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shares with several other evolution equations like the modi¬ed KdV
(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

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