ñòð. 34 |

1 1 1 1

?y ? W ? W 2 W ? yW ? W

W = 0.

3 6 3 3

Equating the expression in the second brackets to zero and integrating it we come

to the second Painlev? transcendent

e

131

(5.15)

W= W + yW + K,

18 3

where K is an arbitrary constant.

To make one more reduction of equation (3.8c) we take U = ? . Then, integrating

the resultant equation, we obtain

2

? ? 3 (? ) ? 2?3 (x?) = C. (5.16)

Then, de?ning

v v

1 C

? = 2 3 2?3 ? + y 2 + , y = 3 2?3 x,

4 2?3

(5.17)

1 ??

?

W = ? ? ? 2 ? y, ? =

2 ?y

we represent (5.16) as

? ? ? ?

W ? 4? W + 2? W ? y W = 0. (5.18)

The trivial solutions of (5.18) correspond to the following Riccati equation for ?:

1

? ? ? 2 ? y = 0. (5.19)

2

Higher symmetries and exact solutions of Schr?dinger equation

o 151

It follows from the above that any solution of equations (5.15) or (5.19) generates

a potential U de?ned by relations (5.12), (5.14) or (5.17). The corresponding Schr?- o

dinger equation admits a third-order SO.

The last of the equations considered, i.e., equation (3.8d), is the most complicated.

The change

1

? = 2f ? ?4 x3 (5.20)

3

reduces it to the following form:

1

? 6(f )2 + 4?4 (f x2 ? xf ) = ?4 + ?5 . (5.21)

f

2

Multiplying (5.21) by f and integrating we obtain the ?rst integral

1 1

(f )2 ? 2(f )3 + 2?4 (f ? xf )2 ? ?4 + ?5 f = C (5.22)

2 2

which is still a very complicated nonlinear equation.

Let us demonstrate that (5.21) can be reduced to the Riccati equation. To realize

this we rewrite (5.21) as follows

1

F + 2f F ? 4f F = ?5 ? ?4 , (5.23)

2

where

F = f ? f 2 ? ?4 x2 .

Choosing ?5 = 2?4 we conclude that any solution of the Riccati equation

f = f 2 + ?4 x2 (5.24)

generates a solution of equation (3.8d), given by relation (5.20).

One more possibility in solving of equation (3.8d) consists in its reduction to

v

the Painlev? form. Making the change of variables ? = ?w4 ?, x = v??4 y and

1

e

di?erentiating equation (3.8d) w.r.t. y, we obtain

? ? ? ? ?

U ? 3U 2 = 4x2 , (5.25)

+ 6U + 6xU + +2U

?

where U = ?? = ? ?4 U .

1

?y

Using the following generalized Miura ansatz

?

U = ?V + V 2 + 2V y + y 2 ? 1, (5.26)

we reduce equation (5.25) to the form

?y (?y ? 2V ? 2y ? 2) ?

? V ? 6V 2 V ? 4V2 ? 12yV V ? 4yV ? 4V y 2 ? 2V = 0.

Equating the expression in the right brackets to zero, integrating and dividing it

by 2V , we come to the fourth Painlev? transcendent

e

2

V 3 b

+ V 3 + 8yV 2 + 2y 2 ? 1 V + . (5.27)

V =

2V 2 V

152 W.I. Fushchych, A.G. Nikitin

We note that the double di?erentiation and consequent change of variables

?4 1 1

x= v

? =? ? + y2 , y

4

3 6 4?4

transform equation (3.8d) to the form

1

?4? + ? ? + ? ? ? 8? + x2 ? + 7x? =0

3

which coincides with the reduced Boussinesq equation [3, 12]. The procedures outlined

above reduces the equation either to the fourth Painlev? transcendent (5.27) or to the

e

Riccati equation (5.24).

Thus, the third-order SO are admitted by a very extended class of potentials descri-

bed above. We should like to emphasize that in general the corresponding Schr?dinger

o

equation does not possesses any nontrivial (distinct from time displacements) Lie

symmetry.

6 Equations for time-dependent potentials

Consider brie?y the case of time-dependent potentials U = U (x, t). The determining

equations (2.6) are valid in this case also. Moreover, the compatibility condition for

system (2.6) takes the form

? ??

F (a, b, c; x, U ) + 12aU ? 4(b ? 2ax)U = 0, (6.1)

where F (a, b, c; x, U ) is de?ned in (2.7).

Equation (6.1) is much more complicated than (2.7) due to the time dependence

of U , which makes it impossible to separate variables. For any ?xed set of functions

a(t), b(t), and c(t), formula (6.1) de?nes a nonlinear equation for potential. Moreover,

any of these equations admits the Lax representation

?Q

(6.2)

[H, Q] = i ,

?t

cf. (2.3). Refer to Refs. [30, 31] for the general results connected with arbitrary ordi-

nary di?erential operators satisfying (6.2).

We will not analyze equations (6.1) here, but present a few simple examples

concerning particular choices of arbitrary functions a, b, and c.

a = const, b = c = 0:

?

?12U + U ? 6(U U ) = 0; (6.3)

a, b are constants, c = 0:

? ?

12U ? (4bU ? U (6.4)

+ 6U U ) = 0;

?

a = c = 0, b = ?3 a:

?

? ?

12U ? 4(?3 t ? 2x)U + (U ? 3U 2 ) + 2?3 (xU + 2U ) = 0; (6.5)

a = exp(t), b = c = 0:

? ?

12U + 8xU + (U ? U 2 ) ? 12(U x) ? 2x2 U ? 4x2 = 0. (6.6)

Higher symmetries and exact solutions of Schr?dinger equation

o 153

Formula (6.3) de?nes the Boussinesq equation. The Lax representation (6.2) for

this equation is well known [15]. Formulae (6.4)–(6.6) present other examples of non-

linear equations admitting this representation and arise naturally under the analysis

of third-order SOs of the Schr?dinger equation.

o

7 Exact solutions

Let us regard the case of potentials satisfying (3.8a) or (5.4), (5.6). Taking into account

commutativity of the corresponding SO (3.12a) with Hamiltonian (2.1) it is convenient

to search for solutions of the Schr?dinger equation in the form

o

(7.1)

?(t, x) = exp(?iEt)?(x),

where ?(x) are eigenfunctions of the commuting operators H and Q

(7.2a)

H?(x) = E?(x),

(7.2b)

Q?(x) = ??(x).

Using (7.2a), (3.12a), and (5.4) we reduce (7.2b) to the ?rst-order equation

V 1

(7.3)

2E + + ?0 ? =

? V + i? ?

2 4

whose general solution has the form

dx

(7.4)

? = A V + 4E + 2? 0 exp 2i?

? ,

V + 4E + 2? 0

?

where A is an arbitrary constant. Then, expressing ? via ? in accordance with (7.3)

and using (5.6), we reduce (7.2a) to the following algebraic relation for E and ?

(compare with (4.1b)):

?2 = 8E 2 (E + ?0 ). (7.5)

?

Thus there exists a remarkably simple way to integrate the Schr?dinger equation

o

which admits a third order SO. The integration reduces to the problem of solving the

?rst-order ordinary di?erential equation (7.3) and algebraic equation (7.5).

Let us show that the existence of a third-order SO for the linear Schr?dinger

o

equation enables one to ?nd exact solutions for the following nonlinear equation:

1? 1 ?? ? ?

?

i?t ? = p2 ? + (7.6)

(? ?)?.

2A2

2

Indeed, if ?2 > 0, solutions (7.1), (7.4) satisfy the following relations

?? ? = A2 (V + 4E + 2? 0 ). (7.7)

?

Using (7.2a) and (7.7) we make sure that the functions

?

? = exp(i?t)?(x), ? = ?3E ? ?0 (7.8)

?

(where ?(x) are functions de?ned in (7.4)) are exact solutions of (7.6).

Thus, we obtain a wide class of exact solutions of the nonlinear Schr?dinger equati-

o

on, which depend on arbitrary parameters ?, ?0 , ?1 , k (see (7.8), (7.4), (5.6), (5.8)).

??

Properties of these (and some more general) solutions are discussed in the following

section.

154 W.I. Fushchych, A.G. Nikitin

8 Lie symmetries and generation of solutions

It is well known that equation (7.6) is invariant under the Galilei transformations

(refer, e.g., to Refs. [2, 3])

x > x = x ? vt,

v2 (8.1)

?(t, x) > ? (t, x ) = exp i vx ? + ?0 ?(t, x),

2

where v and ?0 are real parameters. Using (8.1) it is possible to generate a more

extended family of solutions starting with (7.8)

?

? = A V (x ? k ? vt) + 4E + 2? 0 ?

?

x?k?vt (8.2)

t dy

? exp i (2? ? v 2 ) + vx + ?0 + 2? .

2 V (y) + 4E + 2? 0

?

0

Here, V is an arbitrary solution of equation (5.6), v, ?0 , ?1 , k, ?0 and E are real

??

parameters, ? and ? are de?ned in (7.5), (7.8).

In order for ? to be real we require ? ? 0, other parameters are arbitrary.

Solutions (8.2) are qualitatively di?erent for di?erent values of free parameters

enumerated in (5.7). If ?0 and ?1 satisfy (5.7a) or (5.7c), possible V are given by

? ?

formulae (5.9a), (5.9a ) or (5.9c). Solutions (8.2), (5.9a) are bounded for any x and t,

whereas solutions (8.2), (5.9a ) and (8.2), (5.9c) are singular at x ? k ? vt = 0.

For ?0 and ?1 satisfying (5.7b) the modulus of the complex function (8.2), (5.9b)

? ?

is periodic and singular at x ? k ? vt = (2n + 1)?/2?. All the above mentioned

singularities are simple poles. If ?0 and ?1 satisfy relations (5.8a), the solutions (8.2)

? ?

are expressed via the two-periodic Weierstrass function ? (refer to (5.10)) and are,

generally speaking, unbounded. But if we restrict ourselves to solutions (5.11) for

potential, the corresponding solutions (8.2) are periodic and bounded.

To inquire into a physical content of the obtained solutions let us consider in more

detail the cases (8.2), (5.9a) and (8.2), (5.11).

For potentials (5.9a) the corresponding relation (7.5) reduces to

= 2E ? ? 2 ,

?2 = 4E 2 , (8.3)

and the integral in (8.2) can be easily calculated. This enables us to represent solutions

(8.2), (5.9a) as follows

? 2 ? v2

A?

? (8.4)

?= exp i t + vx + ?0 , E = 0;

cosh[?(x ? k ? vt)] 2

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