant with respect to all elements of g.

2.4. Prolongations. The idea of prolongation originates from a simple

observation that, a di¬erential equation given, not all relations between

dependent variables are explicitly encoded in this equation. To reconstruct

these relations, it needs to analyze “di¬erential consequences” of the initial

equations.

Example 1.16. Consider the system

2

uxxy = vy , uxyy = vx + uy .

Then, di¬erentiating the ¬rst equation with respect to y and the second one

with respect to x, we obtain

uxxyy = 2vy vyy , uxxyy = vxx + uxy

and consequently

2vy vyy = vxx + uxy .

Example 1.17. Let

1

v t = u2 + u x .

vx = u,

2

Then

ut = uux + uxx

by a similar procedure.

Example 1.18. Consider equations (1.19) from Example 1.8 on p. 15.

Then as consequences of these equations we obtain equations (1.20) which

may be viewed at as compatibility conditions for equations (1.19). One can

see that if the functions j satisfy (1.20), i.e., if the connection is ¬‚at,

i

then these conditions are void; otherwise we obtain functional relations on

the variables uj .

i

Geometrically, the process of computation of di¬erential consequences is

expressed by the following de¬nition:

Definition 1.19. Let E ‚ J k (π) be a di¬erential equation of order k.

De¬ne the set

E 1 = {θk+1 ∈ J k+1 (π) | πk+1,k (θk+1 ) ∈ E, Lθk+1 ‚ Tπk+1,k (θk+1 ) E}

and call it the ¬rst prolongation of the equation E.

2. NONLINEAR PDE 29

If the ¬rst prolongation E 1 is a submanifold in J k+1 (π), we de¬ne the

second prolongation of E as (E 1 )1 ‚ J k+2 (π), etc. Thus the l-th prolongation

is a subset E l ‚ J k+l (π).

Let us rede¬ne the notion of l-th prolongation directly. Namely, take a

point θk ∈ E and consider a section • ∈ “loc (π) such that the graph of jk (•)

is tangent to E with order l. Let πk (θk ) = x ∈ M . Then [•]k+l is a point of

x

k+l (π) and the set of all points obtained in such a way obviously coincides

J

with E l , provided all intermediate prolongations E 1 , . . . , E l’1 be well de¬ned

in the sense of De¬nition 1.19.

Assume now that locally E is given by the equations

F 1 = 0, . . . , F r = 0, F j ∈ Fk (π)

and θk ∈ E is the origin of the chosen special coordinate system. Let u1 =

•1 (x1 , . . . , xn ), . . . , um = •m (x1 , . . . , xn ) be a local section of the bundle π.

Then

‚ |σ| •±

— j j

jk (•) F = F (x1 , . . . , xn , . . . , ,...)

‚xσ

n

‚F j ‚ |σ|+1 •±

‚F j

= + xi + o(x),

‚u± ‚xσ+1i

‚xi σ

±,σ

i=1 θk

where the sums are taken over all admissible indices. From here it follows,

that the graph of jk (•) is tangent to E at the point under consideration if

and only if

n

‚F j ‚ |σ|+1 •±

‚F j

+ = 0.

‚u± ‚xσ+1i

‚xi σ

±,σ

i=1 θk

Hence, the equations of the ¬rst prolongation are

n

‚F j ‚F j ±

+ u = 0, i = 1, . . . , n.

‚u± σ+1i

‚xi σ

±,σ

i=1

From here and by comparison with the coordinate representation of prolon-

gations for nonlinear di¬erential operators (see Subsection 1.2), we obtain

the following result:

Proposition 1.14. Let E ‚ J k (π) be a di¬erential equation. Then

(i) If the equation E is determined by a di¬erential operator ∆ : “(π) ’

“(π ), then its l-th prolongation is given by the l-th prolongation

∆(l) : “(π) ’ “(πl ) of the operator ∆.

(ii) If E is locally described by the system of equations

F 1 = 0, . . . , F r = 0, F j ∈ Fk (π),

then the system

Dσ F j = 0, |σ| ¤ l, j = 1, . . . , r, (1.41)

30 1. CLASSICAL SYMMETRIES

def σ

where Dσ = D1 1 —¦ · · · —¦ Dnn , corresponds to E l . Here Di stands for

σ

the i-th total derivative (see (1.35)).

From the de¬nition it follows that for any l ≥ l ≥ 0 one has the

embeddings πk+l,k+l (E l ) ‚ E l and consequently one has the mappings

πk+l,k+l : E l ’ E l .

Definition 1.20. An equation E ‚ J k (π) is called formally integrable,

if

(i) all prolongations E l are smooth manifolds

and

(ii) all the mappings πk+l+1,k+l : E l+1 ’ E l are smooth ¬ber bundles.

In the sequel, we shall mostly deal with formally integrable equations.

The rest of this chapter is devoted to classical symmetries of some par-

ticular equations of mathematical physics.

3. Symmetries of the Burgers equation

As a ¬rst example, we shall discuss the computation of classical symme-

tries for the Burgers equation, which is described by

ut = uux + uxx . (1.42)

The equation holds on J 2 (x, t; u) = J 2 (π) for the trivial bundle π : R —

R2 ’ R2 with x, t being coordinates in R2 (independent variables) and u a

coordinate in the ¬ber (dependent variable). The total derivative operators

are given by

‚ ‚ ‚

Dx = + ux + uxx

‚x ‚u ‚ux

‚ ‚ ‚ ‚

+ ··· ,

+ uxt + uxxx + uxxt + uxtt

‚ut ‚uxx ‚uxt ‚utt

‚ ‚ ‚

Dt = + ut + uxt

‚t ‚u ‚ux

‚ ‚ ‚ ‚

+ ···

+ utt + uxxt + uxtt + uttt (1.43)

‚ut ‚uxx ‚uxt ‚utt

We now introduce the vector ¬eld V of the form

‚ ‚ ‚ ‚

V =Vx +Vt +Vu + · · · + V utt , (1.44)

‚x ‚t ‚u ‚utt

where in (1.44) V x , V t , V u are functions depending on x, t, u, while the

components with respect to ‚/‚ux , ‚/‚ut , ‚/‚uxx , ‚/‚uxt , ‚/‚utt , which

are denoted by V ux , V ut , V uxx , V uxt , V utt , are given by formula (1.34) and

are of the form

V ux = Dx (V u ’ ux V x ’ ut V t ) + uxx V x + uxt V t ,

V ut = Dt (V u ’ ux V x ’ ut V t ) + uxt V x + utt V t ,

3. SYMMETRIES OF THE BURGERS EQUATION 31

V uxx = Dx (V u ’ ux V x ’ ut V t ) + uxxx V x + uxxt V t ,

2

V uxt = Dx Dt (V u ’ ux V x ’ ut V t ) + uxxt V x + uxtt V t ,

V utt = Dt (V u ’ ux V x ’ ut V t ) + uxtt V x + uttt V t .

2

(1.45)

The symmetry condition (1.37) on V , which is just the invariance condition

of the hypersurface E ‚ J 2 (x, t; u) given by (1.42) under the vector ¬eld V ,

results in the equation

V ut ’ ux V u ’ uV ux ’ V uxx = 0. (1.46)

Calculation of the quantities V ut , V ux , V uxx required in (1.46) yields

‚V u ‚V u ‚V x ‚V x ‚V t ‚V t

ux

’ ux ’ ut

V = + ux + ux + ux ,

‚x ‚u ‚x ‚u ‚x ‚u

‚V u ‚V u ‚V x ‚V x ‚V t ‚V t

V ut ’ ux ’ ut

= + ut + ut + ut ,

‚t ‚u ‚t ‚u ‚t ‚u

‚2V u ‚2V u 2u ‚V u

2‚ V

V uxx = + 2ux + ux + uxx

‚x2 ‚u2

‚x‚u ‚u

x x t ‚V t

‚V ‚V ‚V

’ 2uxx ’ 2uxt

+ ux + ux

‚x ‚u ‚x ‚u

‚2V x ‚2V x 2x ‚V x

2‚ V

’ ux + 2ux + ux + uxx

‚x2 ‚u2

‚x‚u ‚u

‚2V t ‚2V t 2t ‚V t

2‚ V

’ ut + 2ux + ux + uxx . (1.47)

‚x2 ‚u2

‚x‚u ‚u

Substitution of these expressions (1.47) together with

ut = uux + uxx ,

uxt = u2 + uuxx + uxxx , (1.48)

x

into (1.46) leads to a polynomial expression with respect to the variables

uxxx , uxx , ux , the coe¬cients of which should vanish.

The coe¬cient at uxxx , which arises solely from the term uxt in V uxx ,

leads to the ¬rst condition

‚V t ‚V t

+ ux = 0, (1.49)

‚x ‚u

from which we immediately obtain that ‚V t /‚x = 0, ‚V t /‚u = 0, or

V t (x, t, u) = F0 (t), (1.50)

i.e., the function V t is dependent just on the variable t.

Remark 1.11. Although V t is a function dependent just on one variable

t, we prefer to write in the sequel partial derivatives instead of ordinary

derivatives.

32 1. CLASSICAL SYMMETRIES

Now, using the obtained result for the function V t (x, t, u) we obtain

from (1.46), (1.47), (1.48), (1.49) that the coe¬cients at the corresponding

terms vanish:

‚V x

uxx ux : 2 = 0,

‚u

‚V t ‚V x

uxx : ’ +2 = 0,

‚t ‚x

‚2V x

3

ux : = 0,

‚u2

‚2V u ‚2V x

2

ux : ’ +2 = 0,

‚u2 ‚x‚u

‚V x ‚V t ‚V x ‚2V u ‚2V x

u

ux : ’ ’u ’V +u ’2 + = 0,

‚x2

‚t ‚t ‚x ‚x‚u

‚V u ‚V u ‚ 2 V u

’u ’

1: = 0. (1.51)

‚x2

‚t ‚x

From the ¬rst and the fourth equation in (1.51) we have

V x = F1 (x, t), V u = F2 (x, t) + F3 (x, t)u. (1.52)

Substitution of this result into the second, ¬fth and sixth equation of (1.51)

leads to

‚F0 (t) ‚F1 (x, t)

’2 = 0,

‚t ‚x

‚F1 (x, t) ‚F0 (t)

+u + F2 (x, t) + uF3 (x, t)

‚t ‚t

‚F3 (x, t) ‚ 2 F1 (x, t)

‚F1 (x, t)

’u ’

+2 = 0,

‚x2

‚x ‚x

‚F2 (x, t) ‚ 2 F2 (x, t)

‚F2 (x, t)

’u ’

‚x2

‚t ‚x

‚F3 (x, t) ‚ 2 F3 (x, t)

‚F3 (x, t)

’u ’

+u = 0. (1.53)

‚x2

‚t ‚x

We now ¬rst solve the ¬rst equation in (1.53):

x ‚F0 (t)

F1 (x, t) = + F4 (t), (1.54)

2 ‚t

The second equation in (1.53) is an equation polynomial with respect to u,

so we obtain from this the following relations:

‚F0 (t)

+ 2F3 (x, t) = 0,

‚t

‚ 2 F0 (t)

‚F3 (x, t) ‚F4 (t)

4 +2 +x + 2F2 (x, t) = 0, (1.55)

‚t2

‚x ‚t

while from the third equation in (1.53) we obtain

‚F3 (x, t)

= 0,

‚x

3. SYMMETRIES OF THE BURGERS EQUATION 33

‚F3 (x, t) ‚ 2 F3 (x, t) ‚F2 (x, t)

’ ’ = 0,

‚x2

‚t ‚x

‚F2 (x, t) ‚ 2 F2 (x, t)

’ = 0. (1.56)

‚t ‚x

From (1.55) we can obtain the form of F3 (x, t) and F2 (x, t), i.e.,

1 ‚F0 (t)

F3 (x, t) = ’ ,

2 ‚t

‚F4 (t) x ‚ 2 F0 (t)

F2 (x, t) = ’ ’ . (1.57)

2 ‚t2

‚t

The ¬rst and second equation in (1.56) now ful¬ll automatically, while the

third equation is a polynomial with respect to x; hence we have

‚ 2 F4 (t) x ‚ 3 F0 (t)

+ = 0, (1.58)

‚t2 2 ‚t3

from which we ¬nally arrive at

F0 (t) = c1 + c2 t + c3 t2 , F4 (t) = c4 + c5 t. (1.59)

Combining the obtained results we ¬nally have:

1

V x (x, t, u) = c4 + c2 x + c5 t + c3 xt,

2

V (x, t, u) = c1 + c2 t + c3 t2 ,

t

1

V u (x, t, u) = ’c5 ’ c3 x ’ c2 u ’ c3 tu,

2

which are the components of the vector ¬eld V , whereas c1 , . . . , c5 are arbi-

trary constants.

From (1.59) we have that the Lie algebra of classical symmetries of the

Burgers equation is generated by ¬ve vector ¬elds

‚

V1 = ,

‚t

1‚ ‚ 1‚

+t ’ u ,

V2 = x

2 ‚x ‚t 2 ‚u

‚ ‚ ‚

+ t2 ’ (x + tu) ,

V3 = xt

‚x ‚t ‚u

‚

V4 = ,

‚x

‚ ‚

’

V5 = t . (1.60)

‚x ‚u

The commutator table for the generators (1.60) is presented on Fig. 1.1.

Note that the generating functions •i = Vi (du ’ ux dx ’ ut dt) corre-

sponding to symmetries (1.60) are

•1 = ’ut ,

1

•2 = ’ (u + xux + 2tut ),

2

34 1. CLASSICAL SYMMETRIES

[Vi , Vj ] V1 V2 V3 V4 V5

V1 0 V1 2V2 0 V4

1 1

0 V 3 ’ 2 V4

V2 2 V5

’V5

V3 0 0

V4 0 0

V5 0

Figure 1.1. Commutator table for classical symmetries of

the Burgers equation

•3 = ’(x + tu + xtux + t2 ut ),

•4 = ’ux ,

•5 = ’(tux + 1). (1.61)

The computations carried through in this application indicate the way

one has to take to solve overdetermined systems of partial di¬erential equa-

tions for the components of a vector ¬eld arising from the symmetry con-

dition (1.37). We also refer to Chapter 8 for description of computer-based

computations of symmetries.