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equations of the second order

? ? 2 4b(b ? 1) + ? ?[4b2 ? 2b(m + 1) + 4a(1 ? 2b)] +
+ 2a?(2a + m ? 1) = ??F (?, ?? ), (29)
F = (??? )k for (10), F = (?2 + ??2 )k for (11).

We can adduce the parametrical solutions of (29) when
2a
? = ?? (?2 = 0, ? = ?1 ),
b= ,
m + 4a ? 1
?1/2
? 2k+2 ?c2
+ 21 + c2 + Br2
?= r dr + c3 ,
k+1 r
(30)
?1/2
? 2k+2 ?c2
r?2 + 21 + c2 + Br2
? = c1 r dr + c4 ,
k+1 r
1
(r + 4a ? 1)2 ,
B=
4
? = ?1 and the representation (14) for ? is taken for (10), and ? = 1 and the
representation (15) for ? is taken for (11).
3. The conformally invariant families of solutions
Let us consider the conformally invariant system of the form (2) (n = 2)

u R u R
2u? = u?3 F ?
2u = u3 F (31)
, , , ,
u? (uu? )3 u? (uu? )3
where R is defined in (4).
We obtain here the conformally invariant families of solutions of (31) with certain
F using the formulae of conformal reproduction of solutions.
We used the ans?tze (20), where
a

f = (x2 )?1/2 , (32a)
? = ?x,

f = (x2 )?1/2 ,
? = ?x/x2 , (32b)

f = [x2 ? 2?x?x + ? 2 (?x)2 ]?1/2 , (33a)
? = ?x,

f = [2?x?x ? ? 2 (?x)2 + c(?x)]?1/2 , (33b)
? = ?x,

where ?2 = ?2 = ?? = ?? = 0, ?? = ?? = 1.
When u, u? are defined from the ans?tze (20), (32) and (33), R ? 0. Then the
a
reduced equations have the form

? ?
??? ? 2?? ? = ??3 F ?
? ?
?? ? 2?? = ?3 F (34)
, ,
?? ??
where ? = ?1 for (32), ? = 1 for (33).
The solution of (34) in parametrical form can be obtained for arbitrary F .
The symmetry and exact solutions of the non-linear d’Alembert equation 571

The multiparametrical conformally invariant families of solutions we adduce for
the equations are
2u = (u2 + u?2 )(g1 u + ig2 u? ),
(35)
2u? = (u2 + u?2 )(g1 u? ? ig2 u),

2u = (g1 + ig2 )u(uu? ),
(36)
2u? = (g1 ? ig2 )u? (uu? ),

where g1 , g2 are real functions of R(u2 + u?2 )?3 .
Their families of solutions are non-reproducible by conformal transformations and
given by the following formulae. The solutions of (35) are
1+i
u = f (x)? ?/2 c2 |c1 + ?? ? A1 |?(A1 +A2 )/2A1 +
4
1?i
+ c?1 |c1 + ?? ? A1 |?(A1 ?A2 )/2A1 ,
2
4
cj ? R,
Aj = g j (o), A2 = 0, c1 = 0 (j = 1, 2)
and the solutions of (36) are
A2
u = f (x)? ?/2 |A1 ?? ? + c1 |?1/2 exp i c2 ? ln |c1 + A1 ?? ? | ,
2A1
f (x) and ? being substituted from Table 2 and ? being defined from the corresponding
ans?tze (32) or (33).
a
Table 2

{f (x)}?2
?
Ansatz ?
?x + ?? x2 + ab?(?, x)
?(?, x)[x2 + 2bx + 2b? x2 + b2 ?(?, x)]
?1
(32b)
x2 + 2bx + 2b? x2 + b2 ?(?, x)
?1
(32a)
??(?, x)[2(?x + ?? x2 ) ? ? 2 (?x+
?1 2
1 (?(?, x)) [?x+?? x +?b?(?, x)]
(33b)
+?? x2 ) + (c + 2b? ? ? 2 ?b)?(?, x)]
?(?x + ?? x2 )(?x + ?? x2 ) + ? 2 (?x+
+?? x2 )2 + ?(?, x)[x2 + 2bx + 2b? x2 ?
?2b?(?x + ?? x2 ) ? 2b?(?x + ?? x2 )+
1
(33a)
+? 2 2?b(?x + 2?? x2 )+
+?(?, x)(b2 ? 2b?b? + ? 2 (?b)2 ]

?(?, x) = 1 + 2? x + ? 2 x2 , bµ , ?µ are arbitrary parameters.

1. Bogolubov N.N., Shirkov D.V., Introduction into the theory of quantized fields, Moscow, Nauka,
1973.
2. Collins C.B., Math. Proc. Camb. Phil. Soc., 1976, 80, 165–187.
3. Fushchych W.I., in Algebraic-Theoretical Studies in Mathematical Physics, Kyiv, Institute of Ma-
thematics, 1981, 6–28.
4. Fushchych W.I., Serov N.I., J. Phys. A: Math. Gen., 1983, 16, 3645–3656.
572 W.I. Fushchych, I.A. Yehorchenko

5. Fushchych W.I., Yehorchenko I.A., Dokl. Akad. Nauk USSR, 1988, 298, 347–351.
6. Grundland A.M., Tuszynski J.A., J. Phys. A: Math. Gen., 1987, 20, 6243–6258.
7. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978.
8. Winternitz P., Grundland A.M., Tuszynski J.A., J. Math. Phys., 1987, 28, 2194–2212.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 573–576.

On some new exact solutions of
the nonlinear d’Alembert–Hamilton system
W.I. FUSHCHYCH, R.Z. ZHDANOV
Some new exact solutions of the d’Alembert–Hamilton system of partial differential
equations are obtained. The necessary conditions of integrability of an over-determined
d’Alembert–Hamilton system are established.

Since Euler (1734–1740) the method of reduction of partial differential equations
(PDEs) to ordinary differential equations (ODEs) is one of the most effective ways to
construct partial solutions of PDEs.
In refs. [1–4] the symmetry reduction of the d’Alembert equation,

2u = F1 (u), 2 = ?x0 ? ?x1 ? ?x2 ? ?x3
2 2 2 2
(1)

(where F1 (u) is an arbitrary smooth function), to an ODE has been carried out. So
the four-dimensional PDE (1) with the ansatz

(2)
u(x) = ?(?),

where ? ? C 2 (R1 , R1 ), and ? = ?(x) ? C 2 (R4 , R1 ) being the new variable, is reduced
to an ODE having variable coefficients,

(3)
(?µ ?µ )? + (2?)? = F1 (?),
? ?

where ?µ ? ??/?xµ , µ = 0, . . . , 3, ? ? d?/d?. Hereafter the summation over
?
repeated indices in Minkowski space R(1, 3) having the metric gµ? = diag (1, ?1, ?1,
?1) is supposed, i.e.

?µ ?µ = gµ? ?µ ?? = ?1 ? ?1 ? ?2 ? ?3 .
2 2 2 2

In refs. [3, 4] using the symmetry properties of eq. (1) and the subgroup structure
of the Poincar? group P (1, 3) new variables have been constructed for eq. (3) dependi-
e
ng on ? only.
In the present paper we suggest a more general approach to the problem of
reduction of the PDE (1) to an ODE than the approach based on the employment of
its symmetry properties [1–4].
We say that the ansatz (2) reduces the PDE (1) to an ODE if the new variable
?(x) satisfies both the d’Alembert and the Hamilton equation,

2? = F2 (?), (4)

(5)
?µ ?µ = F3 (?),

where F2 , F3 are arbitrary smooth functions.
Physics Letters A, 1989, 141, ¹ 3–4, 113–115.
574 W.I. Fushchych, R.Z. Zhdanov

Evidently for every ?(x) satisfying the system (4), (5) the ODE (3) depends on ?
only

(6)
F3 (?)? + F2 (?)? = F1 (?)
? ?

(one can be easily convinced that the invariants obtained by Winternitz et al. [4]
satisfy this system). Thus the problem of finding the ansatz (2) reducing the PDE (1)
to an ODE leads to the construction of solutions of the d’Alembert–Hamilton system
(4), (5).
In the present paper the compatibility of the overdetermined system (4), (5) is
investigated, i.e. all smooth functions ensuring the compatibility of the d’Alembert–
Hamilton system are described. Besides wide classes of exact solutions of the system
(4), (5) are presented.
System (4), (5) via the change of the dependent variable Z = Z(?) can be reduced
to the allowing system:

2? = F (?), (7)

(8)
?µ ?µ = ?, ? = const.

The ODE (6) then takes the form

(9)
?? + F (?)? = F1 (?).
? ?

Before formulating the principal result of the paper we adduce without proof some
auxiliary statements.
Lemma 1. Solutions of the system (7), (8) satisfy the identities
?
?µ?1 ??1 µ = ??F (?),
1 ?
?µ?1 ??1 ?2 ??2 µ = ?2 F (?), . . . , (10)
2!
dn F (?)
1
?µ?1 ??1 ?2 · · · ??n µ = ?n ,
d? n
n!
where ??? ? ? 2 ?/?x? ?x? , ?, ? = 0, . . . , 3, n ? 1.
Lemma 2. Solutions of the system (7), (8) satisfy the following equality:

(10 )
det(?µ? ) = 0.

Let us now formulate the principal statement.
Theorem 1. The necessary condition of compatibility of the overdetermined system
(7), (8) is
?
?0,
?
?
??(? + C )?1 ,
1
(11)
F (?) =
?2?(? + C1 )[(? + C1 )2 + C2 ]?1 ,
?
?
?
3?[(? + C1 )2 + C2 ][(? + C1 )3 + 3C2 (? + C1 ) + C3 ]?1 .

where C1 , C2 , C3 are arbitrary constants.
On some new exact solutions of the nonlinear d’Alembert–Hamilton system 575

Proof. By direct (and rather tiresome) verification one can be convinced that the
following identity holds,
6(?µ?1 ??1 ?2 ??2 ?3 ??3 µ ) ? 8(2?)(?µ?1 ??1 ?2 ??2 µ ) ? 3(?µ?1 ??1 µ )2 +
(12)
+ 6(2?)2 (?µ?1 ??1 µ ) ? (2?)4 = 24 det(??? ).
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