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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 ].
v
In formulas (10), k, k = 1 ? k 2 are the moduli of the corresponding elliptic
Jacobi functions and k is an arbitrary constant satisfying the inequality 0 < k < 1.
Theorem 5. The equation 2u + ch?2 x(m1 + m2 sh x)u = 0 is separated in four
coordinate systems

g2 = ch?2 ?2 (m1 + m2 sh ?2 );
(1) ?1 = t, ?2 = x, g1 = 0,
1/2 1/2
k k
x
= ? ln ? ln cn (?1 ? ?2 ) ,
(2) cn (?1 + ?2 )
t k k
g1 = m1 (k )2 (dn2?1 )?2 + m2 cn 2?1 (dn2?1 )?2 ,
g2 = m1 (k )2 (dn2?2 )?2 + m2 cn 2?2 (dn2?2 )?2 ;

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

1 1 (11)
x
= ln tn (?1 + ?2 ) ± ln dn (?1 + ?2 ).
(4)
t 2 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 .
v
Here, k, k = 1?k 2 are the moduli of the corresponding elliptic functions, 0 < k < 1.
Separation of variables in the two-dimensional wave equation with potential 295

Theorem 6. The equation 2u + sh?2 x(m1 + m2 chx)u = 0 is separated in eleven
coordinate systems
?1 = t, ?2 = x, g1 = 0, g2 = sh?2 ?2 (m1 + m2 ch?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 ?1 , g2 = (m1 + m2 ) sin?2 ?2 ;
?2

1 1
x
= ? ln sh (?1 + ?2 ) ? ln sh (?1 ? ?2 ),
(4)
t 2 2
g1 = sh ?1 (m1 + m2 ch ?1 ), g2 = sh?2 ?2 (m1 ? m2 ch ?2 );
?2

1 1
x
= ? ln ch (?1 + ?2 ) ? ln ch (?1 ? ?2 ),
(5)
t 2 2
g1 = sh ?1 (m1 ? m2 ch ?1 ), g2 = sh?2 ?2 (m1 ? m2 ch ?2 );
?2

1 1
x
= ln th (?1 + ?2 ) ± ln th (?1 ? ?2 ),
(6)
t 2 2
g1 = ch ?1 (m2 ? m1 ), g2 = ?ch?2 ?2 (m2 + m1 );
?2

1 1
x
= ln tg (?1 + ?2 ) ± ln tg (?1 ? ?2 ),
(7)
t 2 2 (12)
g1 = cos?2 ?1 (m1 + m2 ), g2 = cos?2 ?2 (m1 ? m2 );
x
= arth cn (?1 + ?2 ) ± arth 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
= arth dn (?1 + ?2 ) ± arth 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
= arth sn (?1 + ?2 ) ± arth sn (?1 ? ?2 ),
(10)
t
g1 = (m1 + m2 ) sn?2 ?1 + (m1 ? m2 )k 2 sn2 ?1 ,
g2 = (m1 + m2 )k 2 cn2 ?2 dn?2 ?2 + (m1 ? m2 ) 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 is the modulus of the corresponding elliptic functions, 0 < k < 1.
Theorem 7. The equation 2u + ex (m1 + m2 ex )u = 0 is separated in six coordinate
systems
g2 = e?2 (m1 + m2 e?2 );
(1) ?1 = t, ?2 = x, g1 = 0,
296 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

x
= ? ln cos(?1 + ?2 ) ? ln cos(?1 ? ?2 ),
(2)
t
m2
g1 = ?2m1 cos 2?1 ? cos 4?1 ,
2
m2
g2 = ?2m1 cos 2?2 ? cos 4?2 ;
2
x
= ln sh (?1 + ?2 ) ± ln sh (?1 ? ?2 ),
(3)
t
m2
g1 = ?2m1 ch2?1 ? ch 4?1 ,
2
m2
g2 = ?2m1 ch 2?2 ? ch 4?2 ;
2
x
= ln ch(?1 + ?2 ) ± ln ch(?1 ? ?2 ),
(4)
t (13)
m2
g1 = ?2m1 ch 2?1 ? ch 4?1 ,
2
m2
g2 = ?2m1 ch2?2 ? ch 4?2 ;
2
x
= ln ch(?1 + ?2 ) ± ln sh(?1 ? ?2 ),
(5)
t
m2
g1 = ?2m1 sh 2?1 ? ch 4?1 ,
2
m2
g2 = ?2m1 sh 2?2 ? ch 4?2 ;
2
x
= ln(?1 + ?2 ) ± ln(?1 ? ?2 ),
(6)
t
2 2
g1 = 2m1 + 2m2 ?1 , g2 = 2m1 + 2m2 ?2 .
Theorem 8. The equation 2u + (m1 + m2 x?2 )u = 0 separated 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
g1 = 4m1 exp 2?1 , g2 = m2 ch?2 ?2 ;
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
= sh(?1 + ?2 ) ± sh(?1 ? ?2 ),
(4)
t (14)
?2
g1 = 2m1 sh2?1 + m2 sh ?1 ,
g2 = ?2m1 sh2?2 ? m2 sh?2 ?2 ;
x
= ch(?1 + ?2 ) ± ch(?1 ? ?2 ),
(5)
t
g1 = 2m1 ch2?1 ? m2 ch?2 ?1 , g2 = 2m1 ch2?2 ? m2 ch?2 ?2 ;
x
= (?1 + ?2 )2 ± (?1 ? ?2 )2 ,
(6)
t
?2 ?2
g1 = ?16m1 ?1 + m2 ?1 , g2 = ?16m1 ?2 + m2 ?2 .
2 2
Separation of variables in the two-dimensional wave equation with potential 297

It was established in [13] that the Euler–Poisson–Darboux equation
Vtt ? Vxx ? x?1 Vx + m2 x?2 V = 0
is separated in nine coordinate systems. Since the above equation is reduced to the
equation utt ? uxx + (m2 ? 1/4)x?2 u = 0 by the change of dependent variable ?(t, x) =
x?1/2 u(t, x), equation (1) with V (x) = ?x?2 is also separated in nine coordinate
systems.
It has been understood not long ago [6, 14] that SV is intimately connected with
the symmetry properties of the equation under the study. Therefore, it is important
to investigate the symmetry of equation (1).
Clearly, equation (1) with an arbitrary V (x) is invariant under the two-dimensional
Lie algebra that has the basis elements Q1 = ?t , Q2 = u?u . Below, we adduce without
a proof the assertion which gives a complete description of the potentials V (x) that
provide an extension of the symmetry algebra admitted by equation (1).
belonging to the algebra ?t , u?u ) iff the potential V (x) is given by one of the
following formulas:
(1) V (x) = m exp x;
V (x) = mx?2 ;
(2)
V (x) = m sin?2 x;
(3)
V (x) = m sh?2 x;
(4)
V (x) = m ch?2 x;
(5)
V (x) = m, m ? R1 ,
(6)
with the additional symmetry operators having the form
1 1
(t ? x) (?x ? ?t ), Q4 = exp ? (x + t) (?x + ?t );
(1) Q3 = exp
2 2
= x?x + t?t , Q4 = (x2 + t2 )?t + 2tx?x ;
(2) Q3
= sin t cos x?t + sin x cos t?x , Q4 = ? cos t cos x?t + sin x sin t?x ;
(3) Q3
(4) Q3 = sh t ch x?t + sh x ch t?x , Q4 = ch t ch x?t + sh t sh x?x ;
(5) Q3 = sh x ch t?t + sh t ch x?x , Q4 = sh t sh x?t + ch t ch x?x ;
(6) Q3 = ?x , Q4 = t?x + x?t .
This theorem is proved by the standard Lie method (see, e.g., [15, 16]).
Corollary. If equation (1) admits additional symmetry operators, then it is locally
equivalent to one of the equations 2u + mu = 0 or 2u + mx?2 u = 0.
Thus, separability of equations 1, 3–7 from (6) is not connected with their Lie
symmetry. To explain this fact one has to take into account the second-order (non-
Lie) symmetry operators of equation (1). This problem will be briefly discussed in the
last section.
3. Proof of Theorems 1–8. To prove the assertions listed in the previous section
one has to apply the above described procedure of SV to equation (1).
By substituting ansatz (2) into equation (1), expressing the functions ?i in terms
?
of the functions ?i , ?i , with the help of equalities (3), and splitting the obtained
?
298 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

equation with respect to independent variables ?i , ?i , we get the following system of
?
nonlinear partial differential equations:

1) A2?1 + 2(At ?1t ? Ax ?1x ) + AA1 (?1 , ?)(?1t ? ?1x ) = 0,
2 2
(15)

2) A2?2 + 2(At ?2t ? Ax ?2x ) + AA2 (?2 , ?)(?2t ? ?2x ) = 0,
2 2
(16)

3) 2A + A[B1 (?1 , ?)(?1t ? ?1x ) + B2 (? 2 , ?)(?2t ? ?2x ) + AV (x) = 0,
2 2 2 2
(17)

4) ?1t ?2t ? ?1x ?2x = 0. (18)
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