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rem 1.
Theorem 2. Provided equation (1) admits non-trivial SV in the sense of Definition 1,
it is locally equivalent to one of the following equations:

2u + mxu = 0;
(1)
2u + mx?2 u = 0;
(2)
2u + cos?2 x(m1 + m2 sin x)u = 0;
(3)
2u + cosh?2 x(m1 + m2 sinh x) = 0;
(4)
(12)
2u + sinh?2 x(m1 + m2 cosh x) = 0;
(5)
2u + exp x(m1 + m2 exp x)u = 0;
(6)
2u + (m1 + m2 x?2 )u = 0;
(7)
2u + mu = 0.
(8)

Thus, there exist eight inequivalent types of equations of the form (1) admitting
non-trivial SV.
It is well known that there are 11 coordinate systems providing separability of the
Klein–Gordon–Fock equation 2u + mu = 0 into two second-order ODE [6]. Besides
130 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

that, in [14] it was established that the Euler–Poisson–Darboux equation (6), which
is equivalent to the second equation of (12), separates in nine coordinate systems.
That is why cases V (x) = m and V (x) = mx?2 are not considered here.
As is shown below, the general form of solution with separated variables of (12) is
as follows:

(13)
u(t, x) = ?1 (?1 (t, x))?2 (?2 (t, x)),

where ?1 (?1 ), ?2 (?2 ) are arbitrary solutions of the separated ODE

(14)
?i = (? + gi (?i ))?i ,
? i = 1, 2

and explicit forms of the functions ?i (t, x), gi (?i ) are given below.
Theorem 3. Equation 2u + mxu = 0 separates in two coordinate systems
(1) ?1 = t ?2 = x, g1 = 0, g2 = m?2 ;
?1 = (x + t)1/2 + (x ? t)1/2 , ?2 = (x + t)1/2 ? (x ? t)1/2 ,
(2) (15)
1 1
g1 = ? m?1 , g2 = ? m?2 .
4 4
4 4
Theorem 4. Equation 2u + sin?2 x(m1 + m2 cos x)u = 0 separates in four coordinate
systems
?1 = t, ?2 = x, g1 = 0, g2 = cos?2 ?2 (m1 + m2 sin ?2 );
(1)
x
= arctan sinh(?1 + ?2 ) ± arctan sinh(?1 ? ?2 ),
(2)
t
g1 = (m1 + m2 ) sinh?2 ?1 , g2 = ?(m1 ? m2 ) cosh?2 ?2 ;
sn (?1 ? ?2 )
sn (?1 + ?2 )
x
± arctan
(3) = arctan ,
cn (?1 ? ?2 )
t cn (?1 + ?2 )
(16)
g1 = m1 dn2 ?1 cn?2 ?1 sn?2 ?1 + m2 [cn?2 ?1 ? dn2 ?1 cn?2 ?1 ],
g2 = m1 k 4 sn2 ?2 cn2 ?2 dn?2 ?2 + m2 k 2 [cn2 ?2 dn?2 ?2 ? sn2 ?2 ];
1/2 1/2
k k
x
cn (?1 + ?2 ) ± arctan cn (?1 ? ?2 ),
(4) = arctan
t k k
g1 = m1 [dn2 ?1 cn?2 ?1 + k 2 sn2 ?1 ] + m2 [(k )2 cn?2 ?1 + k 2 cn2 ?1 ],
g2 = m1 [dn2 ?2 cn?2 ?2 + k 2 sn2 ?2 ] + m2 [(k )2 cn?2 ?2 + k 2 cn2 ?2 ].

In the above formulae (16) k, k = (1?k 2 )1/2 are the moduli of corresponding elliptic
Jacobi functions, and k is an arbitrary constant satisfying the inequality 0 < k < 1.
Theorem 5. Equation 2u + cosh?2 x(m1 + m2 sinh x)u = 0 separates in four coordi-
nate systems

g2 = cosh?2 ?2 (m1 + m2 sinh ?2 );
(1) ?1 = t, ?2 = x, g1 = 0,
1/2 1/2
k k
t
= ? ln cn (?1 + ?2 ) ± ln cn (?1 ? ?2 ) ,
(2)
x k k
g1 = m1 (k )2 dn?2 2?1 + m2 cn 2?1 dn?2 2?1 ,
g2 = m1 (k )2 dn?2 2?2 + m2 cn 2?2 dn?2 2?2 ;
Orthogonal and non-orthogonal separation of variables in the wave equation 131

1 1
x
= ? ln sinh (?1 + ?2 ) ± ln cosh (?1 ? ?2 ),
(3)
t 2 2
g1 = cosh ?1 (m1 ? m2 sinh ?1 ), g2 = cosh?2 ?2 (m1 ? m2 sinh ?2 );
?2

sn 1 (?1 + ?2 ) 1 (17)
x
± ln dn (?1 + ?2 ),
2
(4) = ln 1
t 2
cn 2 (?1 + ?2 )
g1 = ?m1 k 2 sn2 ?1 + k 2 m2 sn ?1 cn ?1 ,
g2 = ?m1 k 2 sn2 ?2 + k 2 m2 sn ?2 cn ?2 .

Here k, k = (1?k 2 )1/2 are the moduli of corresponding elliptic functions, 0 ? k ? 1.
Theorem 6. Equation 2u + sinh?2 x(m1 + m2 cosh x)u = 0 separates in eleven
coordinate systems:

?1 = t, ?2 = x, g1 = 0, g2 = sinh?2 ?2 (m1 + m2 cosh ?2 );
(1)
1 1
x
= ? ln (?1 + ?2 ) ± ln (?1 ? ?2 ),
(2)
t 2 2
?2 ?2
g1 = (m1 ? m2 )?1 , g2 = (m1 + m2 )?2 ;
1 1
x
= ? ln sin (?1 + ?2 ) ± ln sin (?1 ? ?2 ),
(3)
t 2 2
g1 = (m1 ? m2 ) sin?2 ?1 , g2 = (m1 + m2 ) sin?2 ?2 ;
1 1
t
= ? ln sinh (?1 + ?2 ) ± ln sinh (?1 ? ?2 ),
(4)
x 2 2
g1 = sinh ?1 (m1 + m2 ) cosh ?1 ), g2 = sinh?2 ?2 (m1 ? m2 cosh ?2 );
?2

1 1
t
= ? ln cosh (?1 + ?2 ) ± ln cosh (?1 ? ?2 ),
(5)
x 2 2
g1 = sinh?2 ?1 (m1 ? m2 cosh ?1 ), g2 = sinh?2 ?2 (m1 ? m2 cosh ?2 );
1 1
x
= ln tanh (?1 + ?2 ) ± ln tanh (?1 ? ?2 ),
(6)
t 2 2 (18)
g1 = cosh?2 ?1 (m1 ? m2 ), g2 = ? cosh?2 ?2 (m1 + m2 );
1 1
x
= ln tan (?1 + ?2 ) ± ln tan (?1 ? ?2 ),
(7)
t 2 2
g1 = cos ?1 (m1 + m2 ), g2 = cos?2 ?2 (m1 ? m2 );
?2

x
= arctanh cn (?1 + ?2 ) ± arctanh cn (?1 ? ?2 ),
(8)
t
g1 = (m1 + m2 ) dn2 ?1 cn?2 ?1 + (m1 ? m2 )k 2 sn2 ?1 ,
g2 = (m1 ? m2 ) dn2 ?2 cn?2 ?2 + (m1 + m2 )k 2 sn2 ?2 ;
x
= arctanh dn (?1 + ?2 ) ± arctanh dn (?1 ? ?2 ),
(9)
t
g1 = (m1 + m2 )k 2 cn2 ?1 dn?2 ?1 + (m ? m2 )k 2 sn2 ?1 ,
g2 = (m1 ? m2 )k 2 cn2 ?2 cn?2 ?2 + (m1 + m2 )k 2 sn2 ?2 ;
x
= arctanh sn (?1 + ?2 ) ± arctanh sn (?1 ? ?2 ),
(10)
t
132 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

g1 = (m1 + m2 ) sn?2 ?1 + (m1 ? m2 )k 2 sn2 ?1 ,
g2 = (m1 + m2 )k 2 cn2 ?2 dn?2 ?2 + (m1 ? m2 )k 2 dn2 ?2 cn?2 ?2 ;
x
= ± ln cn (?1 + ?2 ) ± ln cn (?1 ? ?2 ),
(11)
t
g1 = ?m1 sn?2 ?1 ? m2 cn ?1 sn?2 ?1 ,
g2 = ?m1 sn?2 ?2 ? m2 cn ?2 sn?2 ?2 .

Here k are the moduli of corresponding elliptic functions, 0 < k < 1.
Theorem 7. Equation 2u + exp x(m1 + m2 exp x)u = 0 separates in six coordinate
systems:

(1) ?1 = t, ?2 = x, g1 = 0, g2 = exp ?2 (m1 + m2 exp ?2 );
x
= ? ln cos(?1 + ?2 ) ± ln cos(?1 ? ?2 ),
(2)
t
1
g1 = ?2m1 cos 2?1 ? m2 cos 4?1 ,
2
1
g2 = ?2m1 cos 2?2 ? m2 cos 4?2 ;
2
x
= ln sinh(?1 + ?2 ) ± ln sinh(?1 ? ?2 ),
(3)
t
1
g1 = ?2m1 cosh 2?1 ? m2 cosh 4?1 ,
2
1
g2 = ?2m1 cosh 2?2 ? m2 cosh 4?2 ;
2
x
= ln cosh(?1 + ?2 ) ± ln cosh(?1 ? ?2 ), (19)
(4)
t
1
g1 = ?2m1 cosh 2?1 ? m2 cosh 4?1 ,
2
1
g2 = ?2m1 cosh 2?2 ? m2 cosh 4?2 ;
2
x
= ln cosh(?1 + ?2 ) ± ln sinh(?1 ? ?2 ),
(5)
t
1
g1 = ?2m1 sinh 2?1 ? m2 cosh 4?1 ,
2
1
g2 = ?2m1 sinh 2?2 ? m2 cosh 4?2 ;
2
x
= ln(?1 + ?2 ) ± ln(?1 ? ?2 ),
(6)
t
g1 = 2m1 + 2m2 ?1 , g2 = ?2m1 + 2m2 ?2 .
2 2

Theorem 8. Equation 2u + (m1 + m2 x?2 )u = 0 separates in six coordinate systems:
?2
(1) ?1 = t, ?2 = x, g1 = 0, g2 = m1 + m2 ?2 ;
x
= exp(?1 + ?2 ) ± exp(?1 ? ?2 ),
(2)
t
g2 = m2 cosh?2 ?2 ;
g1 = 4m1 exp 2?1 ,
Orthogonal and non-orthogonal separation of variables in the wave equation 133

x
= sin(?1 + ?2 ) ± sin(?1 ? ?2 ),
(3)
t
g1 = 2m1 cos 2?1 + m2 sin?2 ?1 , g2 = ?2m1 cos 2?2 + m2 cos?2 ?2 ;
x
= sinh(?1 + ?2 ) ± sinh(?1 ? ?2 ),
(4)
t
g1 = 2m1 sinh 2?1 + m2 sinh?2 ?1 ,
g2 = ?2m1 sinh 2?2 ? m2 sinh?2 ?2 ; (20)
x
= cosh(?1 + ?2 ) ± cosh(?1 ? ?2 ),
(5)
t
g1 = 2m1 cosh 2?1 ? m2 cosh?2 ?1 , g2 = 2m1 cosh 2?2 ? m2 cosh?2 ?2 ;
x
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