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10 0 0 0 11 0 11 1 1 0 00 0 00
? (u0 u10 ? u0 u10 )(u1 u0 + u0 u1 )]),
10 01 10 10
(31)
f 7 (u0 , ?? ?3 · [(u0 )2 (u0 u1 ? u1 u0 ) + (u0 )2 (u0 u1 ? u1 u0 ) ?
11 0 1 11 1 11 1 1 00 1 00
? 2u1 u0 (u1 u10 ? u1 u10 )]),
00 01 10

f 8 (u0 , ?? ?3 [(u1 )2 (u0 u1 ? u1 u0 ) + (u1 )2 (u0 u1 ? u1 u0 ) ?
00 0 1 11 1 11 1 1 00 1 00
? 2u1 u0 (u1 u10 ? u1 u10 )]),
11 01 10

f 9 (u0 , ?? ?3 [u0 u1 (u0 u1 ? u1 u0 ) + u0 u1 (u0 u1 ? u1 u0 ) ?
10 0 0 1 11 1 11 1 1 1 00 1 00
? (u1 u10 ? u1 u10 )(u1 u0 + u0 u1 )]).
01 10 10 10

All functions f k , k = 1, 9 are arbitrary smooth and symmetric.
So, we now are able to construct the hodograph-invariant system of second order
PDEs

F ? ({f k }) = 0, (32)
k = 1, 9, ? = 1, 2, . . . , N.
110 W.I. Fushchych, V.A. Tychynin

(2) (2) (2)
We construct a new solution u = ( u 0 , u 1 ) of system (32) via known solution
(1) (1) (1)
u = ( u 0 , u 1 ) according to the formula
(2) (1)
u (x) = ?, x = u (? ). (33)
Here x = (x0 , x1 ), ? = (? 0 , ? 1 ), ? µ are parameters to be eliminated out of system (33).
Example 3. Let us consider the simplest hodograph-invariant system of first order
PDE
u1 ? u0 = 0, u1 ? u0 = 0. (34)
0 1 1 0

It is easily to verify, that pair of functions
(1)0 (1)1
u ? x2 + x2
u = 2x0 x1 + c, 0 1

is the solution of system (34). Making use of formula (33) one obtain the new solution
of this system
1
2
1
(2)1
u = ± v x1 ± x2 + (x0 ? c)2 ,
1
2
(35)
?1
x ?c 2
(2)0
u =± 0
v x1 ± x2 + (x0 ? c)2 .
1
2
Let us consider the linear system of first order PDEs
b?? (y)vµ + b?? (y)v ? + c? (y) = 0.
?
(36)
µ

Here b?? , b?? , c? are arbitrary smooth functions of y = (y0 , y1 ), summation over
µ
?
repeated indices is understood in the space with metric gµ? = diag (1, 1). This
system (36) under transformation (28) reduces into system of nonlinear PDEs
b?0 (u)? ?1 u1 ? b?0 (u)? ?1 u0 ? b?1 (u)? ?1 u1 +
1 1 1 0 0
(37)
?1 0
+ b1 (u)? u0 + b (u)x0 + b (u)x1 + c? (u) = 0.
?1 ?0 ?1

The solutions superposition formula for the system (37) has the form
(3) (1) (1) (2)
u 0 (? 0 , ? 1 ) = u 0 (x0 ? ? 0 , x1 ? ? 1 ),
u 0 (x0 , x1 ) = u 0 (? 0 , ? 1 ),
(38)
(3) (1) (1) (2)
u (? , ? ) = u (x0 ? ? , x1 ? ? ).
1 1 0 1 1 0 1 1 0 1
u (x0 , x1 ) = u (? , ? ),

Making use of designations u = (u0 , u1 ), x = (x0 , x1 ), ? = (? 0 , ? 1 ), one can rewrite
the formula (38) in another way:
(3) (1) (1) (2)
u (? ) = u (x ? ? ).
u (x) = u (? ), (38a)
Example 4. It is obviously, that two pairs of functions

1 12
(1)
(1)
? = (2?)?1 x ? x1 ,
u= x0 ,
40
2 (39)
1 (2)
(2)
x?1 ?1
? = (2?x0 )
u= c1 + x1 , c0
0
2
Hodograph transformations and generating of solutions 111

give two partial solutions of the system
u0 + uu1 + 4?2 ??1 = 0,
(40)
?0 + u1 ? + u?1 = 0.
(3) (3)
Let us apply the formula (38) to construct a new solution u , ? via (39). Finally we
get
1
(3)2 (3) (3)
u (x0 , x1 ) ? c2 (x0 ? 2 u (x0 , x1 ))?2 ? x0 u (x0 , x1 ) + x1 + c1 = 0,
2
2
1
2
1
(3) (3) (3)2
? (x0 , x1 ) = (2?)?1 x0 u (x0 , x1 ) ? u (x0 , x1 ) ? x1 ? c1 .
2

1. Forsyth A.R., Theory of differential equations, New York, Dover Publication, 1959, Vol. 5, 478 p.;
Vol. 6, 596 p.
2. Ames W.F., Nonlinear partial differential equations in engineering, New York, Academic Press,
1965, Vol. 1, 511 p.; 1972, Vol. 2, 301 p.
3. Курант Р., Уравнения в частных производных, М., Мир, 1964, 830 с.
4. Фущич В.И., Серов Н.И., Негрупповая симметрия некоторых нелинейных волновых уравнений,
Докл. АН УССР, 1991, № 9, 45–49.
5. Fushchych W.I., Serov N.I., Tychynin V.A., Amerov Т.К., On nonlocal symmetries of nonlinear
heat equation, Докл. АН Украины, Сер. A, 1992, № 11, 27–33.
6. Фущич В.И., Тычинин В.А., О линеаризации некоторых нелинейных уравнений с помощью
нелокальных преобразований, Препринт № 82.33, Киев, Ин-т математики АН УССР, 1982, 53 c.
7. Фущич В.И., Тычинин В.А., Жданов Р.З., Нелокальная линеаризация и точные решения не-
которых уравнений Монжа–Ампера, Дирака, Препринт № 85.88, Киев, Ин-т математики АН
УССР, 1985, 28 c.
8. Тычинин В.А., Нелокальная линеаризация и точные решения уравнения Борна–Инфельда и
некоторых его обобщений, в сб. Теоретико-групповые исследования уравнений математической
физики, Киев, Ин-т математики АН УССР, 1986, 54–60.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 112–115.

New conditionally invariant solutions
for non-linear d’Alembert equation
W.I. FUSHCHYCH, I.A. YEGORCHENKO
We describe all ansatzes of a specific form that reduce the non-linear d’Alembert equa-
tion. In this way we obtain some new solutions of the equation with a polynomial
non-linearity.

1. Introduction. Let us consider a non-linear d’Alembert equation of the form
2u = ?uk , (1)
where u = u(x0 , x1 , x2 , x3 ) is a real function; k = 1, ? are parameters,
?2u ?2u ?2u ?2u
2u ? ? ? ? .
?x2 ?x2 ?x2 ?x2
0 1 2 3

?
Equation (1) is invariant under the Poincar? algebra AP (1, 3) + D with the follo-
e
wing basis operators:
Jab = xa ?b ? xb ?a ,
?0 , ?a ,J0a = x0 ?a + xa ?0 ,
(2)
2
D = x0 ?0 + xa ?a + u?u ,
1?k
when k is arbitrary, k = 1. Here a, b = 1, 2, 3, and we imply summation over the
repeated indices from 1 to 3. We shall not consider here the special case k = 3 when
equation (1) is invariant under the conformal algebra.
All similarity solutions for equation (1) are adduced in [1, 2]. The similarity
ansatzes corresponding to three-dimensional subalgebras of the algebra (2) have the
form
(3)
u = f (x)?(?),
where f (x) is some function, ? = ?(x) is a new invariant variable.
In this paper we try to search for a wider class of solutions than similar ones
by means of the ansatz (3). Some ansatzes of this form were described in [3]. An
example of such ansatz is
u = (x2 )?1/2 ?(?x), (4)
where x2 = x2 ? xa xa , ?0 ? ?a ?a = 0.
2
0
The substitution (3) reduces equation (1) to an ordinary differential equation of the
functions f and ? satisfy the following set of equations:
2f = f k S(?),
(5)
2fµ ?µ + f 2(?) = f k T (?), ?µ ?µ = R(?)f k?1 .
Preprint LiTH-MAT-R-93-07, Department of Mathematics, Link?ping University, Sweden, 9 p. (revised
o
version).
New conditionally invariant solutions for non-linear d’Alembert equation 113

Here fµ ? ?xµ , the summation over the repeated Greek indices is as follows:
?f

fµ ?µ ? f0 ?0 ? fa ?a , a = 1, 2, 3; S, T , R are some functions; T and R do not
vanish simultaneously.
Further we shall consider the system (5) for the case ?µ ?µ = 0.
2. New ansatzes for the d’Alembert equation (1). We succeeded to find all
solutions of the system (5) for ? = ?x, ?2 = 0. In this case the system (5) reduces
to the equations
2f = f k S(?x), 2fµ ?µ = f k T (?x).
Its solutions have the following form:
1
f = [h(?, ?x, ?x) + ?x)] 1?k , (6)
where the parameters ?µ , ?µ , ?µ , ?µ satisfy the relations ?? = ?? = ? 2 = ?? = 0,
?? = ?? 2 = ?? 2 = 1.
1 (?x)2 (? + B1 ) + 2B3 (?x)(?x) + (?x)2 (? + B2 )
2
(7)
h= ,
(? + B1 )(? + B2 ) ? B3
2
2
(?x)2
(8)
h= ,
2? + B1
2
B1
(9)
h = B1 ?x + B2 + ?.
2
Here B1 , B2 , B3 are some constants. If B1 = B2 , B3 = 0 we get an ansatz that is
equivalent to (4).
3. Operators of conditional symmetry for equation (1). The notion of conditi-
onal symmetry had been defined in [2, 4–6]. This approach enabled to construct wide
classes of exact solutions for nonlinear partial differential equations of mathematical
physics (see [2, 4–6, 8]). In this paper we do not search specially for operators of
conditional symmetry but for ansatzes of the form (3) explicitly.
The following statement describes the operators of conditional invariance corres-
ponding to ansatzes of the form (3) with ? = ?x, ?2 = 0, f being of the form (6), (7).
Theorem 1. Equation (1) with the additional conditions
L1 = f ?µ uµ ? ?µ fµ u = 0,
L2 = f ?µ uµ ? ?µ fµ u = 0, (10)
L3 = 2?µ uµ (1 ? k) ? f k?1 u = 0
is invariant under operators:
Q1 = r(x)(f ?µ ?xµ ? ?µ fµ u?u) = 0,
Q2 = r(x)(f ?µ ?xµ ? ?µ fµ u?u = 0, (11)
Q3 = r(x)(2?µ ?xµ ? 1?k f k?1 u?u = 0,
1

where r(x) is an arbitrary non-zero function, f satisfies the equations
1
2f = f k S(?),
f k, (12)
fµ ?µ =
1?k
where S is some function.
114 W.I. Fushchych, I.A. Yegorchenko

The above theorem can be proved by means of the Lie algorithm (see e.g. [7]).
Note 1. The same ansatzes may also be obtained from the Lie symmetry operators.
4. Exact solutions of equation (1). The ansatz (3) with ? = ?x, ?2 = 0, f of the
form (6), (7) reduces equation (1) to the following ordinary differential equation:
2
+ S(?)? = ??k , (13)
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