ñòð. 121 |

? ?A?1 (xµ Aµ + B) + A?2 A0 (xµ Aµ + B) ? h(u) =

? ? ?

0 0

= ?(?Aµ Aµ )?1/2 (xµ Aµ + B) ? h(u).

?? ? ?

510 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

??

The only thing left is to prove that Aµ Aµ < 0. Since Aµ Aµ = 0, the equality

? ??

Aµ Aµ = 0 holds. Consequently, by force of the Lemma ?Aµ Aµ ? 0, and what is

?? ?

more, the equality Aµ Aµ = 0 holds if and only if Aµ = k(u)Aµ . General solution of

the above system of ordinary differential equations reads Aµ = l(u)?µ , where l(u) is

an arbitrary function, ?µ are arbitrary real parameters obeying the equality ?µ ?µ = 0.

3

Hence it follows that ?a = Aa A?1 = ?a ?0 = const, and the condition

?1

?2

?a = 0

0

a=1

??

does not hold. We come to the contradiction, whence it follows that Aµ Aµ < 0.

Thus, we have obtained the formula (44). Derivation of the remaining formulas

from (43), (47) is carried out in the same way. The theorems are proved.

Substitution of the results obtained above into the formula (34) yields the following

collection of Ans?tze for the nonlinear d’Alembert equation (33):

a

?1 2

? ? ? ?

?A? (u)A? (u) Aµ (u)xµ + B(u)

(1) w(x) = ? +

?3 2 1/2

? ? ? ?

+ ?A? (u)A? (u) ?µ??? Aµ (u)A? (u)A? (u)x? + C(u) ,u ;

1/2

? ? ? ?

?A? (u)A? (u) Aµ (u)xµ + B(u) , u ;

(2) w(x) = ?

(70)

2 2 1/2

x1 + C1 (x0 ? x3 ) + x2 + C2 (x0 ? x3 ) , x0 ? x3 ;

(3) w(x) = ?

?3/2

? ? ? ?

?A? (u)A? (u) ?µ??? Aµ (u)A? (u)A? (u)x? + C(u) , u ;

(4) w(x) = ?

(5) w(x) = ? x1 cos C1 (x0 ? x3 ) + x2 sin C1 (x0 ? x3 ) + C2 (x0 ? x3 ), x0 ? x3 .

Here B, C, C1 , C2 are arbitrary smooth functions of the corresponding arguments,

Aµ (u) are arbitrary smooth functions satisfying the condition Aµ Aµ = 0 and the

function u = u(x) is determined by JSSF (10) with C1 (u) = B(u), n = 4. Note

that arbitrary functions h contained in the functions v(x) (see above the formulas

(43), (44), (46)) are absorbed by the function ?(v, u) at the expense of the second

argument.

Substitution of the expressions (70) into (33) gives the following equations for

? = ?(u, v):

3

(1) ?vv + ?v = ?F (?), (71)

v

2

(2) ?vv + ?v = ?F (?), (72)

v

1

(3) ?vv + ?v = ?F (?), (73)

v

(4) ?vv = ?F (?), (74)

(5) ?vv = ?F (?), (75)

Equations (4), (5) from (71)–(75) are known to be integrable in quadratures.

Therefore, any solution of the d’Alembert-eikonal system (2) corresponds to some

class of exact solutions of the nonlinear wave equation (33) that contains arbitrary

functions. Saying it in another way, the formulas (70) make it possible to construct

General solution of the d’Alembert equation with a nonlinear eikonal constraint 511

wide families of exact solutions of the nonlinear PDE (33) using exact solutions of the

linear d’Alembert equation 24 u = 0 satisfying an additional constraint uxµ uxµ = 0.

It is interesting to compare our approach to the problem of reduction of Eq. (33)

with the classical Lie approach. Within the framework of the Lie approach functions

?1 (x), ?2 (x) from (34) are looked for as invariants of the symmetry group of the

equation under study (in the case involved it is the Poincar? group P (1, 3)). Since the

e

group P (1, 3) is a finite-parameter group, its invariants cannot contain an arbitrary

function (a complete description of invariants of the group P (1, 3) had been carried

out in [19]). Therefore, the Ans?tze (70) cannot, in principle, be obtained by means

a

of the Lie symmetry of the PDE (33).

All Ans?tze listed in (70) correspond to a conditional invariance of the nonlinear

a

d’Alembert equation (33). It means that for each Ansatz from (70) there exist two

differential operators Qa = ?aµ (x)?xµ , a = 1, 2 such that

Qa w(x) ? Qa ?(?1 , ?2 ) = 0, a = 1, 2

and besides, the system of PDE

24 w ? F (w) = 0, Qa w = 0, a = 1, 2

is invariant in Lie’s sense under the one-parameter groups with the generators Q1 , Q2 .

For example, the fourth Ansatz from (16) is invariant with respect to the operators:

?

Q1 = Aµ (u)?µ , Q2 = Aµ (u)?µ . A direct computation shows that the following rela-

tions hold:

Qi (24 ?) = ?(A? x? + B)?1 Aµ ?µ Qi w,

? ? i = 1, 2,

2

[Q1 , Q2 ] = 0,

where Qi stands for the second prolongation of the operator Qi . Hence it follows

2

that the nonlinear d’Alembert equation (33) is conditionally-invariant under the two-

dimensional commutative Lie algebra having the basis elements Q1 , Q2 (for more

details about conditional symmetry of PDE see [20, 21]). It should be said that the

notion of conditional symmetry of PDE is closely connected with the “non-classical

reduction” [22–24] and “direct reduction” [25] methods.

5 On the new exact solutions

of the nonlinear d’Alembert equation

According to [26], general solutions of Eqs. (74), (75) are given by the following

quadrature:

?1/2

?(u,v) ?

?2 (76)

v + D(u) = F (z)dz + C(u) d?,

0 0

where D(u), C(u) ? C 2 (R1 , R1 ) are arbitrary functions.

Substituting the expressions for u(x), v(x) given by the formulas (4), (5) from

(70) into (76) we obtain two classes of exact solutions of the nonlinear d’Alembert

equation (33) that contain several arbitrary functions of one variable.

512 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Equations (71) and (72) with F (?) = ??k are Emden–Fowler type equations. They

were investigated by many authors (see, e.g. [26]). In particular, it is known that the

equations

?vv + 2v ?1 ?v = ???5 , (77)

?vv + 3v ?1 ?v = ???3 (78)

are integrated in quadratures. In the paper [27] it has been established that Eqs. (77),

(78) possess a Painlev? property. This fact makes it possible to integrate these by

e

applying rather complicated technique. In [28] we have integrated Eqs. (77), (78) using

a standard technique based on their Lie symmetry. Substituting the results obtained

into the corresponding Ans?tze from (70) we get exact solutions of the nonlinear PDE

a

(33) with F (w) = ?w , ?w5 , which include an arbitrary solution of the system (2)

3

with n = 4. Consequently, we have constructed principally new exact solutions of the

nonlinear d’Alembert equation (33) depending on several arbitrary functions. Let us

stress that following the classical Lie symmetry reduction procedure one can not in

principle obtain solutions with arbitrary functions since the maximal symmetry group

of Eq. (33) is finite-dimensional (see, e.g. [16]).

Below we give new exact solutions of the nonlinear d’Alembert equation (33)

obtained with the use of the technique described above. We adduce only those ones

that can be written down explicitly

1. F (w) = ?w3

1

(1) w(x) = v (x2 + x2 + x2 ? x2 )?1/2 ?

1 2 3 0

a2

v

2

? tan ? ln C(u)(x2 + x2 + x2 ? x2 ) ,

1 2 3 0

4

where ? = ?2a2 < 0,

v

22 ?1

C(u) 1 ± C 2 (u)(x2 + x2 + x2 ? x2 )

(2) w(x) = ,

1 2 3 0

a

where ? = ±a2 ;

2. F (w) = ?w5

1/2

w(x) = a?1 (x2 + x2 ? x2 )?1/4 sin ln C(u)(x2 + x2 ? x2 )?1/2 + 1 ?

(1) 1 2 0 1 2 0

?1/2

? 2 sin ln C(u)(x2 + x2 ? x2 )?1/2 ? 4 ,

1 2 0

where ? = a4 > 0,

31/4 ?1/2

w(x) = v C(u) 1 ± C 4 (u)(x2 + x2 ? x2 )

(2) ,

1 2 0

a

where ? = ±a2 .

In the above formulas C(u) is an arbitrary twice continuously differentiable functi-

on on

x0 x1 ± x2 x2 + x2 ? x2

1 2 0

u(x) = ,

x2 + x2

1 2

a = 0 is an arbitrary real parameter.

General solution of the d’Alembert equation with a nonlinear eikonal constraint 513

6 Conclusion

The present paper demonstrates once more that possibilities to construct in explicit

form new exact solutions of the nonlinear d’Alembert equation (33) (as compared

with those obtainable by the standard symmetry reduction technique [16, 19, 27]) are

far from being exhausted. A source of new (non-Lie) reductions is the conditional

symmetry of Eq. (33).

Roughly speaking, a principal idea of the method of conditional symmetries is the

following: to be able to reduce given PDE it is enough to require an invariance of

a subset of its solutions with respect to some Lie transformation group. And what is

more, this subset is not obliged to coincide with the whole set. This specific subsets

can be chosen in different ways: one can fix in some way an Ansatz for a solution to

be found (the method of Ans?tze [16, 17] or the direct reduction method [25]) or one

a

can impose an additional differential constraint (the method of non-classical [22–24]

or conditional symmetries [20, 21]). But all the above approaches have a common

feature: to find new (non-Lie) reduction of a given PDE one has to solve some

nonlinear over-determined system of differential equations. For example, to describe

Ans?tze of the form (34) reducing Eq. (33) to ODE one has to integrate system of five

a

nonlinear PDE (37). This is a “price” to be paid for the new possibilities to reduce a

given nonlinear PDE to equations with less number of independent variables and to

construct its explicit solutions.

As mentioned in the Introduction, the Ansatz (34) can also be interpreted as a map

(more exactly, a family of maps) from the set of solutions of the linear d’Alembert

equation,

24 u = 0 (79)

into the set of solutions of the nonlinear d’Alembert equation (33).

Really, we started with a subset of solutions of Eq. (79) which was chosen by

an additional eikonal constraint uxµ uxµ = 0. Then, we constructed the functions

v(x) and ?(v, u) in such a way that the function w(x) determined by the equality

w = ?(v(x), u(x)) satisfied the nonlinear d’Alembert equation (33) (see below the

Fig. 1).

There is an analogy between the map described above and B?cklund transforma-

a

tions of partial differential equations. System of PDE (38)–(40) and the Ansatz (34)

(level 2 of the Fig. 1) can be interpreted as a B?cklund transformation of a set of

a

solutions of linear PDE (level 1 of the Fig. 1) into a set of solutions of nonlinear

PDE (level 3). A principal difference is that a classical B?cklund transformation acts

a

on the whole spaces of solutions of equations under study and the above map acts

on subsets of solutions of the linear and nonlinear d’Alembert equations. It is known

that technique of linearization of PDE with the use of B?cklund transformations

a

can be effectively applied to two-dimensional equations only. The results obtained in

the present paper imply the following way of extension of applicability of B?cklund

a

transformations: one should consider B?cklund transformations connecting subsets of

a

solutions of linear and nonlinear equations. And these subsets may not coincide with

the whole sets of solutions.

514 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

As an illustration we consider the case when in (33) F (w) = 0, i.e. the case when

the map constructed above transforms a subset of solutions of the linear d’Alembert

equation into another subset of solutions of the same equation. Integrating ODE (71)–

(75) we obtain explicit forms of functions ?(v, u)

?(v, u) = f1 (u)v ?2 + f2 (u),

(1)

?(v, u) = f1 (u)v ?1 + f2 (u),

(2)

(3) ?(v, u) = f1 (u) ln v + f2 (u),

(4) ?(v, u) = f1 (u)v + f2 (u),

(5) ?(v, u) = f1 (u)v + f2 (u),

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