стр. 35 |

?

? = A ? tanh[?(x ? k ? vt)] ± i ?

v

? ?v

2 2 (8.5)

? exp i ? 3E t + (v ? )x + ?0 ? 0.

, E = 0,

2

For potentials (5.11) we obtain from (8.2)

v

? ?

? = ?1 = A B cn [D(x ? vt) + k] exp[if1 (t, x)], (8.6)

E = 0;

? ?

? = ?2 = A B cn2 [D(x ? vt) + k] + F exp(if2 (t, x)], (8.7)

E + ?0 = 0,

?

Higher symmetries and exact solutions of Schr?dinger equation

o 155

where

v2

3

F?

f1 (t, x) = f2 (t, x) + F t = t + vx + ?0 ,

2 2

B, D and F are parameters de?ned in (5.11b).

For another values of E solutions (8.2), (5.11) are also reduced to the form (8.7)

where the phase f2 (t, x) is expressed via elliptic integrals.

Formula (8.4) presents a fast decreasing one-soliton solution [31]. Relation (8.5)

de?nes a soliton solution whose behavior at x > ? is typical of solitons with a ?nite

density. Formulae (8.6), (8.7) describe “cnoidal” solutions for the nonlinear Schr?dinger

o

equation.

9 Conditional symmetry and generation of solutions

Let us return to the linear Schr?dinger equation (2.1) with the potential U satisfying

o

(3.8a). Generally speaking it possesses no non-trivial (distinct from time displace-

ments) Lie symmetry. Nevertheless, its solutions can be generated within the frame-

work of the concept of conditional symmetry [2, 3, 12, 14, 32]. Indeed, these solutions

satisfy (7.7), and equation (2.1) with the additional condition (7.7) is invariant under

the Galilei transformations (8.1) (i.e., condition (7.7) extends the symmetry of equa-

tion (2.1)).

This conditional symmetry enables us to generate new solutions. Starting with

(7.1), (7.4) and using (8.1) we obtain

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

?

?? ??

? ? (9.1)

x?k?vt

t dy

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

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

2 ?

0

Functions (9.1) satisfy the Schr?dinger equation with a potential V (x ? k ? vt)

o

where V (x) is a solution of equation (5.6). In the particular case E = ? ?0 these

?

2

functions are reduced to solutions (8.2) of the nonlinear equation (7.6).

One more generation of solutions can be made using a third-order SO. Inasmuch

as V (x) satis?es (5.6), then V (x ? vt) satis?es the Boussinesq equation (6.3). It

means that the corresponding linear Schr?dinger equation admits a third-order SO.

o

In accordance with (2.2), (2.6) this SO can be represented in the form

1 3

Q = p3 + {3V + 2? 0 + 6v 2 , p} + vV ?

?

4 2 (9.2)

1 3 i

? 2pH + (V + 2? 0 + 6v )p + vV + V .

2

?

2 2 4

Formula (9.2) generalizes (3.12a) to the case of time-dependent potential.

Acting by operator (9.2) on ? in (9.1) we obtain a new family of solutions

? = Q? = a? + iv 2 ?1 , (9.3)

where a = ? + 4Ev + ?0 v ? 4v 3 , ? is the initial solution (9.1),

?

V + 4i?

(9.4)

?1 = ?.

2(4E + V + 2? 0 )

?

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

We note that if ? is a soliton solution

v2

?A

exp i ? t + vx + ?0 (9.5)

?=

cosh[?[x ? vt)] 2

(the corresponding potential is present in (5.9a)), then (9.4) is a soliton solution too:

? 2 A sinh[?(x ? vt)] v2

exp i ? t + vx + ?0 (9.6)

?1 = .

cosh2 [?(x ? vt)] 2

Starting with the potential (5.11) we obtain from (9.1) a particular solution

v2

+ F exp i ? t + vx + ?0 z = D(x ? vt).

cn2 z (9.7)

?=A B ,

2

The corresponding generated solution (9.4) reads

v2

ABD cn z sn z dn z

?1 = ? exp i ? t + vx + ?0 (9.8)

B cn2 z + 2F 2

and is also bounded.

Acting by SO (9.2) on solutions (9.3), (9.8) we again obtain new solutions. Mo-

reover, this procedure can be repeated. In particular, in this way it is possible to

construct multisoliton solutions of the linear Schr?dinger equation.

o

We see that higher order SOs present e?cient possibilities for solving equations of

motion and generating new solutions starting with known ones.

10 Conclusion

Higher order SOs present a powerful tool for analyzing and solving the Schr?dinger

o

equation. The concept of higher symmetries enables us to extend the class of privileged

potentials (2.4) and to investigate invariance algebras of the equations whose poten-

tials satisfy one of relations (3.8).

We note that potentials (5.9) can be represented in the form V = W 2 + W where

W = ? tanh[?(x ? k)] for solution (5.9a) (superpotentials W for solutions (5.9a)–

?

(5.9c) can be also easily calculated). Moreover, the corresponding superpartners V =

W 2 ?W reduce to constants, therefore it is possible to integrate easily the Schr?dinger

o

equation with potentials (5.9) using the Darboux transformation [33].

It is worth to note that invariance condition (2.3) for operators (2.1), (3.12) can

be treated as a zero curvature condition for equations associated with the eigenvalue

problem for operator Q, or as the Lax condition where a role of the Lax operator L

is played by a SO, refer to (6.2). The reasons stimulating our research of such a well-

studied subject and distinguishing features of our approach are the following:

(1) The main goal of our paper is to present a constructive description of potentials

for the Schr?dinger equation which admit higher symmetries. In this way we extend

o

the fundamental results [19–21] connected with the search for potentials admitting

usual Lie symmetries.

To solve the deduced determining equations for potentials we use direct reductions

to the Painlev? or Riccati forms. The obtained results can be used for analysis and

e

Higher symmetries and exact solutions of Schr?dinger equation

o 157

solution of the Schr?dinger equation as well as for construction of exact solutions of

o

the Boussinesq equation, see item 5 in the following.

In the method of inverse problem, description of pairs of operators (2.1), (2.8) sati-

sfying the Lax condition (6.2) is reduced to the Gelfand–Marchenko–Levitan equati-

ons [34] or to the Riemann problem [15, 31] which can be solved explicitly for a

restricted class of potentials.

(2) We use non-Lie symmetries of the Schr?dinger equation for construction and

o

generation of exact solutions. Moreover, we are interested not so much in ?nding

new solutions as in developing a new method of their derivation, which consists in

simultaneous using of higher order and conditional symmetries. Nevertheless, the

cnoidal solutions (9.7), (9.8) and (8.6), (8.7) for the linear and nonlinear Schr?dinger

o

equations can be of interest for physicists as well as in?nite series of soliton and cnoidal

solutions generated by a repeated application of the procedure described in Section 9.

We believe that the combination “higher order symmetries + conditional symmet-

ries” may be used e?ectively in the investigations and analysis of other equations of

mathematical physics.

(3) Our approach admits a direct generalization to multidimensional Schr?dinger o

equations. Note that higher symmetries of the three-dimension Schr?dinger equation

o

were investigated in [18, 35] for particular potentials.

(4) Algebraic relations (4.1)–(4.4) are valid for extended classes of potentials. They

open additional possibilities in the application of algebraic methods to investigate the

Schr?dinger equation, in particular, the use of raising and lowering operators for this

o

equation with potentials satisfying (3.8d). We note that relations (3.8d) are valid also

?

for time-independent operators Q± = exp(?i?t)Q± where Q± are given by relations

(3.12d).

(5) Equations (3.8) which describe potentials that admit third-order symmetries

are equivalent to the reduced versions of the Boussinesq equation, which appear under

the similarity reduction [36] (this is the case for (3.8a,d)) and the reduction with using

symmetries [14, 25, 26] (the last is valid for (3.8b,c)). Thus, the results obtained in

Section V can be used to construct exact solutions of the Boussinesq equation.

A systematic study of higher symmetries of multidimensional Schr?dinger equa-

o

tions is planned to be carried out elsewhere.

Acknowledgements. We are indebted to our anonymous referee for the rigorous

criticism and helpful suggestions. This work is partly supported by the Ukrainian

DFFD foundation (Project 1.4/356).

1. Anderson R.L., Ibragimov N.H., Lie–B?cklund Transformations in Applications, Philadelphia,

a

SIAM, 1979.

2. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations

of nonlinear mathematical physics, Kiev, Naukova Dumka, 1990; D.Reidel, Dordrecht, 1993.

3. Fushchych W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, Nauka, Mos-

cow, 1990; Allerton Press Inc., New York, 1994.

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5. Beckers J., Debergh N., Nikitin A.G., Mod. Phys. Lett. A, 1992, 7, 1609.

6. Beckers J., Debergh N., Nikitin A.G., Mod. Phys. Lett. A, 1993, 8 , 435.

7. Beckers J., Debergh N., Nikitin A.G., Phys. Lett. B, 1992, 279, 333.

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

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1983; D. Reidel, Dordrecht, 1987.

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in Mathematical Physics, Kiev, Inst. of Math., 1981, 6–28 (in Russian).

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equation, in Symmetry and Solutions of Nonlinear Equations of Mathematical Physics, Kiev,

Inst. of Math., 1989, 96–103 (in Russian).

15. Krichever I.M., Funk. Analiz, 1978, 12, 20;

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16. Fushchych W.I., Segeda Yu.N., Dokl. AN SSSR, 1977, 232, 800.

17. Beckers J., Debergh N., Nikitin A.G., J. Phys. A, 1992, 24, L1269.

18. Nikitin A.G., Onufriichuk S.P., Fushchych W.I., Teor. Math. Phys., 1992, 91, 268 (in Russian);

English translation in Theor. Math. Phys., 1992, 91, 514.

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20. Boyer C., Helv. Phys. Acta, 1974, 47, 589.

21. Niederer U., Helv. Phys. Acta, 1972, 45, 802.

4

22. This formula is present in [17] with a misprint: the coe?cient 2 for ?t is missing there.

23. Burchnall J.L., Chaundy T.W., Proc. London Math. Soc., 1922, 21, 420; Proc. Royal Soc.

London, 1928, 118, 557.

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27. Bagrov V.G., Gitman D.M., Exact solutions of relativistic wave equations, Dordrecht, Kluwer

Academic, 1990.

28. Whittaker E.T., Watson G.N., A Course of modern analysis, Cambridge University Press,

1927.

29. Miura R.M., J. Math. Phys., 1968, 9, 1202.

30. Gelfand I.M., Dikyi L.A., Uspekhi Mat. Nauk, 1975, 30, 67–100; Funk. Analiz, 1976, 10, 13.

31. Takhtadjian L.A., Faddeev L.D., Hamiltonian Approach in soliton theory, Moscow, Nauka,

1986 (in Russian).

32. Fushchych W.I., Tsyfra I.M., J. Phys. A, 1987, 20, 45.

33. Darboux G., Compt. Rend., 1882, 94, 1456.

34. Gelfand I.M., Levitan B.M., Izv. AN SSSR, Mat. Ser., 1951, 15, 309;

Marchenko V.A., Proc. Moskow Math. Soc., 1952, 1, 357.

35. Nikitin A.G., Ukrain. Math. J., 1991, 43, 1521 (in Russian); English translation in Ukr. Math.

J., 1991, 43, 1413.

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W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 159–165.

Умовна симетрiя рiвнянь Нав’є–Стокса

В.I. ФУЩИЧ, М.I. СЄРОВ, Л.О. ТУЛУПОВА

The conditional symmetry of the Navier–Stokes equations is studied. The multipara-

meter families of exact solutions of the Navier–Stokes equations are constructed.

Вивчена умовна симетрiя рiвнянь Нав’є–Стокса. Побудованi багатопараметричнi

сiм’ї точних розв’язкiв рiвнянь Нав’є–Стокса.

Розглянемо систему рiвнянь Нав’є–Стокса

1

u0 + (u?)u + ??u = ? ?p,

? (1)

?0 + div (?u) = 0, p = f (?),

де u = u(x) ? Rn , ? = ?(x) ? R, p = p(x) ? R, x = (x0 , x) ? R1+n .

Лiївська симетрiя рiвнянь (1) добре вивчена (див., наприклад, [1]). Результати

стр. 35 |