ñòð. 63 |

?

?

Ga = t?xa ? i?Va + i?Va .

?

2. F = C|?|k , where C is an arbitrary complex constant, C = 0, k is an arbitrary

real number, k = 0 and k = ?1. The maximal invariance algebra is P0 , Pa , Jab , Ga ,

D(1) , where

2

D(1) = 2t?t + xc ?xc ? Vc ?Vc ? Vc? ?Vc? ? (??? + ? ? ??? ).

1+k

C

3. F = , where C is an arbitrary complex constant, C = 0. The maximal

|?|

invariance algebra is P0 , Pa , Jab , Ga , Z = Z1 + Z2 , where

Z = ??? + ? ? ??? , Z2 = ? ? ??? .

Z1 = ??? ,

4. F = C = 0, where C is an arbitrary complex constant. The maximal invariance

algebra is P0 , Pa , Jab , Ga , D(1) , Z3 , Z4 , where

Z3 = ?? , Z4 = ??? .

284 W.I. Fushchych, Z.I. Symenoh, I.M. Tsyfra

5. F = 0. The maximal invariance algebra is P0 , Pa , Jab , Ga , D, A, Z1 , Z2 , Z3 , Z4 ,

where

D = 2t?t + xc ?xc ? Vc ?Vc ? Vc? ?Vc? ,

A = t2 ?t + txc ?xc ? (ixc + tVc )?Vc + (ixc ? tVc? )?Vc? .

6 Contact transformations

Consider the two-dimensional Schr?dinger equation

o

(24)

i?t + ?xx = V (t, x, ?, ?x , ?t ).

We seek the in?nitesimal operators of contact transformations in the class of the

?rst-order di?erential operators of the form [1, 9]

X = ? ? (t, x, ?, ?t , ?x )?x? + ?(t, x, ?, ?t , ?x )?? +

(25)

+ ? ? (t, x, ?, ?t , ?x )??? + µ(t, x, ?, ?t , ?x , V )?V ,

where

?W ?W ?W ?W

?? = ? ? = W ? ?? ?? = (26)

, , + ??

??? ??? ?x? ??

for a function W = W (t, x, ?, ?x , ?t ). The condition of invariance of equation (24)

under operators (25), (26) implies that the unknown function W has the form

W = F 1 (t)?t + F 2 (t, x, ?, ?x ),

where F 1 and F 2 are arbitrary functions of their arguments.

Then

? 0 = ?F 1 (t), ? 1 = ?F?x (t, x, ?, ?x ),

2

? = F 2 ? ? x F ?x ,

2

? 0 = Ft1 ?t + Ft2 + ?t F? ,

2

? 1 = Fx + ? x F? ,

2 2

µ = i(Wt + ?t W? ) + Wxx + 2Wx? ?x ?

? (i?t ? V ) (Wx?x + W? + ?x W??x ) + (?x )2 W?? ?

? (i?t ? V ) (Wx?x + ?x W??x ? (i?t ? V )W?x ?x ) .

Thus, equation (24) is invariant under the in?nite-dimensional group of contact trans-

formations with the in?nitesimal operators:

QF 1 = ?F 1 ?t + Ft1 ?t ??t + iFt1 ?t ?V ,

QF 2 = ?F?x ?x + (F 2 ? ?x F?x )?? + (Ft2 + ?t F? )??t +

2 2 2

+ (Fx + ?x F? )??x + iFt2 + i?t F? + Fxx + 2Fx? ?x + (?x )2 F?? ?

2 2 2 2 2 2

? (i?t ? V )(2Fx?x + 2?x F??x + F? ) + (i?t ? V )2 F?x ?x ?V ,

2 2 2 2

where F 1 = F 1 (t) and F 2 = F 2 (t, x, ?, ?x ) are arbitrary functions.

Consider the special case. Let F 1 (t) = 1, F 2 (t, x, ?, ?x ) = ?(?x )2 . Then W =

?t ? (?x )2 . The operators of the contact transformations have the form

QF 2 = 2?x ?x + (?x )2 ?? ? 2(i?t ? V )2 ?V . (27)

QF 1 = ?t ,

The Schr?dinger equation with variable potential

o 285

The operator (27) generate the ?nite transformations:

x = 2?x ? + x, t = t,

? = (?x )2 ? + ?, ?x = ?x , ?t = ? t ,

(28)

2i?(V ? i?t )?t + V

V= .

2?(V ? i?t ) + 1

Transformations (28) can be used for generating exact solutions of equation (24) from

the known solution and for constructing nonlocal ansatzes reducing the given equation

to the system of ordinary di?erential equations.

1. Fushchych W., Shtelen V., Serov N., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.

2. Fushchych W., Nikitin A., Symmetry of equations of quantum mechanics, New York, Allerton

Press, 1994.

3. Boyer C., The maximal ‘kinematical’ invariance group for an arbitrary potential, Helv. Phys.

Acta, 1974, 47, 589–605.

4. Truax D.R., Symmetry of time-dependent Schr?dinger equations. I. A classi?cation of time-

o

dependent potentials by their maximal kinematical algebras, J. Math. Phys., 1981, 22, ¹ 9,

1959–1964.

5. Fushchych W., How to extend symmetry of di?erential equations?, in Symmetry and Solutions

of Nonlinear Equations of Mathematical Physics, Kyiv, Inst. of Math., 1987, 4–16.

6. Fushchych W., New nonlinear equations for electromagnetic ?eld having the velocity di?erent

from c, Dopovidi Academii Nauk Ukrainy, 1992, ¹ 4, 24–27.

7. Fushchych W., Ansatz ’95, J. Nonlinear Math. Phys., 1995, 2, ¹ 3–4, 216–235.

8. Ovsyannikov L.V., Group analysis of di?erential equations, New York, Academic Press, 1982.

9. Olver P., Application of Lie groups to di?erential equations, New York, Springer, 1986.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 286–290.

Stationary mKdV hierarchy and integrability

of the Dirac equations by quadratures

R.Z. ZHDANOV, I.V. REVENKO, W.I. FUSHCHYCH

Using the Lie’s in?nitesimal method we establish that the Dirac equation in one

variable is integrable by quadratures if the potential V (x) is a solution of one of the

equations of the stationary mKdV hierarchy.

Consider the eigenvalue problem for the Dirac operator L = i?1 d/dx ? V (x)?2 ,

du

(L ? ?)u ? i?1 ? (V (x)?2 + ?)u = 0, (1)

dx

where ?1 , ?2 are 2 ? 2 Pauli matrices, u = (u1 (x), u2 (x))T , V (x) is a real-valued

function and ? is a real parameter. We remind that Eq. (1) is one of two equations

composing the Lax pair for the mKdV equation,

vt + vxxx ? 6v 2 vx = 0, (2)

integrable by the inverse scattering method (see, e.g., Refs. [1, 2]). Next, as the identity

d2 dV

(L ? ?)(L + ?) = ? + V 2 ? ?3 ? ?2 ,

2

dx dx

holds, components of the vector-function u ful?ll the stationary Schr?dinger equation,

o

d2 u i dV

? V 2 + ?2 ui = 0,

+ (?1)i+1 (3)

i = 1, 2.

2

dx dx

The aim of the present Letter is to show that there exists an initimate connection

between integrability of system (1) (in what follows we will call it the Dirac equation)

by quadratures and solutions of the stationary mKdV hierarchy.

Integrability of system (1) will be studied with the use of its Lie symmetries. As

usual, we call a ?rst-order di?erential operator

d

X = ?(x) + ?(x),

dx

where ? is a real-valued function and ? is a 2 ? 2 matrix complex-valued function,

a Lie symmetry of system (1) if commutation relation

(4)

[L, X] = R(x)L,

holds with some 2 ? 2 matrix function R(x) (for details, see, e.g., Ref. [3]).

A simple computation shows that if X is a Lie symmetry of system (1), then an

operator X + r(x)L with a smooth function r(x) is its Lie symmetry as well. Hence

we conclude that without loss of generality we can look for Lie symmetries within the

Physics Letters A, 1998, 241, P. 155–158.

Stationary mKdV hierarchy and integrability of the Dirac equations 287

class of matrix operators X = ?(x). Furthermore, inserting X = ?(x) into Eq. (4)

and computing the commutator yield that the matrix ?(x) is necessarily of the form

f (x) g(x)

(5)

?= ,

?f (x)

h(x)

where f (x), g(x), h(x) are arbitrary solutions of the following system of ordinary

di?erential equations,

df dg dh

= i?(g ? h), = ?2i?f ? 2hV. (6)

= 2i?f + 2gV,

dx dx dx

With a solution of system (6) in hand we can integrate the initial equations (1) by

quadratures using the classical results by Elie Cartan [4]. Since these results are well-

known we will give them without derivation in the form of the following lemma.

Lemma 1. Let the functions f (x), g(x), h(x) satisfy system (6). Then the general

solution of the Dirac equation is given by the formulae

u1 (x) = C1 (R(x) + f (x))(h(x))?1/2 (R2 (x) ? ?)?1/2 ,

(7)

u2 (x) = C1 (h(x))1/2 (R2 (x) ? ?)?1/2 ,

where ? = f 2 (x) + g(x)h(x) is constant on the solution variety of system (6),

?v

? ? tanh C ? i?v? dx

?

? , ? > 0,

? 2

? g(x)

?

?

? ?1

dx

C ? i?

R(x) = , ? = 0,

?2

? g(x)

?

?v

? v

? dx

? ?? tan C2 + i? ??

? , ? < 0,

g(x)

and C1 , C2 are arbitrary complex constants.

However, solving system of ordinary di?erential equations (6) is by no means

easier than solving the initial Dirac equation. This is a common problem in applying

Lie symmetries to integration of ordinary di?erential equations. The key idea of our

approach is to restrict a priori the class within which Lie symmetries are looked for

and suppose that they are polynomials in ? with variable matrix coe?cients.

Introducing the new dependent variables ?1 (x), ?2 (x),

i d?1

?

f (x) = + 2V ?2 ,

4? dx (8)

1 1

g(x) = (?1 (x) + ?2 (x)), h(x) = (?1 (x) ? ?2 (x)),

2 2

we rewrite Eq. (6) in the following equivalent form,

d2 ? 1 d?2 dV d?2

= ?4?2 ?1 + 2V (9)

+ 2?2 , = 2V ?1 .

dx2 dx dx dx

As mentioned above solutions of system (9) are looked for as polynomials in ?,

namely

n n

2k

rk (x)(2?)2k . (10)

?1 (x) = pk (x)(2?) , ?2 (x) =

k=1 k=1

288 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Inserting the expressions (10) into (9) and equating the coe?cients by the powers

of ? yield pn = 0 and

drk

(11)

= 2V pk , k = 1, . . . , n

dx

d 2 pk drk dV

rk ? pk?1 , k = 1, . . . , n ? 1, (12)

= 2V +2

dx2 dx dx

where we set by de?nition p?1 (x) = 0. Eliminating from Eqs. (11), (12) the functions

rk (x), we get recurrent relations for the functions pk (x),

d2 dV ?1

? 2 +4 k = n, n ? 1, . . . , 0.

D V + 4V 2 pk (x), (13)

pk?1 (x) =

dx x

dx

Q

?1

Here Dx denotes integration by x.

A reader familiar with the theory of solitons will immediately recognize the opera-

tor Q as the recursion operator for the mKdV equation (2) (see, e.g., Refs. [5, 6]).

Acting repeatedly with this operator on the trivial symmetry S0 = 0 yields an in?nite

number of higher symmetries S1 , S2 , . . . admitted by the mKdV equation [5]. Hence it

is not di?cult to derive that the functions pk , k = 0, . . . , n ? 1 are linear combinations

of the higher symmetries S1 , . . . , Sn with arbitrary constant coe?cients Ci ,

k

(14)

pn?k (x) = Ci Sk+1?i , k = 1, . . . , n,

ñòð. 63 |