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shares with several other evolution equations like the modiп¬Ѓed 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

В© 2001 by Chapman & Hall/CRC
 we express u(x, t) in terms of a function П‰(x, t) deп¬Ѓned by
22
u = П‰ + П‰x + П‰ (7.5)

where is a parameter. The left-hand-side of (7.2) can be factorized
in a manner
в€‚
+ 2 2П‰
ut в€’ 6uux + uxxx = 1+
в€‚x
22
П‰t в€’ 6 П‰ + П‰ П‰x + П‰xxx (7.6)

implying that if u is a solution of the KdV equation then the function
П‰ needs to satisfy
22
П‰t в€’ 6 П‰ + П‰ П‰x + П‰xxx = 0 (7.7)

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

П‰t + в€’3П‰ 2 в€’ 2 2 П‰ 3 + П‰xx =0 (7.8)
x

and look for an expression of П‰ like

П‰ = ОЈв€ћ n П‰n (u) (7.9)
n=0

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

П‰0 = u
П‰1 = в€’П‰0x = в€’ux
П‰2 = в€’П‰1x в€’ П‰0 = uxx в€’ u2
2
(7.10)

etc.
On the other hand, combining (7.8) with (7.9) and matching for
the coeп¬ѓcients , 2 , and so on we are also led to
2
П‰0t + в€’3П‰0 + П‰0xx =0
x
П‰1t + (в€’6П‰0 П‰1 + П‰1xx )x = 0
2 3
П‰2t + в€’6П‰0 П‰2 в€’ 3П‰1 в€’ 2П‰0 + П‰2xx =0 (7.11)
x

etc.

В© 2001 by Chapman & Hall/CRC
In this way we see that the KdV admits of an inп¬Ѓnity of conser-
vation laws. Exploiting (7.10), the conserved densities and п¬‚ows for
the KdV can be arranged as follows
T0 = u
П‡0 = uxx в€’ u2

T1 = u2
П‡1 = 2uuxx в€’ 4u3 в€’ u3
x

1
T2 = u3 + u2
2x
9 1
П‡2 = в€’ u4 + 3u2 uxx в€’ 6uu2 + ux uxxx в€’ u2 (7.12)
x
2 xx
2
etc.
Note that the set (T0 , П‡0 ) corresponds to the KdV equation itself.
However, (T1 , П‡i ), i = 1, 2, . . . yield higher order KdV equations.
Indeed a recursion operator can be found  by which the inп¬Ѓnite
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 ОЁ(x, t) О¦(x, t)
ОЁx в€’ О»ОЁ = f О¦
О¦x + О»О¦ = gОЁ (7.13)
where f and g also depend upon x and t. The question we ask is un-
der what conditions the eigenvalues О» are rendered time-independent.
Suppose that the time evolutions of ОЁ and О¦ are given by the
forms
ОЁt = a(x, t, О»)ОЁ + b(x, t, О»)О¦
О¦t = c(x, t, О»)ОЁ в€’ a(x, t, О»)О¦ (7.14a, b)
where as indicated a, b, c, d are certain functions of x, t, and О». The
conditions for the time-independence of О» may be worked out to be

ax = f c в€’ gb
bx в€’ 2О»b = ft в€’ 2af
cx + 2О»c = gt + 2ag (7.15)

В© 2001 by Chapman & Hall/CRC
By taking diп¬Ђerent 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 Оѕ and О· to
be
О¦ ОЁ
Оѕ= , О·= (7.16)
ОЁ О¦
The set of equations (7.13) and (7.14) can be expressed in terms
of these variables as

Оѕx = g в€’ 2О»Оѕ в€’ f Оѕ 2
Оѕt = c в€’ 2aОѕ в€’ bОѕ 2
О·x = f + 2О»О· в€’ gО· 2

О·t = b + 2aО· в€’ cО· 2 (7.17a, b, c, d)
Eliminating c between (7.17b) and (7.14b) we get

1 в€‚ОЁ
a + bОѕ = (7.18)
ОЁ в€‚t
This implies
в€‚ в€‚
(a + bОѕ) = (f Оѕ) (7.19a)
в€‚x в€‚t
Similarly eliminating b between (7.17d) and (7.14a) we arrive at

в€‚ в€‚
(в€’a + cО·) = (gО·) (7.19b)
в€‚x в€‚t
(7.19a) and (7.19b) are the required conservation relations.
We have already provided the п¬Ѓrst few conserved densities and
the п¬‚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 в€’ 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 xВ± = 1 (x В± t) reads
2

в€‚2П€
= в€’ sin П€ (7.21a)
в€‚x+ в€‚xв€’

В© 2001 by Chapman & Hall/CRC
Table 7.1
A summary of nonlinear equations KdV, MKdV and SG for various choices of the coeп¬ѓcient
functions deп¬Ѓned in the text
'S<QQMSbySChapmanS>SHall/CRC

Equation f g a b c
uxx + 2О»u в€’ 2u2 2
KdV : ut = 6uux в€’ uxxx в€’u в€’1 в€’4О»3 4О» в€’ 2u

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

Л™ 1
в€’ 1 ux 1 1 1
SG : П€ = в€’ sin П€ 2 ux cos u sin u sin u
2 О» 4О» 4О»
That is
Л™
П€ = в€’ sin П€ (7.21b)
where the overdot (prime) denotes a partial derivative with respect
to x+ (xв€’ ).
The results for (Ti , П‡i ) corresponding to (7.20) and (7.21) are

MKdV

T1 = v 2 ,
П‡1 = в€’3v 4 в€’ 2vvxx + vx
2

T2 = v 4 в€’ 6k 2 v 2 + vx ,
2

2
vx vxxx vx
6 3
3v 2 vx
2
П‡2 = v + v vxx в€’ + в€’ (7.22)
2 4
etc.

SG
2
T0 = П€
П‡0 = 2 cos П€

4 2
T1 = П€ в€’ 4П€ ,
2
П‡1 = 4П€ cos П€

16 3
6 2 2 2
T2 = П€ в€’ 4П€ П€ + П€ П€ + 8П€
3
14 2
П‡2 = 2 П€ в€’ 4П€ cos П€ (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
в€ћ
I[f ] = T [f (x, t)] dx (7.24)
в€’в€ћ

where f (x, t) and its derivatives are assumed to decrease to zero
suп¬ѓciently rapidly, that is as |x| в†’ в€ћ. Here is a summary of I[f ]
for the KdV, MKdV, and SG systems

В© 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 в€’ 6k 2 v 2
2
I2 = (7.26)

etc.

SG
2
I0 = dxв€’ П€
4 2
I1 = dxв€’ П€ в€’ 4П€
2 2 2
dxв€’ П€ 6 в€’ 20П€ П€
I2 = + 8П€ (7.27)

etc.
In writing I2 of SG we have integrated by parts and discarded a
total derivative.

7.3 Lax Equations
LaxвЂ™s idea of an operator formulation  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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