(MKdV), Sine-Gordon (SG), Liouville equation, etc. Obviously in

all these equations conservation laws exist.

Let us show how conservation laws can be derived for the KdV

equation. Following the treatment of Miura, Gardner, and Kruskal

© 2001 by Chapman & Hall/CRC

[17] we express u(x, t) in terms of a function ω(x, t) de¬ned by

22

u = ω + ωx + ω (7.5)

where is a parameter. The left-hand-side of (7.2) can be factorized

in a manner

‚

+ 2 2ω

ut ’ 6uux + uxxx = 1+

‚x

22

ωt ’ 6 ω + ω ωx + ωxxx (7.6)

implying that if u is a solution of the KdV equation then the function

ω needs to satisfy

22

ωt ’ 6 ω + ω ωx + ωxxx = 0 (7.7)

(7.7) is called the Gardner equation. We may rewrite (7.7) as

ωt + ’3ω 2 ’ 2 2 ω 3 + ωxx =0 (7.8)

x

and look for an expression of ω like

ω = Σ∞ n ωn (u) (7.9)

n=0

Substitution of (7.9) in (7.5) gives the correspondence

ω0 = u

ω1 = ’ω0x = ’ux

ω2 = ’ω1x ’ ω0 = uxx ’ u2

2

(7.10)

etc.

On the other hand, combining (7.8) with (7.9) and matching for

the coe¬cients , 2 , and so on we are also led to

2

ω0t + ’3ω0 + ω0xx =0

x

ω1t + (’6ω0 ω1 + ω1xx )x = 0

2 3

ω2t + ’6ω0 ω2 ’ 3ω1 ’ 2ω0 + ω2xx =0 (7.11)

x

etc.

© 2001 by Chapman & Hall/CRC

In this way we see that the KdV admits of an in¬nity of conser-

vation laws. Exploiting (7.10), the conserved densities and ¬‚ows for

the KdV can be arranged as follows

T0 = u

χ0 = uxx ’ u2

T1 = u2

χ1 = 2uuxx ’ 4u3 ’ u3

x

1

T2 = u3 + u2

2x

9 1

χ2 = ’ u4 + 3u2 uxx ’ 6uu2 + ux uxxx ’ u2 (7.12)

x

2 xx

2

etc.

Note that the set (T0 , χ0 ) corresponds to the KdV equation itself.

However, (T1 , χi ), i = 1, 2, . . . yield higher order KdV equations.

Indeed a recursion operator can be found [18] by which the in¬nite

hierarchy of the corresponding equations can be derived.

To touch upon other evolution equations it is convenient to refor-

mulate the inverse method in terms of the following coupled system

involving the functions Ψ(x, t) ¦(x, t)

Ψx ’ »Ψ = f ¦

¦x + »¦ = gΨ (7.13)

where f and g also depend upon x and t. The question we ask is un-

der what conditions the eigenvalues » are rendered time-independent.

Suppose that the time evolutions of Ψ and ¦ are given by the

forms

Ψt = a(x, t, »)Ψ + b(x, t, »)¦

¦t = c(x, t, »)Ψ ’ a(x, t, »)¦ (7.14a, b)

where as indicated a, b, c, d are certain functions of x, t, and ». The

conditions for the time-independence of » may be worked out to be

[19]

ax = f c ’ gb

bx ’ 2»b = ft ’ 2af

cx + 2»c = gt + 2ag (7.15)

© 2001 by Chapman & Hall/CRC

By taking di¬erent choices of a, b, and c the corresponding evolution

equations can be derived. A few are listed in Table 7.1.

The structure of the conservation laws following from (7.13), and

(7.14) can be made explicit if we introduce the quantities ξ and · to

be

¦ Ψ

ξ= , ·= (7.16)

Ψ ¦

The set of equations (7.13) and (7.14) can be expressed in terms

of these variables as

ξx = g ’ 2»ξ ’ f ξ 2

ξt = c ’ 2aξ ’ bξ 2

·x = f + 2»· ’ g· 2

·t = b + 2a· ’ c· 2 (7.17a, b, c, d)

Eliminating c between (7.17b) and (7.14b) we get

1 ‚Ψ

a + bξ = (7.18)

Ψ ‚t

This implies

‚ ‚

(a + bξ) = (f ξ) (7.19a)

‚x ‚t

Similarly eliminating b between (7.17d) and (7.14a) we arrive at

‚ ‚

(’a + c·) = (g·) (7.19b)

‚x ‚t

(7.19a) and (7.19b) are the required conservation relations.

We have already provided the ¬rst few conserved densities and

the ¬‚uxes for the KdV. We now furnish the same for the MKdV and

SG equations. Note that the generalized form of the MKdV equation

is

vt + 6(k 2 ’ v 2 )vx + uxxx = 0 (7.20)

where k is a constant. On the other hand, the SG equation when

expressed in light-cone coordinates x± = 1 (x ± t) reads

2

‚2ψ

= ’ sin ψ (7.21a)

‚x+ ‚x’

© 2001 by Chapman & Hall/CRC

Table 7.1

A summary of nonlinear equations KdV, MKdV and SG for various choices of the coe¬cient

functions de¬ned in the text

'S<QQMSbySChapmanS>SHall/CRC

Equation f g a b c

uxx + 2»u ’ 2u2 2

KdV : ut = 6uux ’ uxxx ’u ’1 ’4»3 4» ’ 2u

MKdV : vt = 6v 2 vx ’ vxxx ’4»3 ’ 2»v 2 vxx + 2»vx + 4»2 v + 2v 3 ’vxx + 2»vx ’ 4»2 v ’ 2u3

’v v

™ 1

’ 1 ux 1 1 1

SG : ψ = ’ sin ψ 2 ux cos u sin u sin u

2 » 4» 4»

That is

™

ψ = ’ sin ψ (7.21b)

where the overdot (prime) denotes a partial derivative with respect

to x+ (x’ ).

The results for (Ti , χi ) corresponding to (7.20) and (7.21) are

MKdV

T1 = v 2 ,

χ1 = ’3v 4 ’ 2vvxx + vx

2

T2 = v 4 ’ 6k 2 v 2 + vx ,

2

2

vx vxxx vx

6 3

3v 2 vx

2

χ2 = v + v vxx ’ + ’ (7.22)

2 4

etc.

SG

2

T0 = ψ

χ0 = 2 cos ψ

4 2

T1 = ψ ’ 4ψ ,

2

χ1 = 4ψ cos ψ

16 3

6 2 2 2

T2 = ψ ’ 4ψ ψ + ψ ψ + 8ψ

3

14 2

χ2 = 2 ψ ’ 4ψ cos ψ (7.23)

3

etc.

Now given a conservation law in the form (7.4) we can identify

the corresponding constants of motion in the manner

∞

I[f ] = T [f (x, t)] dx (7.24)

’∞

where f (x, t) and its derivatives are assumed to decrease to zero

su¬ciently rapidly, that is as |x| ’ ∞. Here is a summary of I[f ]

for the KdV, MKdV, and SG systems

© 2001 by Chapman & Hall/CRC

KdV

I0 = dxu

dxu2

I1 =

1

dx u3 + u2

I2 = (7.25)

2x

etc.

MKdV

dxv 2

I1 =

dx v 4 + vx ’ 6k 2 v 2

2

I2 = (7.26)

etc.

SG

2

I0 = dx’ ψ

4 2

I1 = dx’ ψ ’ 4ψ

2 2 2

dx’ ψ 6 ’ 20ψ ψ

I2 = + 8ψ (7.27)

etc.

In writing I2 of SG we have integrated by parts and discarded a

total derivative.

7.3 Lax Equations

Lax™s idea of an operator formulation [13] of evolution equations gives

much insight into the rich symmetry structure of nonlinear systems.

The KdV equation admits a Lax representation which implies that

(7.2) can be represented as