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?1 x0
v
Q3 = ?0 + u ? + ???1 ??u + ??? ,
?
?
Q4 = ?0 + u ? + ??u + ?2 ?2 ?? ,
?
Q5 = ??0 ? (?2 u2 ? ?)??u ,
?
Q6 = f (?)? + ??? ,
?
Q7 = ??0 + ??u?u + ??2 ?? ,
Q8 = ?0 + ??u + ?uf ?1 (?)?? ,
?
where ? is arbitrary constant unit vector. By analogy with one-dimensional case,
?
using operators Qi , i = 1, 8 we can construct ans?tze which reduce system (1) to
a
systems with lesser number of variables.
Let us show some examples. The ans?tze for system (1) that were constructed by
a
? 7 , Q8 , Q5 , respectively, are of the form
??
means of operators Q
? = (?0 (x) ? ?x0 )?1 ,
a)
ua = ?a (x)(?0 (x) ? ?x0 )?1 , a = 1, n;
1
f d? = ?0 (x) + ?a ?a (x)x0 + ? 2 x2 ,
b) 0
2
ua = ?a (x) + ?a x0 , a = 1, n;
v
? = ??0 (?), ? = {?1 , ?2 , ?3 } = {a x, b x, c x},
c)
?
c
?a?1 + b?2 + v (x ?0 + ?3 )?1 , ? = 0,
?
?
? 0
? ?
v
u= 1 2 0 3
?a? + b? + c th ?(x0 ? + ? ), ? = 1,
?
?
?1 v
?
a? + b?2 + c tg ?(?x0 ?0 + ?3 ), ? = ?1,
where x = (x1 , x2 , x3 ); ? ? (?1 )2 + (?2 )2 ? (?0 )?2 ; a, b, c are arbitrary orthonormal
vectors.
456 W.I. Fushchych, N.I. Serov, T.K. Amerov

These ans?tze reduce (1) to the following systems
a
?0 = const,
a)
(? ? + ?)? = 0,
div ? + ? = 0;
? + ??0 + (? ?)? = 0,
b)
div ? = 0,
?a + ?b = 0, a, b = 1, n, a = b;
a
b
for ? = 0; 1; ?1 respectively:
c)
?0 = ?s = ?3 = 0, s = 1, 2,
3 3 3
? ?s = ?? ,
s3 0

?s ?? ?? = 0, s, ? = 1, 2,
s
?1 ? ?1 = 0, ?1 + ?2 = 0.
2
2 1 2
?v
Having defined the potential v = v(?1 , ?2 ), ?s = ??s , s = 1, 2, we can rewrite the
system c) in the following form
?0 = ?3 = ?s = 0, s = 1, 2,
3 3 3
vs ? = ?? ,
3 0
?
? 0,
0 ?2
?(? ) + vs vs = 1,
?
?1,
?v = 0,
vs v? vs ? = 0.
Having got a solution of the system
(6)
?v = 0, vs v? vs ? = 0,
we can write down the solution of system (6) and, using corresponding ansatz, to
construct a solution for system (1). For example we can consider the particular
solution of system (7)
v = ?1 .
It leads to the solution of system (6) v = ?1 , ?3 = ?1 + ?(?2 ), ?0 = 1, ?1 = 1,
?2 = 0.
Using the corresponding ansatz we obtain a solution of system (1) which depends
on arbitrary function ?
v c
? = ?, u = a + v (x0 + ?1 + ?(?2 ))?1 ,
?
where ?1 = a x, ?2 = b x, a, b, c are arbitrary orthonormal vectors.
1. Овсянников Л.В., Лекции по основам газовой динамики, М., Наука, 1981, 368 c.
2. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 c.
3. Фущич В.И., Серова М.М., О максимальной группе инвариантности и общем решении одно-
мерных уравнений газовой динамики, Докл. АН СССР, 1983, 268, № 5, 1102–1104.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 457–463.

On non-local symmetries
of nonlinear heat equation
W.I. FUSHCHYCH, N.I. SEROV, V.A. TYCHININ, T.K. AMEROV
Для нелинейного уравнения теплопроводности приведены нелокальные формулы ра-
змножения и суперпозиции его решений.

1. Introduction. L.V. Ovsiannikov [1] gave the group classification of nonlinear
one-dimensional heat equation
(1)
u0 = ?1 (F (u)u1 ),
?
where u = u(x), x = (x0 , x1 ), uµ = ?µ u, ?µ = ?xµ , µ = 0, 1; F (u) is arbitrary
differentiable function. These results can be formulated as follows:
Theorem 1. The widest algebra of invariance of equation (1) with F (u) = const in
class of S. Lie operators is given by the following basis elements
? ?
(2)
a) ?0 = , ?1 = , D1 = 2x0 ?0 + x1 ?1 ,
?x0 ?x1
if F (u) is arbitrary differentiable function;
2
(3)
b) ?0 , ?1 , D1 , D2 = x1 ?1 + u?u ,
k
if F (u) = ?uk , ?, k are arbitrary constants, not equal to zero;
(4)
c) ?0 , ?1 , D1 , D3 = x1 ?1 + 2?u ,
if F (u) = ? exp u;
3
d) ?0 , ?1 , D1 , D4 = x1 ?1 ? u?u , ? = x2 ?1 ? 3x1 u?u , (5)
1
2
4
if F (u) = ?u? 3 .
It is well known (see for example [3]) that the sequence of transformations
?v(x0 , x1 )
(6)
u(x0 , x1 ) = ,
?x1
(7)
x0 = t, x1 = w(t, x), v = x,
?w(t, x)
(8)
= z(t, x)
?x
do not take out of the equations class (1), i.e. if the sequence of transformations (6),
(7), (8) is carried out then equation (1) goes to the form
zt = ?x (F ? (z)zx ), (9)
Доклады АН Украины, 1992, № 11, С. 27–33.
458 W.I. Fushchych, N.I. Serov, V.A. Tychinin, T.K. Amerov

where

F ? (z) = z ?2 F (z ?1 ). (10)

In this paper transformations (6)–(8) are used to construct nonlocal ans?tzes,
a
which reduce equation (1) to ordinary differential equations (ODE). The generating
and superposition formulas for solutions of equation (1) are given for corresponding
nonlinearities F (u).
2. The equation u0 = ?1 (u?2 u1 ). In the case when F (u) = u?2 the equation
(1) takes the form

u0 = ?1 (u?2 u1 ). (11)

It follows from (10) that equation (11) can be reduced to the linear heat equation by
means of transformations (6)–(8):

(12)
zt = zxx .

As it was established by S. Lie, the widest algebra of invariance of equation (12)
consists of the operators:
? ? 1
G = t?x ? xz?z ,
?t = , ?x = , I = z?z ,
?t ?x 2
(13)
x2
1
P = t t?t + x?x ? z?z ? z?z .
D = 2t?t + x?x ,
2 4
The symmetry of equation (11) is given by only four operators (3), whereas the
symmetry of equation (12) is given by six operators (13). It means that nonlinear
equation (11) has some non-Lie symmetry which cannot be obtained by Lie’s method.
Let us use this fact to construct nonlocal ans?tzes for nonlinear equation (11), i.e.
a
having used operators G, P (13) we will construct non-local ans?tzes for equation
a
(11) by means of transformations (6)–(8). Below we will show only those ans?tzes,
a
which cannot be obtained from Lie symmetry of equation (11)
1
? = ? + x2 ,
u(x0 , x1 ) = , 0
x0 x1 + x1 h(?)
(14)
2
exp x0 ? + x3 ?(?) = x1 ;
30

2(x2 + 1) 1
? = ? (x2 + 1)? 2 ,
0
u(x0 , x1 ) = , 0
2 + 1)1/2 h(?) ? x ?
x1 2(x0 0
(15)
x0 ? 2
exp ? arctg x0 ? ?(?) = x1 (x2 + 1)1/4 .
0
4(x2 + 1)
0

In formulas (14), (15) ? = ? (x0 , x1 ) is functional parameter, functions ?(?) and h(?)
are connected by the relation
?(?)
?
h(?) = .
?(?)
On non-local symmetries of nonlinear heat equation 459

Ans?tzes (14), (15) reduce equation (11) to Riccati equations for unknown function h:
a
?
h + h2 = ?, (16)
2
?
?
h + h2 = ? (17)
+ ?,
4
respectively. Equations (16), (17) being written down for function ?(?), have the form
?2
? ? ?? = 0, ? ? ? = 0.
? ?+
?
4
The solutions of these equations can be expressed only in terms of special functions.
As it follows from transformations (6)–(7) the relation between the solutions of equa-
tions (11) and (12) is given by following formula
?1
?z(x0 , ? )
(18)
u(x0 , x1 ) = ,
??
where ? = ? (x0 , x1 ) is functional parameter, which can be obtained from the relation
(19)
z(x0 , ? ) = x1 .
3. Non-Lie generating of equation solutions. Let us illustrate the process of
finding new solutions by means of formulas (18), (19). The function
x2
z(t, x) = t + ,
2
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