™

ψ = Bul + (lu0 B + Bx + b)ul’1 + O[l ’ 2]

2

and let us compute (2.68) for these functions temporary denoting ku0 A +

™ ™ ¯ ¯

A + a and lu0 B + B + b by A and B respectively. Then we have:

1¯ 1 l’1 1¯ ¯

l

{•, ψ} = Dx (Auk + Auk’1 )B + Dx (Auk + Auk’1 )B

2 2 2

1¯ 1 k’1 1¯ ¯

k

’ Dx (Bul + Bul’1 )A ’ Dx (Buk + Bul’1 )A + O[k + l ’ 1]

2 2 2

1 ¯1 1¯

¯ ¯

= (lDx (A)uk+l’2 + Auk+l’1 )B + (Auk+l’1 + Auk+l’2 )B

2 2 2

1 ¯1 1¯

¯ ¯

’ (kDx (B)uk+l’2 + Buk+l’1 )A ’ (Buk+l’1 + Buk+l’2 )A +

2 2 2

O[k + l ’ 3],

or in short,

1™ ™

{•, ψ} = (lAB ’ K BA)uk+l’2 + O[k + l ’ 3]. (2.69)

2

3. THE BURGERS EQUATION 83

3.4. Low order symmetries. These computations were done already

in Section 3 of Chapter 1 (see equation (1.61)). They can also be done

independently taking k = 2 and solving equation (2.64) directly. Then one

obtains ¬ve independent solutions which are

•0 = u 1 ,

1

•1 = tu1 + 1,

1

•0 = u 2 + u 0 u 1 ,

2

1 1

•1 = tu2 + (tu0 + x)u1 + u0 ,

2

2 2

2 2 2

•2 = t u2 + (t u0 + tx)u1 + tu0 + x. (2.70)

3.5. Action of low order symmetries. Let us compute the action

def

Tij = {•j , •} = ’

•j •j

i i i

of symmetries •j on other symmetries of the equation E.

i

0 one has

For •1

‚ ‚

0

’ ’ Dx = ’ .

T1 = = ui+1

u1 u1

‚ui ‚x

i≥0

Hence, if • = Auk + O[k ’ 1] is a function of the form (2.67), then we obtain

1™

0

T1 • = ’ Auk’1 + O[k ’ 2].

2

Consequently, if • is a symmetry, then, since sym(E) is closed under the

Jacobi bracket,

k’1

dk’1 A

1

(T1 )k’1 •

0

’

= u1 + O[0]

dtk’1

2

is a symmetry as well. But from (2.70) one sees that ¬rst-order symmetries

are linear in t. Thus, we have the following result:

Proposition 2.25. If • = Auk + O[k ’ 1] is a symmetry of the Burgers

equation, then A is a k-th degree polynomial in t.

3.6. Final description. Note that direct computations show that the

equation E possesses a third-order symmetry of the form

3 3 3

•0 = u 3 + u 0 u 2 + u 2 + u 2 u 1 .

3

20 40

2

2 0

Using the actions T2 and T3 , one can see that

k’1

k!(k ’ 1)!

3

((T2 )i

2 0

T2 )k’1 )u1

2

—¦ —¦ ’ uk + O[k ’ 1]

(T3 = (2.71)

(k ’ i)!

2

is a symmetry, since u1 is the one.

84 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Theorem 2.26. The symmetry algebra sym(E) for the Burgers equation

E = {ut = uux + uxx }, as a vector space, is generated by elements of the

form

•i = ti uk + O[k ’ 1], k ≥ 1, i = 0, . . . , k,

k

which are polynomial in all variables. For the Jacobi bracket one has

1

{•i , •j } = (li ’ kj)•i+j’1 + O[k + l ’ 3]. (2.72)

k l k+l’2

2

The Lie algebra sym(E) is simple and has •0 , •2 , and •0 as its generators.

1 2 3

Proof. It only remains to prove that all •i are polynomials and that

k

sym(E) is a simple Lie algebra. The ¬rst fact follows from (2.71) and from

2 0

the obvious observation that coe¬cients of both T2 and T3 are polynomials.

Let us prove that sym(E) is a simple Lie algebra. To do this, let us

introduce an order in the set {•i } de¬ning

k

def

¦ k(k+1) +i = •i .

k

2

Then any symmetry may be represented as s »± ¦± , » ∈ R.

±=1

Let I ‚ sym(E) be an ideal and ¦ = ¦s + s’1 »± ¦± be its element.

±=1

i for some k ≥ 1 and i ¤ k.

Assume that ¦s = •k

Note now that

‚ ‚ ‚

1 ±

’ tDx = ’t

T1 = Dx (tu1 + 1)

‚u± ‚u0 ‚x

±≥0

and

‚ ‚

0 ± 2

’ D x ’ u 0 Dx ’ u 1 = ’ .

T2 = Dx (u2 + u0 u1 )

‚u± ‚t

±≥0

Therefore,

((T1 )k’1 —¦ (T2 )i )¦ = c•0 ,

1 0

1

where the coe¬cient c does not vanish. Hence, I contains the function •0 . 1

But due to (2.71) the latter, together with the functions •2 and •0 , generates

2 3

the whole algebra.

Further details on the structure of sym(E) one can ¬nd in [60].

4. The Hilbert“Cartan equation

We compute here classical and higher symmetries of the Hilbert“Cartan

equation [2]. Since higher symmetries happen to depend on arbitrary func-

tions, we consider some special choices of these functions [38].

4. THE HILBERT“CARTAN EQUATION 85

4.1. Classical symmetries. The Hilbert“Cartan equation is in e¬ect

an underdetermined system of ordinary di¬erential equations in the sense of

De¬nition 1.10 of Subsection 2.1 in Chapter 1. The number of independent

variables, n, is one while the number of dependent variables, m, is two. Local

coordinates are given by x, u, v in J 0 (π), while the order of the equations is

two, i.e.,

2

ux = vxx (2.73)

The representative morphism (see De¬nition 1.6 on p. 6) ¦ is given by

2

¦∆ (x, u, v, ux , vx , uxx , vxx ) = ux ’ vxx . (2.74)

The total derivative operator Dx is given by the formula

‚ ‚ ‚ ‚ ‚

+ ···

D = Dx = + ux + vx + uxx + vxx (2.75)

‚x ‚u ‚v ‚ux ‚vx

To construct classical symmetries for (2.73), we start from the vector ¬eld

X, given by

‚ ‚ ‚

X = X(x, u, v) + U0 (x, u, v) + V0 (x, u, v)

‚x ‚u ‚v

‚ ‚

+ U1 (x, u, x, ux , vx ) + V1 (x, u, x, ux , vx )

‚ux ‚vx

‚ ‚

+ U2 (x, u, x, ux , vx , uxx , vxx ) + V2 (x, u, x, ux , vx , uxx , vxx ) .

‚uxx ‚vxx

The de¬ning relations (1.34) (see p. 26) for U1 , V1 , U2 , V2 are

U1 = D(U0 ) ’ ux D(X) = D(U0 ’ ux X) + uxx X,

V1 = D(V0 ) ’ vx D(X) = D(V0 ’ vx X) + vxx X,

U2 = D(U1 ) ’ uxx D(X) = D 2 (U0 ’ ux X) + uxxx X,

V2 = D(V1 ) ’ vxx D(X) = D 2 (V0 ’ vx X) + vxxx X. (2.76)

From (2.76) we derive the following explicit expressions for U1 , V1 , U2 , V2 :

U1 = U0,x + U0,u ux + U0,v vx ’ ux (X0,x + X0,u ux + X0,v vx ),

V1 = V0,x + V0,u ux + V0,v vx ’ ux (X0,x + X0,u ux + X0,v vx ),

U2 = U0,xx + 2U0,xu ux + 2U0,xv vx + U0,uu u2 + 2U0,uv ux vx + U0,u uxx

x

2

+ U0,vv vx + U0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ ux (X0,xx + 2X0,xu ux + 2X0,xv vx

+ X0,uu u2 + 2X0,uv ux vx + X0,u uxx + X0,vv vx + X0,v vxx ),

2

x

V2 = V0,xx + 2V0,xu ux + 2V0,xv vx + V0,uu u2 + 2V0,uv ux vx + V0,u uxx

x

2

+ V0,vv vx + V0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ vx (X0,xx + 2X0,xu ux + 2X0,xv vx + X0,uu u2

x

2

+ 2X0,uv ux vx + X0,u uxx + X0,vv vx + X0,v vxx ). (2.77)

86 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Now the symmetry-condition X(¦∆ ) |E = 0 results in

2

U1 ’ 2vxx V2 = »(ux ’ vxx ) (2.78)

which is equivalent to

1

U1 ’ 2(ux ) 2 V2 = 0 mod ¦∆ = 0, (2.79)

which results in

U0,x + U0,u ux + U0,v vx ’ ux (X0,x + X0,u ux + X0,v vx )

’ V0,xx + 2V0,xu ux + 2V0,xv vx + V0,uu u2 + 2V0,uv ux vx + V0,u uxx

x

2

+ V0,vv vx + V0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ vx (X0,xx + 2X0,xu ux + 2X0,xv vx + X0,uu u2 + 2X0,uv ux vx

x

+ X0,u uxx + X0,vv vx + X0,v vxx ) · 2(ux )1/2 = 0.

2

(2.80)

Equation (2.80) is a polynomial in the “variables” (ux )1/2 , vx , uxx , the

coe¬cients of which should vanish. From this observation we obtain the

following system of equations:

1: U0,x = 0,

u1/2 : ’2V0,xx = 0,

x

u1/2 vx : ’4V0,xv + 2X0,xx = 0,

x

u1/2 uxx : ’2V0,u = 0,

x

u1/2 uxx vx : 2X0,u = 0,

x

u1/2 vx :

2

’2V0,vv + 4X0,xv = 0,

x

u1/2 vx :

3

2X0,vv = 0,

x

U0,u ’ X0,x ’ 2V0,v + 4X0,x = 0,

ux :

’X0,v + 4X0,v + 2X0,v = 0,

u x vx :

u2 : ’X0,u + 4X0,u = 0,

x

u3/2 : ’4v0,xu = 0,

x

u3/2 vx : ’4V0,uv + 4X0,xu = 0,

x

u3/2 vx :

2

4X0,uv = 0,

x

u5/2 : ’2V0,uu = 0,

x

u5/2 vx : 2X0,uu = 0,

x

vx : U0,v = 0. (2.81)

From system (2.81) we ¬rst derive:

X0,u = X0,v = 0, V0,uu = V0,uv = V0,vv = 0 = V0,u = V0,xx ,

4. THE HILBERT“CARTAN EQUATION 87

[Ai , Aj ] A1 A2 A3 A4 A5 A6

A1 0 0 0 0 A1 A3

0 2A2 ’3A2

A2 0 0

A3 0 A3 0 0

’A6

A4 0 0

A5 0 A6

A6 0

Figure 2.1. Commutator table for classical symmetries of

the Hilbert“Cartan equation

which results in the equality X(x, u, v) = H(x) and in the fact that V0 is

independent of u, being of degree 1 in v and of degree 1 in x, i.e.,

X(x, u, v) = H(x), V0 = a0 + a1 x + a2 v + a3 xv.

1/2

Now from the equation labeled by ux vx in (2.81) we derive

H(x) = a3 x2 + a4 x + a5 . (2.82)

From the equations U0,v = 0 and U0,u + 3X0,x ’ 2V0,v = 0 we get

U0 = ’(4a3 x + 2a2 ’ 3a5 )u + G(x). (2.83)

Finally from U0,x = 0 we arrive at a3 = 0, G(x) = a6 , from which the general

solution is obtained as

U0 = (2a2 ’ 3a4 )u + a6 ,

X = a4 x + a5 , V0 = a0 + a1 x + a2 v.

This results in a 6-dimensional Lie algebra, the generators of which are given

by

‚

A1 = ,

‚x

‚

A2 = ,

‚u

‚

A3 = ,

‚v

‚ ‚

A4 = 2u +v ,

‚u ‚v

‚ ‚

’ 3u ,

A5 = x

‚x ‚u

‚

A6 = x ,

‚v

while the commutator table is given on Fig. 2.1.

4.2. Higher symmetries. As a very interesting and completely com-

putable application of the theory of higher symmetries developed in Subsec-

tion 2.1, we construct in this section the algebra of higher symmetries for

88 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

the Hilbert“Cartan equation E

2

ux ’ vxx = 0. (2.84)

First of all, note that E ∞ is given by the system of equations:

Di (ux ’ vxx ) = 0,

2

i = 0, 1, . . . (2.85)

where D is de¬ned by

∞ ∞

‚ ‚ ‚

D= + uk+1 + vk+1 , (2.86)

‚x ‚uk ‚vk

k=0 k=0

and uk = ux . . . x . So from (2.84) we have

k times

D1 F = u2 ’ 2v2 v3 = 0,

D2 F = u3 ’ 2v3 ’ 2v2 v4 = 0,

2

i

i

i

D F = u1+i ’ v2+l v2+i’l = 0,

l

l=0

2

i = 3, . . . , with F (x, u, v, u1 , v1 , u2 , v2 ) = u1 ’ v2 = 0.

In order to construct higher symmetries of (2.84), we introduce internal

coordinates on E ∞ which are

x, u, v, v1 , v2 , v3 , · · · (2.87)

The total derivative operator restricted to E ∞ , again denoted by D, is given

by the following expression

‚ 2‚ ‚ ‚

D= + v2 + v1 + vi+1 ,

‚x ‚u ‚v ‚vi

i>0

n

‚ 2‚ ‚ ‚

D(n) = + v2 + v1 + vi+1 . (2.88)

‚x ‚u ‚v ‚vi

i>0

Suppose that a vertical vector ¬eld V = with the generating function ¦,

¦

¦ = f u (x, u, v, v1 , . . . , vn ), f (x, u, v, v1 , . . . , vn ) ,

v

(2.89)

is a higher symmetry of E. We introduce the notation

f [vk ] = f (x, u, v, v1 , . . . , vk ). (2.90)

Since the vertical vector ¬eld V is formally given by

‚ ‚ ‚ ‚