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реалiзацiї запропонованого алгоритму для систем.
Розглянемо систему рiвнянь Ойлера руху невязкої, нестисливої рiдини
?v ?v
+ vk (18)
= 0,
?x0 ?xk
де v = (v 1 , v 2 , v 3 ), v l = v l (x0 , x1 , x2 , x3 ), l = 1, 2, 3.
Систему (18) можна записати так:
?0 + v k ?k v l = 0, (19)
l = 1, 2, 3
Пiсля замiни змiнних
? a = xa ? v a x0 , a = 1, 2, 3, z l = v l , l = 1, 2, 3
? = x0 ,
28 В.I. Фущич, В.М. Бойко

система (19) матиме вигляд
?? z l = 0, (20)
l = 1, 2, 3.
Iнтегруючи рiвняння (20) i виконавши обернену замiну змiнних, одержуємо роз-
в’язок системи (18) у неявному виглядi
v l = g l x1 ? v 1 x0 , x2 ? v 2 x0 , x3 ? v 3 x0 .
де g l — довiльнi гладкi функцiї. Даний розв’язок системи (18) спiвпадає з розв’яз-
ком, одержаним iншим шляхом в [5].
Розглянемо систему рiвнянь для вектор-потенцiалу
?Aµ
A? (21)
= 0, µ = 0, . . . , 3.
?x?
Вважаємо, що A0 = 0. За допомогою замiни змiнних
x0
? = 0 , ? a = xa A0 ? x0 Aa , a = 1, 2, 3, Aµ = Aµ , µ = 0, 1, 2, 3
A
одержуємо розв’язок системи (21)
Aµ = g µ x1 A0 ? x0 A1 , x2 A0 ? x0 A2 , x3 A0 ? x0 A3 ,
де g µ — довiльнi гладкi функцiї.
Нехай тепер маємо деяку систему рiвнянь в частинних похiдних, що визначає-
ться набором операторiв L1 , . . . , Lr вигляду (2) (u ? u), причому кiлькiсть опера-
торiв повинна не перевищувати кiлькiсть незалежних змiнних. Якщо оператори
утворюють комутативну алгебру Лi i ранг матрицi, складеної з коефiцiєнтiв опе-
раторiв L1 , . . . , Lr , дорiвнює r, тодi iснує локальна замiна змiнних, що приводить
цi оператори до r операторiв диференцiювання вiдносно r перших незалежних
змiнних.
Розглянемо одновимiрну систему
(?t + v?x )u = 0,
(22)
(?t + u?x )v = 0,
де u = u(t, x), v = v(t, x), u = v. Пiсля замiни змiнних
x ? ut x ? vt
(23)
?= , ?= , U = u, V =v
v?u u?v
система (22) матиме простий вигляд
?? U = 0,
(24)
?? V = 0.
Проiнтегрувавши (24) та виконавши обернену до (23) замiну змiнних, одержуємо
розв’язок системи (22)
x ? vt x ? ut
u=f , v=g ,
u?v v?u
де f , g — довiльнi гладкi функцiї.
Робота виконана при фiнансовiй пiдтримцi AMS, фондiв Сороса та INTAS.
Пониження порядку та загальнi розв’язки деяких класiв рiвнянь 29

1. Fushchych W.I., New nonlinear equation for electromagnetic ?eld having the velocity di?erent
from c, Dopovidi Akademii Nauk Ukrainy, 1992, № 1, 24–27.
2. Fushchych W., Symmetry analysis. Preface, in Symmetry Analysis of Equations of Mathemati-
cal Physics, Kiev, Inst. of Math., 1992, 5–6.
3. Fushchych W., Boyko V., Symmetry classi?cation of the one-dimensional second order equa-
tion of hydrodynamical type, Preprint LiTH-MAT-R-95-19, Link?ping University, Link?ping,
o o
Sweden, 1995, 11 p.
4. Boyko V., Symmetry classi?cation of the one-dimensional second order equation of a hydro-
dynamic type, J. Nonlinear Math. Phys., 1995, 2, № 3–4, 418–424.
5. Fairlie D.B., Leznov A.N., General solution of the universal equation in n-dimensional space,
J. Nonlinear Math. Phys., 1994, 1, № 4, 333–339.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 30–35.

On unique symmetry of two nonlinear
generalizations of the Schr?dinger equation
o
W.I. FUSHCHYCH, R.M. CHERNIHA, V.I. CHOPYK
We prove that two nonlinear generalizations of the nonlinear Schr?dinger equation
o
are invariant with respect to a Lie algebra that coincides with the invariance algebra
of the Hamilton–Jacobi equation.

Nowadays many authors, who start from various physical considerations, have
suggested a wide spectrum of nonlinear equations which can be considered as some
nonlinear generalizations of the classical Schr?dinger equation. It is necessary to note
o
that some of the suggested equations do not satisfy the Galilean relativistic principle.
As a rule this requirement is not used in construction of nonlinear generalizations.
Meantime it is well known that the linear Schr?dinger equation is compatible with
o
the Galilean relativistic principle and, besides, is invariant with respect to scale and
projective symmetries (see, e.g. [1] and references cited therein).
In the [1–6] the construction of nonlinear generalizations of the Schr?dinger equa-
o
tion was based on the idea of symmetry and the following problems were solved:
1. Nonlinear Schr?dinger equations, which are compatible with the Galilean relati-
o
vistic principle, are described.
2. All nonlinear equations, which preserve nontrivial AG2 (1, n)-symmetry of the
linear Schr?dinger equation, are constructed.
o
Let us adduce some nonlinear generalizations of the Schr?dinger equation that
o
have AG2 (1, n)-symmetry, namely:

iUt + ?U = ?1 |U |4/n U, [1, 2] (1)

|U |a |U |a
[3, 4] (2)
iUt + ?U = ?1 U,
|U |2

?|U |2
(3)
iUt + ?U = ?1 U, [6]
|U |2

where U = U (t, x) is an unknown di?erentiable complex function, Ut ? ?U , ? ?
v ?t
?2 ?2 ? , |U | ? ?|U | , and ? is the sign of
+ · · · + ?x2 , x = (x1 , . . . , xn ), |U | = U U
?x2 a ?Xa
n
1
complex conjugation.
Consider the generalization of the nonlinear Schr?dinger equations (2)–(3) of the
o
following form
1 ?|U |2 |U |a |U |a 1 U
? ?1 (4)
iUt + ?U = ?0 + ?2 ln ? U,
|U | |U |
2 2
2 2 U
where ?k = ak + ibk , ak and bk ? R, k = 0, 1, 2.
J. Nonlinear Math. Phys., 1996, 3, № 3–4, P. 296–301.
Unique symmetry of two nonlinear generalizations of the Schr?dinger equation
o 31

It is easily seen that some nonlinear equations, which have been suggested by
many authors as mathematical models of quantum mechanical, are particular cases
of this nonlinear generalization of the Schr?dinger equation. Indeed, we obtain from
o
equation (4) (for ?0 = ?1 and ?2 = ib2 ) the following equation
1/2
?|U | U
(5)
iUt + ?U = ?1 + ib2 ln U,
U?
|U |

which was proposed in [7] for the stochastic interpretation of quantum mechanical
vacuum dissipative e?ects.
Equation (5) for b2 = 0 reduces to the form
?|U |
(6)
iUt + ?U = ?1 U,
|U |
which was studied in [7–11]. The term on the right hand side of (6) takes into consi-
deration the e?ect of quantum di?usion. In all these papers the authors, starting from
some physical models, assumed that the parameters Re ?1 and b2 in (5) and (6) are
small (?1 = 0, b2 = 0).
The main purpose of the present paper is to draw attention to equation (5). If we
reject the mentioned assumptions as it was done in all mentioned papers [7–11] and
put ?1 = 1, then the equations
?|U |
(7)
iUt + ?U = U
|U |
and
1/2
?|U | U
(8)
iUt + ?U = + ib2 ln U
U?
|U |

have the unique symmetry, which is the same as symmetry as of the Hamilton–Jacobi
equation [1].
It means that the nonlinear second-order term ?|U |/|U | changes and essentially
extends symmetry of the linear Schr?dinger equation.
o
Let us note that equation (7) for n = 2 can be obtained from the nonlinear
hyperbolic equation [12]

|?| 2? ? ?2 |?| = 0,
?2 ?2 ?2
where ? = ?(y0 , y), y = (y1 , y2 , y3 ), 2 = ? ? ?
?
2, by means of the
2 2 2
?y0 ?y1 ?y2 ?y3
ansatz

? = ?(t, x1 , x2 ) exp(aµ yµ ), t = bµ yµ , x1 = cµ yµ , x2 = dµ yµ ,

where the parameters aµ , bµ , cµ , dµ , µ = 0, 1, 2, 3 satisfy the following conditions:

a2 = d2 = ?1.
aµ bµ = 1, bµ cµ = cµ aµ = aµ dµ = dµ cµ = 0, µ µ

Now let us formulate theorems which give the complete information about local
symmetry properties of equation (4).
32 W.I. Fushchych, R.M. Cherniha, V.I. Chopyk

Statement 1. Equation (4) for arbitrary complex constants ?0 , ?1 and ?2 is invariant
with respect to the Lie algebra with the basic operators
? ? ? ?
+ U?
Pt = , Pa = , I=U ,
?U ?
?t ?xa ?U (9)
= xa Pb ? xa Pb , a, b = 1, . . . , n,
Jab
?
2a2
? I + Q exp b2 t, b2 = 0,
b2 (10)
X=
?
2a2 tI + Q, b2 = 0,

where Q = i U ?U ? U ? ?U ? .
? ?

Statement 2. Equation (4) for ?2 = ib2 is invariant with respect to the Lie algebra
with the basic operators (9) and
b2 1
Ga = exp(b2 t)Pa + (11)
xa Q1 , Q1 = exp(b2 t)Q.
2 2
Note that the algebra AG(1, n) with basic operators (9) (without I) and (11) is
essentially di?erent from the well-known Galilei algebra AG(1, n) in that it contains
commutative relations [Pt , Ga ] = b2 Ga , [Pt , Q1 ] = b1 Q1 , since in the AG(1, n) algebra
[Pt , Ga ] = Pa , [Pt , Q] = 0.
The operators Ga generate the following transformations
t = t, xa = xa + va exp(b2 t), a = 1, . . . , n,
(12)
b2 va va
U = U exp i exp(b2 t) xa va + exp(b2 t) ,
2 2
where v1 , . . . , vn are arbitrary real group parameters.
Some classes of equations with the AG(1, n)-symmetry were constructed and stu-
died in [4] (see the part II), [13].
Statement 3. Equation (4) for ?2 = 0 is invariant with respect to the Lie algebra
with the basic operators (9) and
xa n
D = 2tPt + xa Pa ?
Ga = tPa + Q, Q, I,
2 2
(13)
|x|2 nt
Q ? I.
2
? = t Pt + txa Pa +
4 2
It is clear that operators (9) and (13) generate the well known generalized Galilei
algebra AG2 (1, n) with the additional unit operator I. The linear Schr?dinger equation
o

(14)
iUt + ?U = 0

is invariant with respect to the AG2 (1, n), I algebra, too. It is well known that
operators Ga , a = 1, . . . , n generate the Galilean transformations

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