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v
?
? = 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).

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a
SIAM, 1979.
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of nonlinear mathematical physics, Kiev, Naukova Dumka, 1990; D.Reidel, Dordrecht, 1993.
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158 W.I. Fushchych, A.G. Nikitin

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1983; D. Reidel, Dordrecht, 1987.
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4
22. This formula is present in [17] with a misprint: the coe?cient 2 for ?t is missing there.
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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
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