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? ? ?? ? ? (44)
By differentiating (44) with respect to f and g, we obtain
(P + R)(h1 P ?1 ? h2 R?1 ) = 2(h1 ? h2 ),
?? ??
... ...
where h1 = P P ?1 and h2 = R R?1 .
? ?
Differentiation of the above equation with respect to f and g yields the following
relation:
(h1 P ?1 )· P ?1 = (h2 R?1 )· R?1 .
?? ??
? ? (45)
Since the functions P (f ), R(g) are independent, it follows from (45) that the
equalities
(h1 P ?1 )· = 12?P , (h2 R?1 )· = 12?R.
?? ??
? ? (46)
hold, where ? is an arbitrary real parameter.
Integration of equations (46) yields
?
P = ?P 4 + C1 P 3 + C2 P 2 + C3 P + C4 ,
?
R2 = ?R4 + D1 R3 + D2 R2 + D3 R + D4 ,
where C1 , . . . , C4 , D1 , . . . , D4 are arbitrary real constants. Substituting the above
result into the initial equation (44), we get restrictions on the choice of arbitrary
constants
C1 = ?D1 = ?, C3 = ?D3 = ?,
C2 = D2 = ?, C4 = D2 = ?.
Thus, we have obtained the potential listed in the lemma under number 3.
It is straightforward to verify that the equations obtained by the substitution of
functions V = m sin?2 H, V = m ch?2 with H = P (f ) + R(g) into (37) are reduced
to equation (44) by the following changes of variables:
P > arctg P, R > arctg R,
P > arth P, R > arth R,
P > arcth P, R > arcth R;
i.e., the potentials listed in the lemma under numbers 4–6 are obtained.
Equation (1) with the potential V = m exp H is reduced to the Klein–Gordon–Fock
equation (see case 4 and Note 2 below).
?
Case 3. Let V = 0 and assume, in addition, that equation (41) does not hold.
In this case, we can exclude from equations (37), (40) the third derivatives of the
function H
Hf f ? Hgg + A(H)(Hf ? Hg ) = 0,
2 2
(47a)
where
.... ... ...
A(H) = ( V ?2 V V V ?1 ? 4V 2 V + 5V V 2 V ?2 )(V ?4V V V ?1 + 3V 3 V ?2 )?1 .
? ? ?? ?? ?
Separation of variables in the two-dimensional wave equation with potential 303

It follows from (47a)
? ?
Hf f f = AHf (Hg ? Hf ) ? 2Hf f Hf A, Hggg = AHg (Hf ? Hg ) ? 2Hgg Hg A,
2 2 2 2

(we have used equation (38)).
By taking the first differential consequence of the above equations with account of
equation (38), we get
? ?2
2A(Hf f ? Hgg ) + A(Hf ? Hg ) = 0.
2
(48)

Clearly, equations (47a) and (48) are consistent iff the function A(H) satisfies the
following ordinary differential equation:
? ?
A = 2AA,

the general solution of which is given by one of the formulas (up to scaling H > CH).
A = C, A = tg (H + C), A = ?th (H + C),
A = ?cth (H + C), A = ?(H + C)?1 , C ? R1 .

Next, we consider the above cases separately.
Case 3.1. A(H) = C, C = 0. In this case, equation (47a) takes the form

Pf f ? Rgg + C(Pf ? Rg ) = 0
2 2
(47b)

or

? ? R1 .
2 2
Pf f + CPf = Rgg + CRg = ?,

Finally, we get

Pf f = ?CPf + ?, Rgg = ?CRg + ?.
2 2
(49)

Differentiating the first equation with respect to f , the second equation with
respect to g, and subtracting, we get
?1 ?1
Pf f f Pf ? Pggg Pg = ?2C(Pf f ? Rgg ). (50)

Substituting (49), (50) into equation (37), we come to the equation for V = V (H),
? ?
V ? 3C V + 2C 2 V = 0

the general solution of which reads

m2 , m2 ? R1 . (51)
V = m1 exp CH + m2 exp 2CH,

It is not difficult to check that function (51) satisfies equation (47b) provided that
A(H) = C. Consequently, if the potential is given by formula (51) (after rescaling
H > CH, we can choose C = 1), then the functions P (f ), R(g) are determined by
equations (35).
Case 3.2. A = tg(H + C). Multiplying equation (47) by ctg (H + C) and differen-
tiating the obtained expression with respect to f and g, we arrive at the equation
?1 ?1
(Pf f f Pf ? Pggg Pg ) ? 2ctg (H + C)(Pf f ? Rgg ) = 0. (52)
304 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

After excluding the function ctg (H + C) from (47) and (52), we get an equation
with separated variables
?1 ?1
(Pf f f Pf ? Pggg Pg ) + 2(Pf ? Rg ) = 0,
2 2

whence
?1 ?1
2 2
(53)
Pf f f Pf + 2Pf = ?, Rggg Rg + 2Rg = ?.
In (53), ? is an arbitrary real constant.
Substitution of formulas (52), (53) into equation (37) gives the equation for V =
V (H),
? ?
V ? 3 tg (H + C)V ? 2V = 0
the general solution of which has the form [17]
V = cos?2 (H + C)[m1 + m2 sin(H + C)]. (54)
As a direct check shows, the function V (H) (54) satisfies equation (47b) with
A = tg (H + C).
Integrating equations (53), we get
2 2
(55)
Pf = C1 sin 2P + C2 cos 2P + ?, Pg = D1 sin 2R + D2 cos 2R + ?,
where Ci , Di , and ? are arbitrary real constants.
Substitution of (55) into (47) with A = tg (H + C) yields the following restrictions
on the choice of the constants Ci , Di : C1 = D1 = ?, C2 = D2 = ?.
Thus, provided that the function V (H) is given by (44), the functions P (f ), R(g)
are determined by equations (32).
Case 3.3. A = ?th (H + C). In this case, one can obtain the following differential
consequence of equation (47):
?1 ?1
Pf f f Pf ? Pggg Pg = 2 cth (P + R + C)(Pf f ? Rgg ). (56)
Excluding the function ctg (H + C) from equations (47), (56), we get the equation
?1 ?1
Pf f f Pf ? Pggg Pg = 2(Pf ? Rg ),
2 2

whence
?1 ?1
Pf f f Pf ? 2Pf = ?, Rggg Rg ? 2Rg = ?.
2 2
(57)
In (57), ? is an arbitrary real constant.
Integration of equations (57) gives
2 2
(58)
Pf = C1 sh 2P + C2 ch 2P + ?, Rg = D1 sh 2R + D2 ch 2R + ?,
where Ci , Di , and ? are arbitrary real constants.
Substituting expressions (56), (57) into (37), we obtain an equation for V (H),
? ?
V + 3 th (H + C)V + 2V = 0,
the general solution of which has the form [17]
V = ch?2 (H + C)(m1 + m2 sh(H + C)), mi ? R1 . (59)
Separation of variables in the two-dimensional wave equation with potential 305

It is not difficult to become convinced of the fact that function (59) satisfies
equation (47b) with A = ?th (H + C).
At last, substituting (57) and (58) into (47), we get C1 = D1 = ?, C2 = ?D2 = ?.
Consequently, if the potential V (H) is given by formula (59), then functions P (f )
and R(g) are determined by equations (33).
Case 3.4. A = ?cth (H + C). In this case, one can obtain the following differential
consequence of equation (47):
?1 ?1
Pf f f Pf ? Rggg Rg = 2 th (P + R + C)(Pf f ? Rgg ). (60)
Using equations (37), (47), and (60), we get an equation for V (H),
? ?
V + 3 cth (H + C)V + 2V = 0,
the general solution of which has the form [17]
V = sh?2 (H + C)(m1 + m2 ch(H + C)), mi ? R1 . (61)
By direct computation, one can check that function (61) satisfies equation (47b)
with A = ?cth (H + C).
Next, by eliminating the function th (H + C) from equations (47) and (60), we get
an equation with separated variables
?1 ?1
Pf f f Pf ? Pggg Pg ? 2Pf + 2Rg = 0,
2 2

whence
?1 ?1
Pf f f Pf ? 2Pf = ?, Rggg Rg ? 2Rg = ?.
3 2

Here, ? is an arbitrary real constant.
Integration of the above ordinary differential equations shows that the functions
P (f ) and R(g) are determined by equations (58), where Ci , Di , and ? are arbitrary
real constants. Substituting (58) into equation (47), we have the following restrictions
on the choice of Ci , Di :
C1 = ?D1 = ?, C2 = D2 = ?.
Thus, if the function V (H) is given by (61), then functions P (f ) and R(g) are
determined by equations (34).
Case 3.5. A = ?(H + C)?1 . In this case, it follows from (47a) that the equality
?1 ?1
Pf f f Pf = Rggg Rg holds. Hence, we get equations for P (f ), R(g),
(62)
Pf f f = ?Pf , Rggg = ?Rg
? ?
with arbitrary ? ? R1 . Moreover, the equation for V (H) has the form V +3(H+C)V =
0, whence
V = m1 + m2 (H + C)?2 , mi ? R1 . (63)
It is not difficult to check that function (63) satisfies (47b) with A = ?(H + C)?1 .
Integration of equations (62) yields the following result:
Pf = ?P 2 + C1 P + C2 ,
2
Rg = ?R2 + D1 R + D2 ,
2
(64)
here ?, Ci , and Di are arbitrary real constants.
306 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

Next, substituting (64) into (47), we get C1 = ?D1 = ?, C2 = D2 = ?.
Thus, if the potential V is given by (63), then the functions P (f ), R(g) are
determined by equations (36).
?1
Case 4. V (H) = m = const. In this case, equation (37) reads Pf f f Pf =
?1
Rggg Rg , whence

(65)
Pf f f = ?Pf , Rggg = ?Rg ,

where ? ? R1 is an arbitrary constant.
Integrating (65), we get equations listed in the lemma under number 13.
??
Case 5. V (H) is an arbitrary function. In this case, the coefficients of V , V , V
in (37) must vanish. Consequently, the relations
?1 ?1 2 2
Hf f f Hf = Hggg Hg , Hgg = Hf f , Hf = Hg

hold. Hence, we have Hf = ?, Hg = ?, ? ? R1 . The lemma is proved.
Theorems 1, 2 are direct consequences of the above lemma. To prove Theorems
3–8, one has to integrate ordinary differential equations (30), (32)–(36) and substitute
the obtained expressions into (27),
?1 ? ?2
1 ?1 + ?2 1
(x + t) = P (f ) ? P (x ? t) = R(g) ? R
, ,
2 2 2 2
and (28).
Integration of equations (30), (32)–(36) is carried out in a standard way [17, 18],
the obtained result depends essentially on relations between parameters ?, ?, ?, ?, ?.
This procedure demands very cumbersome computations; this is why we omit details.
With the above remarks, the proof of Theorems 1–8 is completed.
4. Discussion. Let us say a few words about intrinsic characterization of SV in
equation (1). It is well known that the solution of a second-order partial differential
equation with separated variables is a joint eigenfunction of mutually commuting
second-order symmetry operators of the equation under study (for more details, see [6,
10, 14]). Below we construct, in an explicit form, a second-order symmetry operator
of equation (1) such that the solution with separated variables is its eigenfunction and
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