© 2001 by Chapman & Hall/CRC

where the operators L and B are given by

‚2

L = ’ 2 + u(x, t)

‚x

‚3 ‚ ‚

B=4 ’3 u + u (7.29a, b)

‚x3 ‚x ‚x

Equation (7.28) can be solved to obtain

L(t) = S(t)L(0)S ’1 (t)

™

S = ’BS (7.30a, b)

Corresonding to (7.29a) the associated eigenvalue problem is

L¦ = »¦ which means that if ¦ is an initial eigenfunction of L

with an eigenvalue » then it remains so for all times bearing the

same eigenvalue ». The essence of (7.30a) and L¦ = »¦ is that the

spectrum of L is conserved and that it yields an in¬nite sequence of

conservation laws. Note that the conserved quantities may also be

obtained from the de¬nitions

‚2

L(x, y) = ’ 2 + u(x) δ(x ’ y) (7.31a)

‚x

along with

T rL = dxdyδ(x ’ y)L(x, y) (7.31b)

Now since

L2 (x, y) = dzL(x, z)L(x, y)

(7.32a)

‚4 ‚2

’ {u(x) + u(y)} 2 + u2 δ(x ’ y)

= 4

‚x ‚x

we obtain

T rL = ’vδ (0) + δ(0) dxu(x) (7.32b)

T rL2 = ’V δ (0) ’ 2δ (0) dxu2

dxu + δ(0) (7.32c)

∞

where V ≡ ’∞ dx. If we agree to ignore the additive and multi-

plicative in¬nities, T rL and T rL2 lead to the conserved quantities

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(7.25a, b). The extraction of the conserved quantities when the con-

served functional contains more than one term, as with I2 and I3 etc.,

is a little tricky. One invokes the rule of counting each power of u to

be equivalent to two derivatives. Then introducing an arbitrary coef-

™

¬cient with each term which are determined from Ii = 0(i = 2, 3, . . .)

the higher conservation laws can be obtained [see Chodos [20,21] who

also discusses another method of determining the conservation laws

using pseudo-di¬erential operators.]

Like the KdV the SG equation can also be cast in the Lax form

(7.28) with

‚

L = 2σ3 + σ2 ψ

‚x’

B = (σ3 cos ψ + σ2 sin ψ)L’1 (7.33)

The use of light-cone coordinates re¬‚ects that the conserved quanti-

ties will turn out to be the x’ integrals of appropriate functions of

ψ.

7.4 SUSY and Conservation Laws in the

KdV-MKdV Systems

An interesting aspect of (7.29a) is that its time-independent version

furnishes the Schroedinger equation

d2

’ 2 + u(x, 0) ¦ = »¦ (7.34)

dx

having the stationary solution of the KdV equation as the potential.

Thus it is through the L operator that one looks for a correspon-

dence between the KdV and the Schroedinger eigenvalue problem. In

Chapter 2, we noted that the N soliton solutions of the KdV equa-

tion emerge as families of re¬‚ectionless potentials [22-25]. We may

write them in the form (for t = 0) : uN (x, t) = ’N (N +1) b2 sech2 bx,

N = 1, 2, . . . , which is a family of symmetric, re¬‚ectionless poten-

√

tials. The case N = 1, b = 2a corresponds to the one-soliton solution

for t = 0, while the one for all t is given by (7.3).

Turning now to the MKdV equation we notice that the quantities

1

u± = (v 2 “ vx ’ k 2 ) (7.35)

2

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where k is a constant satisfy the KdV equation provided the condition

2v vt + 6(k 2 ’ v 2 )vx + vxxx “ vt + 6(k 2 ’ v 2 )vx + vxxx =0

x

(7.36)

holds, in other words if v is a solution of the MKdV equation. Ac-

tually one of the solutions in (7.35) corresponds (when k = 0) to the

well-known Miura-map between the KdV and MKdV. Observe that

the MKdV equation is invariant under v ’ ’v so both u± correlate

the KdV and MKdV. In this sense the Miura transformation can be

viewed as the supersymmetric square-root [26].

The combinations (7.35) remind one of the partner potentials

V± encountered in Chapter 2. Here v plays the role of the super-

potential. Interestingly, using (7.35) it is possible to work out the

conserved quantities for the MKdV from those of the KdV equation.

Indeed employing (7.35) it is straightforward to verify that the con-

served quantities I0 , I1 , . . . of the KdV get mapped to I1 , I2 , . . . of

the MKdV.

Physically the transformations (7.35) mean that if we de¬ne

12 1

v + vx = k 2 > 0

V’ ≡ (7.37)

2 2

then, as in (2.71), V’ does not have any bound state. On the other

hand, V+ reading V+ ≡ 1 v 2 ’ vx reveals a zero-energy bound state.

2

Thus one can carry out the construction of re¬‚ectionless potentials as

outlined in Chapter 2. Furthermore, in the spirit of SUSY, employing

appropriate boundary conditions on v, the solutions u+ and u’ may

be identi¬ed with the N + 1 and N soliton solutions, respectively.

7.5 Darboux™s Method

It is instructive to describe brie¬‚y the generation of N + 1 soliton so-

lution from the N soliton solution using Darboux™s procedure [27,28]

which is closely related to Backlund transformation. As has been

emphasized in the literature, the factorization method developed in

connection with SUSYQM is a special case of Darboux construction.

Darboux™s method has been generalized by Crum [29] to the case of

an arbitrary number of eigenfunctions.

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The essence [30-34] of Darboux™s method is to notice that if φ is

a particular solution with an eigenvalue of the Schroedinger equation

1 d2

Hψ = ’ + V (x) ψ = Eψ (7.38)

2 dx2

then the general solution of another Schroedinger equation

1 d2

HΨ ≡ ’ + U (x) Ψ = EΨ (7.39)

2 dx2

with E( = is given by

e)

1

Ψ= (ψφ ’ ψ φ) (7.40)

φ

and

d2

U (x) = V (x) ’ 2 lnφ (7.41)

dx

Equation (7.44) closely resembles the expression (2.84) between the

partner potentials for SUSYQM. The relevance of Darboux™s method

to the factorization scheme is brought out by the fact that V+ (V’ )

+ 1

act like U (V ) with the correspondence ψ0 ” φ .

To have a nonsingular U, φ ought not to be vanishing. This

restricts e ¤ E0 , E0 being the ground state energy of H. If we assume

the existence of a solution of (7.38) that satis¬es u ’ 0 as x ’ ’∞

and is nonvanishing for ¬nite x ∈ R, then the general solution of

(7.38) for the case when E = e can be obtained as

dx

φ(x) = u(x) K + (7.42)

u2

x

where K ∈ R+ . We note that the energy spectrum of H de¬ned by

(7.39) is identical to that of H except for the presence of the ground

state eigenvalue e. The corresponding normalized eigenfunction is

√

given by K/φ.

To illustrate how Darboux™s transformation works [19] in relating

N and N + 1 soliton solutions we identify the potentials Q(N ) and

Q(N +1) with the N and N + 1 soliton solutions, respectively, and

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keep in mind that an M soliton solution has M bound states. Then

we have

1 d2 ψi

+ Q(N ) ψi = Ei ψi , i = 1, 2, . . . , N

’ (7.43)

2

2 dx

1 d2 Ψi

+ Q(N +1) Ψi = Ei Ψi , i = 1, 2, . . . , N, N + 1 (7.44)

’

2 dx2

with the usual boundary conditions imposd namely ψi and Ψi ’ 0

as |x| ’ ∞. On the eigenvalues Ei we can, without losing generality,

provide the ordering EN +1 < EN < . . . < E2 < E1 < 0.

We now supplement to (7.43) the Schroedinger equation for ψN +1 .

The latter, of course, cannot be a bounded solution

1 d2 ψN +1

+ Q(N ) ψN +1 = EN +1 ψN +1

’ (7.45)

2

2 dx

To put (7.45) to use let us set E0 = EN +1 , V = Q(N ) , U = Q(N +1)

and identify the particular solution φ to be ΨN +1 . Then ΨN +1 =

1

ψN +1 and we get from (7.41)

d2

(N +1) (N )

Q =Q + 2 lnψN +1 (7.46)

dx

To get an (N +1) solution the strategy is simple. We solve (7.45)

for an unbounded solution corresponding to a given Q(N ) . This gives

ψN +1 which when substituted in (7.46) determines QN +1 . In this

way we avoid solving (7.44). The following example serves to make

the point clear.

Set Ei = ’»2 < 0 and start with the simplest vacuum case

i

(0) = 0. Then from (7.45)

V

1 d2 ψ1

+ »2 ψ1 = 0

N =0:’ (7.47)

1

2

2 dx

The unbounded solution is given by

√ √

2»1 x ’ 2»1 x

ψ1 = c1 e + c2 e (7.48)

where c1 , c2 are arbitrary constants. Equation (7.46) will imply

d2 √ √

(1)

= 2 log c1 e 2»1 x + c2 e’ 2»2 x

Q (7.49)

dx

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Next consider the case N = 1 which reads from (7.45)

1 d2 ψ2

+ Q(1) ψ2 = ’»2 ψ2

’ (7.50)

2

2

2 dx

Equation (7.50) may be solved for an unbounded solution which turns

out to be

»+ ’»’ x »’ ’»+ x

ψ2 = K c3 e»+ x + c4 e’»’ x + A e +B e /ψ1 (7.51)

»’ »+

where

√

2(»1 ± »2 ) = »± (7.52)

A = (c2 c3 )/c1 (7.53)