ñòð. 32 |

(6)

t

?2 ?2

g1 = ?16m1 ?1 + m2 ?1 , g2 = ?16m1 ?2 + m2 ?2 .

2 2

We now give a sketch of the proof of the above assertions. Substituting ansatz (7)

into (1), expressing functions ?i via functions ?1 , ?i by means of equalities (8) and

? ?

splitting the equation obtained with respect to independent variables ?i , ?i we obtain

?

the following system of nonlinear PDE:

Q2?i + 2(Qt ?it ? Qx ?ix ) + QAi (?i , ?)(?it ? ?ix ) = 0,

2 2

(21)

(1) i = 1, 2;

2Q + Q[B1 (?1 , ?)(?1t ? ?1x ) + B2 (?2 , ?)(?2t ? ?2x )] + QV (x) = 0; (22)

2 2 2 2

(2)

?1t ?2t ? ?1x ?2x = 0. (23)

(3)

Here 2 = ?t ? ?x .

2 2

Thus, to separate variables in the linear PDE (1) one has to construct the general

solution of the system of nonlinear equations (21)–(23). The same assertion holds true

for any general linear differential equation, i.e. the problem of SV is an essentially

nonlinear one.

It is not difficult to become convinced of the fact that, from (23), it follows that

(?1t ? ?1x )(?2t ? ?2x ) = 0.

2 2 2 2

(24)

Differentiating (21) with respect to ? and using (24) we obtain

A1? = A2? = 0,

whence B1? B2? = 0. Differentiating (22) with respect to ?, we have

B1? (?1t ? ?1x ) + B2? (?2t ? ?2x ) = 0

2 2 2 2

or

? 2 ? ?2x

2

B1?

= ? 2t 2.

?1t ? ?1x

2

B2?

Differentiation of the above equality with respect to ? yields

B1?? B2? ? B1? B2?? = 0

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

or

B1?? B2??

= .

B1? B2?

Since functions B1 = B1 (?1 ), B2 = B2 (?2 ) are independent, there exists a func-

tion ?(?) such that

Bi?? = ?(?)Bi? , i = 1, 2.

Integrating the above differential equation with respect to ? we obtain

Bi (?i ) = ?(?)fi (?i ) + gi (?i ), i = 1, 2,

where fi , gi are arbitrary smooth functions.

On redefining the parameter ? > ?(?), we have

(25)

Bi (?i ) = ?fi (?i ) + gi (?i ).

Substitution of (25) into (22) with subsequent splitting with respect to ? yields the

following equations:

2Q + Q[g1 (?1t ? ?1x ) + g2 (?1t ? ?1x )] + V (x)Q = 0,

2 2 2 2

(26)

f1 (?1t ? ?1x ) + f2 (?2t ? ?2x ) = 0.

2 2 2 2

(27)

Thus, system (21)–(23) is equivalent to the system of equations (21), (23), (26), (27).

Before integrating, we make a remark: it is evident that the structure of ansatz (7) is

not altered by transformation

Q > Q = Qh1 (?1 )h2 (?2 ), ?i > ?i = Ri (?i ), (28)

i = 1, 2,

where hi , Ri are smooth-enough functions. This is why solutions of the system under

study connected by relations (28) are considered to be equivalent.

Choosing the functions hi , Ri in a proper way, we can put in (21) and (27)

f1 = f2 = 1, A1 = A2 = 0.

Consequently, functions ?1 , ?2 satisfy equations of the form

?1t ?2t ? ?1x ?2x = 0, ?1t ? ?1x + ?2t ? ?2x = 0,

2 2 2 2

whence

(?1 ± ?2 )2 ? (?1 ± ?2 )2 = 0.

t x

Integrating the above equations, we obtain

?2 = F (?) ? G(?), (29)

?1 = F (?) + G(?),

where F, G ? C 2 (R1 , R1 ) are arbitrary functions, ? = (x + t)/2, ? = (x ? t)/2.

Substitution of (29) into (21) with A1 = A2 = 0 yields the following equations:

(ln Q)t = 0, (ln Q)x = 0,

whence Q = 1.

Orthogonal and non-orthogonal separation of variables in the wave equation 135

Finally, substituting the results obtained into (26), we have

dF dG

V (x) = [g1 (F + G) ? g2 (F ? G)] (30)

.

d? d?

Thus, the problem of integrating an over-determined system of nonlinear differen-

tial equations (21)–(23) is reduced to integration of the functional-differential equa-

tion (30).

Let us summarize the results obtained. The general form of solution of (1) with

separated variables is as follows

u = ?(F (?) + G(?))?2 (F (?) ? G(?)) (31)

where ?i are arbitrary solutions of (14), functions F (?), G(?), g1 (F + G), g2 (F ? G)

being determined by (30).

To integrate Eq. (31) we make the hodograph transformation

(32)

? = P (F ), ? = R(G),

? ?

where P ? 0, R ? 0.

After making the transformation (32), we obtain

? ?

g1 (F + G) ? g2 (F ? G) = P (F )R(G)V (P + R). (33)

Evidently, equation (33) is equivalent to the following equation:

? ?

(?F ? ?G )[P (F )R(G)V (P + R)] = 0

2 2

or

... ...

(P P ?1 ? R R?1 )V + 3(P ? R)V + (P 2 ? R2 )V = 0.

? ? ? ?? ? ?? (34)

Thus, to integrate (30) it is enough to construct all functions P (F ), R(G), V (P + R)

satisfying (34) and to substitute them into (33).

In [8] we have proved the following assertion:

Lemma. The general solution of (34) determined up to transformation (10) is given

by one of the following formulae:

? ?

V = V (x) is an arbitrary function, P = ?,

(1) R = ?;

? ?

V = mx, P 2 = ?P + ?, R2 = ?R + ?;

(2)

V = mx?2 , P = Q1 (F ), R = Q2 (G),

(3)

?

Q2 = ?Q4 + ?Q3 + ?Q2 + ?Q1 + ?,

1 1 1 1

(35)

?

Q2 = ?Q2 ? ?Q2 + ?Q2 ? ?Q2 + ?;

2 4 3

2

V = m sinh?2 x,

(4) P = arctanh Q1 (F ), R = tan Q2 (G)

and Q1 , Q2 are determined by (35);

V = m sinh?2 x,

(5) P = arctanh Q1 (F ), R = arctanh Q2 (G)

and Q1 , Q2 are determined by (35);

V = m cosh?2 x,

(6) P = arccoth Q1 (F ), R = arctanh Q2 (G)

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

and Q1 , Q2 are determined by (35);

(7) V = m exp x,

? ?

P 2 = ? exp 2P + ? exp P + ?, R2 = ? exp 2R + ? exp R + ?;

V = cos?2 x(m1 + m2 sin x),

(8)

? ?

P 2 = ? sin 2P + ? cos 2P + ?, R2 = ? sin 2R + ? cos 2R + ?;

V = cosh?2 x(m1 + m2 sinh x),

(9)

? ?

R2 = ? sinh 2R ? ? cosh 2R + ?;

P 2 = ? sinh 2P + ? cosh 2P + ?,

V = sinh?2 x(m1 + m2 cosh x),

(10)

? ?

R2 = ?? sinh 2R + ? cosh 2R + ?;

P 2 = ? sinh 2P + ? cosh 2P + ?,

(11) V = (m1 + m2 exp x) exp x,

? ? ? ?

P = ?P 2 + ?, R = ?R2 + ?;

V = m1 + m2 x?2 ,

(12)

? ?

R2 = ?R2 ? ?R + ?,

P 2 = ?P 2 + ?P + ?,

(13) V = m,

? ?

P 2 = ?P 2 + ?P + ?, R2 = ?R2 + ?R + ?.

Here ?, ?, ?, ?, ?, m1 , m2 , m are arbitrary real parameters; x = ? + ? = P + R.

Theorems 1 and 2 are direct consequences of the above Lemma. To prove Theo-

rems 3–8 one has to integrate the ODE for P (F ), R(G) and substitute the expressions

obtained into formulae (32)

1 1

(x + t) = P (F ) ? P ((?1 + ?2 )/2), (x ? t) = R(G) ? R((?1 ? ?2 )/2)

2 2

and into (33).

Thus, the problem of separation of the wave equation (1) into two second-order

differential equations is completely solved.

Since all coordinate systems ?1 , ?2 satisfy equation (23), we have orthogonal

separation of variables. To obtain non-orthogonal coordinate systems providing sepa-

rability of (1) one has to carry out SV following Definition 2.

3. Non-orthogonal separation of variables in equation (1)

Utilizing the SV procedure in (1) determined by Definition 2, we come to the

following assertions (corresponding computations are omitted).

Theorem 9. Equation (1) admits SV in the sense of Definition 2 iff it is locally-

equivalent to one of the following equations:

2u + mu = 0;

(1)

2u + mx?2 u = 0,

(2)

where m is an arbitrary real constant.

Orthogonal and non-orthogonal separation of variables in the wave equation 137

Theorem 10. Equation 2u + mu = 0 separates in two coordinate systems

(1) ?1 = ?, ?2 = ? + ?,

?1 = ???1 , ?2 = ??2 + m?2 ;

? ? ?

(2) ?1 = ?, ?2 = ln ? + ln ?,

?1

?1 = ???1 ?1 ,

? ?2 = ??2 + m exp(?2 )?2 .

? ?

Theorem 11. Equation 2u + mx?2 u = 0 separates in eight coordinate systems

(1) ?1 = ?, ?2 = ? + ?,

?2

?1 = ???1 , ?2 = ??2 + m?2 ?2 ;

? ? ?

(2) ?1 = ?, ?2 = arctan ? + arctan ?,

?1 = ??(1 + ?1 )?1 , ?2 = ??2 + m sin?2 ?2 ?2 ;

2

? ? ?

(3) ?1 = ?, ?2 = arctanh ? + arctanh ?,

?1 = ??(1 ? ?1 )?1 ?1 , ?2 = ??2 + m sinh?2 ?2 ?2 ;

2

? ? ?

(4) ?1 = ?, ?2 = arccoth ? + arccoth, ?,

?1 = ?(1 ? ?1 )?1 ?1 , ?2 = ??2 + m sinh?2 ?2 ?2 ;

2

? ? ?

(5) ?1 = ?, ?2 = arctanh ? + arctanh ?,

?1 = ??(1 ? ?1 )?1 ?1 , ?2 = ??2 ? m cosh?2 ?2 ?2 ,

2

? ? ?

(6) ?1 = ?, ?2 = arccoth ? + arccoth ?,

?1 = ?(1 ? ?1 )?1 ?1 , ?2 = ??2 ? m cosh?2 ?2 ?2 ;

2

? ? ?

1

?1 = ?, ?2 = (ln ? ? ln ?),

(7)

2

?1 = ??(2?1 ) ?1 , ?2 = ??2 ? m cosh?2 ?2 ?2 ;

?1

? ? ?

?1 = ?, ?2 = ? ?1 + ? ?1 ,

(8)

?2 ?2

?1 = ??1 ?1 ,

ñòð. 32 |