metry algebra isomorphic to sl(2,R) was obtained for the reduced equations. ... tions such as the Telegrapher's equation,10 the EPD equation is the only PDE for ...
JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 45, NUMBER 8
AUGUST 2004
Symmetry operators for Riemann’s method Peter J. Zeitsch School of Mathematics and Statistics, University of Sydney, NSW 2006, Sydney, Australia
共Received 26 January 2004; accepted 13 April 2004; published online 25 June 2004兲 Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1763003兴 I. INTRODUCTION
The most general self-adjoint partial differential equation 共PDE兲 for which the Riemann function is known is that derived by Chaundy,1 U rs ⫹
冋
册
m 1 共 1⫺m 1 兲 m 2 共 1⫺m 2 兲 m 3 共 1⫺m 3 兲 m 4 共 1⫺m 4 兲 ⫺ ⫹ ⫺ U⫽0, 共 r⫹s 兲 2 共 r⫺s 兲 2 共 1⫺rs 兲 2 共 1⫹rs 兲 2
共1兲
where m 1 ,...,m 4 are real valued constants. For this equation the Riemann function, R(r,s,r 0 ,s 0 ), is given by R 共 r,s,r 0 ,s 0 兲 ⫽F B 共 m 1 ,m 2 ,m 3 ,m 4 ,1⫺m 1 ,1⫺m 2 ,1⫺m 3 ,1⫺m 4 ,1,z 1 ,z 2 ,z 3 ,z 4 兲 ,
共2兲
where 共 r⫺r 0 兲共 s⫺s 0 兲 , 共 r⫹s 兲共 r 0 ⫹s 0 兲
z 2⫽
共 r⫺r 0 兲共 s⫺s 0 兲 , 1⫺rs 兲共 1⫺r 0 s 0 兲 共
z 4⫽
z 1 ⫽⫺
z 3 ⫽⫺
共 r⫺r 0 兲共 s⫺s 0 兲 , 共 r⫺s 兲共 r 0 ⫺s 0 兲
共3兲
共 r⫺r 0 兲共 s⫺s 0 兲 , 1⫹rs 兲共 1⫹r 0 s 0 兲 共
共4兲
and F B is a Lauricella hypergeometric function of four variables.2 Recall that for a self-adjoint equation such as 共1兲, the Riemann function must satisfy3 L 关 R 兴 ⫽0,
R ⫽0 r R ⫽0 s
on
s⫽s 0 , 共5兲
on
r⫽r 0 ,
R 共 r 0 ,s 0 ,r 0 ,s 0 兲 ⫽1. 2993
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When m 1 (1⫺m 1 )⫽m 3 (1⫺m 3 )⫽m 4 (1⫺m 4 )⫽0, Chaundy’s PDE simplifies to the Euler– Poisson–Darboux 共EPD兲 equation which is the original problem solved by Riemann. For the EPD equation, the Riemann function F B reduces to the standard hypergeometric function R 共 r,s,r 0 ,s 0 兲 ⫽ 2 F 1 共 m 2 ,1⫺m 2 ,1,z 2 兲 .
共6兲
It is well known4,5 that the EPD equation possesses a three-dimensional Lie algebra of first-order symmetry operators isomorphic to sl共2,R兲. From standard results in Lie theory,5 the group SL(2,R) acts locally on the solution space of the EPD equation by U 共 r,s 兲 →U
冉
冊 冉 冊
␣ r⫹  ␣ s⫹  , , ␥ r⫹ ␦ ␥ s⫹ ␦
␣

␥
␦
苸SL共 2,R 兲 .
共7兲
Using 共7兲, it is straightforward to show that the Riemann function for the EPD equation can be represented in terms of the hypergeoemtric function 共6兲. This was first shown by Bluman.7 Subsequently Daggit8 and Ibragimov9 extended this concept to more general problems. They considered a generic PDE, to which they could impose restrictions on the coefficients, so that a symmetry algebra isomorphic to sl共2,R兲 was obtained for the reduced equations. This led to invertible mappings from the candidate equations to the EPD equation, likewise for the solutions. In this way an extensive equivalence class of Riemann functions was derived. Apart from confluent contractions such as the Telegrapher’s equation,10 the EPD equation is the only PDE for which this approach has been shown to work. A few calculations reveal that 共1兲 possesses no nontrivial first-order symmetry operators whatsoever. Hence the theory developed for sl共2,R兲 is not applicable to Chaundy’s equation. However in this paper we show that if the symmetry operators are extended to second-order,6 then Chaundy’s full PDE in fact admits a two-dimensional vector space that leads to 4 inequivalent orbits. For each of these orbits there exists a separable coordinate system for 共1兲. They constitute a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via higher order symmetries. Two of the separable forms of 共1兲 are completely new while the other two generalize several Riemann functions already found in the literature. In Sec. II, the symmetries for 共1兲 are calculated. In Sec. III the corresponding separable coordinate systems and separable forms of Chaundy’s equation are found. In Sec. IV a new 5 parameter Riemann function is derived by combining two of the separable forms of Chaundy’s equation in an addition formula due to Olevski.11 The five parameter Riemann function is important as it incorporates one more essential parameter than Chaundy’s equation. II. SYMMETRY CALCULATIONS
Standard techniques6 exist in the literature for seeking separable forms of 共1兲. Following Ref. 6 we define the second-order operator S⫽ f 1 rr ⫹ f 2 ss ⫹ f 3 r ⫹ f 4 s ⫹ f 5 .
共8兲
We say that 共8兲 is a symmetry operator for 共1兲 provided 关 S,L 兴 ⫽QL,
共9兲
where L⫽ rs ⫹
冋
m 1 共 1⫺m 1 兲 m 2 共 1⫺m 2 兲 m 3 共 1⫺m 3 兲 m 4 共 1⫺m 4 兲 ⫺ ⫹ ⫺ 共 r⫹s 兲 2 共 r⫺s 兲 2 共 1⫺rs 兲 2 共 1⫹rs 兲 2
册
共10兲
and Q⫽h 1 r ⫹h 2 s ⫹h 3
共11兲
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Symmetry operators for Riemann’s method
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is a first-order differential operator 共here Q may vary with S兲 and f 1 ,...,h 3 are arbitrary functions of r and s. Evaluating 共9兲 produces 10 equations for the unknown functions f 1 ,...,h 3 . The solution of these equations is long and involved. When m 1 (1⫺m 1 ),...,m 4 (1⫺m 4 ) are all nonzero, solving 共9兲 yields a two-dimensional vector space of operators. The symmetries, S 1 and S 2 , form a basis and have differential part S 1 ⫽ 共 r 4 ⫹1 兲 rr ⫹ 共 s 4 ⫹1 兲 ss ⫹2r 3 r ⫹2s 3 s ,
共12兲
S 2 ⫽r 2 rr ⫹s 2 ss ⫹r r ⫹s s .
共13兲
Importantly, there are no first-order symmetries, which means that Chaundy’s equation is Class II in the sense of Miller.6 The dimension of the vector space now varies depending on the pivots for the constants m 1 ,...,m 4 . However the only new case is that stated above, where m 1 (1⫺m 1 ),...,m 4 (1⫺m 4 ) are all nonzero. All other cases reduce to equations for which the symmetries are well documented. The pivots found during the calculation of 共9兲 fall into two types m i 共 1⫺m i 兲 ⫽0,
i⫽1,...,4
and m i 共 1⫺m i 兲 ⫽m j 共 1⫺m j 兲 ,
i, j⫽1,...,4;
i⫽ j
or a combination of both. Letting any one of m 1 (1⫺m 1 ),...,m 4 (1⫺m 4 ) equal to zero does not increase the dimension of the vector space from two. The next possibility is when any two of m 1 (1⫺m 1 ),...,m 4 (1⫺m 4 ) equal zero. This leads to 6 possible equations. However as Chaundy pointed out,1 if we apply the discrete group 共 r,s 兲 ⫽ 共 ⫺R,S 兲 , 共 r,s 兲 ⫽
共 r,s 兲 ⫽
冉
冉 冊
1 ,S , R
共14兲
冊
1⫺R 1⫺S , , 1⫹R 1⫹S
to 共1兲, then the suffixes, 共1,2,3,4兲 on the constants m, become 共2,1,4,3兲, 共4,3,2,1兲 or 共3,2,1,4兲, respectively. Hence the 6 PDEs can all be mapped to the harmonic equation U rs ⫹
冋
册
m 1 共 1⫺m 1 兲 m 2 共 1⫺m 2 兲 ⫺ U⫽0. 共 r⫹s 兲 2 共 r⫺s 兲 2
共15兲
The Riemann function for 共15兲 has been documented by Henrici.12 More recently Iwasaki10 solved 共15兲 in terms of a system of F 4 functions.2 The symmetries for 共15兲 have been studied by Kalnins and Miller.13 In effect the vector space is four-dimensional with 3 second order symmetries and 1 first order symmetry. For more detail, see Ref. 13. The next pivot occurs when we let m 1 (1⫺m 1 )⫽m 2 (1⫺m 2 ) and m 3 (1⫺m 3 )⫽m 4 (1⫺m 4 ) 共or any combination of two constants兲 in 共1兲 to obtain U rs ⫺4rs
冋
册
m 1 共 1⫺m 1 兲 m 3 共 1⫺m 3 兲 ⫺ U⫽0. 共 r 2 ⫺s 2 兲 2 共 1⫺r 2 s 2 兲 2
As Chaundy pointed out, now make the change of variables r⫽R 1/2, s⫽S 1/2 which gives
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U RS ⫺
冋
P. J. Zeitsch
册
m 1 共 1⫺m 1 兲 m 3 共 1⫺m 3 兲 ⫺ U⫽0. 共 R⫺S 兲 2 共 1⫺RS 兲 2
Then applying the discrete group 共14兲, we arrive at the Harmonic equation 共15兲. Setting three constants equal to zero leads to the EPD equation in all cases under the action of the discrete group 共14兲. As shown in Ref. 4 this leads to an eight-dimensional vector space, and nine inequivalent separable coordinate systems. Hence we continue by focusing our attention on the new case when all constants are arbitrary: the two-dimensional vector space with symmetry operators 共12兲 and 共13兲.
III. THE SEPARABLE COORDINATE SYSTEMS
Taking a linear combination of the operators 共12兲 and 共13兲, there are then four inequivalent orbits namely, • S 1 ⫹2qS 2 , • S 1 ⫹2S 2 , • S 1 ⫺2S 2 , • S 1 ⫹2qS 2 ,
q⬎1,
q⬍⫺1,
where q is a real valued constant. Alternatively consider S 2 by itself. In this case the discrete symmetry 共14兲 maps S 2 to S ⬘2 ⫽
R 共 R 2 ⫺1 兲 S 共 S 2 ⫺1 兲 共 R 2 ⫺1 兲 2 共 S 2 ⫺1 兲 2 RR ⫹ SS ⫹ R⫹ S, 4 4 2 2
which is equivalent to S 1 ⫺2S 2 . Also this discrete symmetry maps the case when ⫺1⬍q⬍1 to the case when q⬎1 above. Hence we may ignore these possibilities and conclude that, in total, there are 4 inequivalent orbits for the two symmetries S 1 and S 2 . We are now in a position to calculate the separable coordinate systems and to analyze their effect on 共1兲. System 1: S 1 ⫹2qS 2 , where q⬎1. The separable coordinates are r⫽b
sn 关 a 共 ⫹ 兲兴 , cn 关 a 共 ⫹ 兲兴
s⫽b
sn 关 a 共 ⫺ 兲兴 , cn 关 a 共 ⫺ 兲兴
共16兲
where 2q⫽(1⫹b 4 )/b 2 , k 2 ⫽1⫺b 4 , 0⬍b⬍1 and k is the modulus of the Jacobian elliptic functions.14 When 共16兲 is substituted into 共1兲 the following separable equation is found:
再冋
U ⫺U ⫹ a 2 m 1 共 1⫺m 1 兲 ⫺m 2 共 1⫺m 2 兲
冋
冉
dn 2 a sn 2 a cn 2 a ⫺k 4 2 sn a cn a dn 2 a 2
⫹4a 2 b 2 m 3 共 1⫺m 3 兲 ⫺m 4 共 1⫺m 4 兲
冉
冉
dn 2 a sn 2 a cn 2 a 4 ⫺k sn 2 a cn 2 a dn 2 a
冉
冊册
冊
dn 2 a dn 2 共 a 兲 ⫹ ⫺1 共 cn 2 a ⫺b 2 sn 2 a 兲 2 共 cn 2 a ⫹b 2 sn 2 a 兲 2
dn 2 a dn 2 a ⫹ ⫺1 共 cn 2 a ⫹b 2 sn 2 a 兲 2 共 cn 2 a ⫺b 2 sn 2 a 兲 2
冊 册冎
冊
U⫽0.
The Riemann function for 共17兲 is obtained by substituting 共16兲 into 共3兲, 共4兲 and 共2兲. Hence
共17兲
J. Math. Phys., Vol. 45, No. 8, August 2004
Symmetry operators for Riemann’s method
R 共 , , 0 , 0 兲 ⫽F B 共 m 1 ,m 2 ,m 3 ,m 4 ,1⫺m 1 ,1⫺m 2 ,1⫺m 3 ,1⫺m 4 ,1,z 11 ,z 12 ,z 13 ,z 14兲 ,
2997
共18兲
where z 5⫽
共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 ⫺ 共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 , 4sn a sn a 0 cn a cn a 0 dn a dn a 0
z 6⫽
共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 ⫺ 共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 , 4sn a sn a 0 cn a cn a 0 dn a dn a 0
z 7 ⫽b 2
共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 ⫺ 共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 , 共 cn 2 a ⫺b 2 sn 2 a 兲共 cn 2 a ⫹b 2 sn 2 a 兲共 cn 2 a 0 ⫺b 2 sn 2 a 0 兲共 cn 2 a 0 ⫹b 2 sn 2 a 0 兲
z 8 ⫽b 2
共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 ⫺ 共 sn a cn a 0 dn a 0 ⫺sn a 0 cn a dn a 兲 2 . 共 cn 2 a ⫹b 2 sn 2 a 兲共 cn 2 a ⫺b 2 sn 2 a 兲共 cn 2 a 0 ⫹b 2 sn 2 a 0 兲共 cn 2 a 0 ⫺b 2 sn 2 a 0 兲
To the best of the author’s knowledge, Chaundy’s equation written as 共17兲 and the associated Riemann function 共18兲 have not previously been published, including subcases. They are completely new. System 2: For the symmetry S 1 ⫹2S 2 we find the separable coordinate system
冋
r⫽tan a
册
共⫹兲 , 2
冋
s⫽tan a
册
共⫺兲 . 2
共19兲
Substituting 共19兲 into 共1兲 yields U ⫺U ⫹a 2
冋
册
m 1 共 1⫺m 1 兲 m 2 共 1⫺m 2 兲 m 3 共 1⫺m 3 兲 m 4 共 1⫺m 4 兲 ⫺ ⫹ ⫺ U⫽0. sin2 a sin2 a cos2 a cos2 a
共20兲
The Riemann function for 共20兲 is then R 共 , , 0 , 0 兲 ⫽F B 共 m 1 ,m 2 ,m 3 ,m 4 ,1⫺m 1 ,1⫺m 2 ,1⫺m 3 ,1⫺m 4 ,1,z 9 ,z 10 ,z 11 ,z 12兲 ,
共21兲
where z 9⫽
cos a 共 ⫺ 0 兲 ⫺cos a 共 ⫺ 0 兲 , 2 sin a sin a 0
z 10⫽
cos a 共 ⫺ 0 兲 ⫺cos a 共 ⫺ 0 兲 , 2 cos a cos a 0
z 12⫽
z 11⫽
cos a 共 ⫺ 0 兲 ⫺cos a 共 ⫺ 0 兲 , 2 sin a sin a 0
共22兲
cos a 共 ⫺ 0 兲 ⫺cos a 共 ⫺ 0 兲 . 2 cos a cos a 0
共23兲
System 3: For the symmetry S 1 ⫺2S 2 we find the separable coordinate system
冋
r⫽tanh a
册
共⫹兲 , 2
冋
s⫽tanh a
册
共⫺兲 . 2
共24兲
Substituting 共24兲 into 共1兲 yields U ⫺U ⫹a 2
冋
册
m 1 共 1⫺m 1 兲 m 2 共 1⫺m 2 兲 m 3 共 1⫺m 3 兲 m 4 共 1⫺m 4 兲 ⫺ ⫹ ⫺ U⫽0. sinh2 a sinh2 a cosh2 a cosh2 a
共25兲
The Riemann function for 共25兲 is now R 共 , , 0 , 0 兲 ⫽F B 共 m 1 ,m 2 ,m 3 ,m 4 ,1⫺m 1 ,1⫺m 2 ,1⫺m 3 ,1⫺m 4 ,1,z 13 ,z 14 ,z 15 ,z 16兲 ,
共26兲
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P. J. Zeitsch
where
z 13⫽
cosh a 共 ⫺ 0 兲 ⫺cosh a 共 ⫺ 0 兲 , 2 sinh a sinh a 0
z 14⫽
cosh a 共 ⫺ 0 兲 ⫺cosh a 共 ⫺ 0 兲 , 2 sinh a sinh a 0
共27兲
z 15⫽
cosh a 共 ⫺ 0 兲 ⫺cosh a 共 ⫺ 0 兲 , 2 cosh a cosh a 0
z 16⫽
cosh a 共 ⫺ 0 兲 ⫺cosh a 共 ⫺ 0 兲 . 2 cosh a cosh a 0
共28兲
For 共25兲 the Riemann function when m 2 (1⫺m 2 )⫽m 3 (1⫺m 3 )⫽m 4 (1⫺m 4 )⫽0 was first published by Cohn15 but no connection to Chaundy or the EPD equation was made. The two parameter equation that results when m 3 (1⫺m 3 )⫽m 4 (1⫺m 4 )⫽0 was derived by Kalnins13 although the focus was not on Riemann functions but rather separation of variables. The full equation 共25兲 was first derived by Papadakis and Wood16 but no connection to symmetry operators was made. System 4: For the symmetry S⫽S 1 ⫹2qS 2 , where q⬍⫺1 we find the separable coordinate system r⫽b sn a 共 ⫹ 兲 ,
s⫽b sn a 共 ⫺ 兲 ,
共29兲
where k⫽b 2 , 2q⫽⫺(1⫹b 4 )/b 2 , and 0⬍b⬍1. As in system 1, a is arbitrary and k is the modulus of the elliptic functions. Substituting 共29兲 into 共1兲 yields
再冋
U ⫺U ⫹ a 2 m 1 共 1⫺m 1 兲
冉
冊
cn 2 a dn 2 a sn 2 a 2 2 ⫺ 1⫺k ⫺m 2 共 1⫺m 2 兲 兲 共 sn 2 a cn 2 a dn 2 a
⫻
冉
⫹
cn 2 a dn 2 a cn 2 a dn 2 a cn 2 a dn 2 a ⫺1 ⫺m 1⫺m ⫹ ⫺1 兲 共 4 4 共 1⫹b 2 sn 2 a 兲 2 共 1⫹b 2 sn 2 a 兲 2 共 1⫺b 2 sn 2 a 兲 2
cn 2 a dn 2 a sn 2 a ⫺ 共 1⫺k 2 兲 2 2 2 sn a cn a dn 2 a
冊
冉
冊册
冋
⫹4a 2 b 2 m 3 共 1⫺m 3 兲
冉
cn 2 a dn 2 a 共 1⫺b 2 sn 2 a 兲 2
冊 册冎
U⫽0. 共30兲
The Riemann function for 共30兲 is then R 共 , , 0 , 0 兲 ⫽F B 共 m 1 ,m 2 ,m 3 ,m 4 ,1⫺m 1 ,1⫺m 2 ,1⫺m 3 ,1⫺m 4 ,1,z 17 ,z 18 ,z 19 ,z 20兲 , where z 17⫽
共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 ⫺ 共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 , 4sn a sn a 0 cn a cn a 0 dn a dn a 0
z 18⫽
共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 ⫺ 共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 , 4sn a sn a 0 cn a cn a 0 dn a dn a 0
z 19⫽b 2
共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 ⫺ 共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 , 共 1⫺b 2 sn2 a 兲共 1⫹b 2 sn2 a 兲共 1⫺b 2 sn2 a 0 兲共 1⫹b 2 sn2 a 0 兲
z 20⫽b 2
共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 ⫺ 共 sn a cn a dn a 0 ⫺sn a 0 cn a 0 dn a 兲 2 . 共 1⫹b 2 sn2 a 兲共 1⫺b 2 sn2 a 兲共 1⫹b 2 sn2 a 0 兲共 1⫺b 2 sn2 a 0 兲
共31兲
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Symmetry operators for Riemann’s method
2999
As for system 1, Chaundy’s equation written as 共30兲 is completely new. We have now established all possible separable coordinate systems for Chaundy’s full equation. IV. A FIVE PARAMETER RIEMANN FUNCTION
In Ref. 11 Olevski showed that the Riemann function R 1 ⫺ 2 , for the equation U y y ⫺U xx ⫹ 共 1 共 y 兲 ⫺ 2 共 x 兲兲 U⫽0,
共32兲
can be given by R 1 ⫺ 2 共 x,y,x 0 ,y 0 兲 ⫽R 1 共 x,y,x 0 ,y 0 兲 ⫹
冕
y⫺y 0
x⫺x 0
R 1 共 t,y,0,y 0 兲
R 共 x,t,x 0 ,0兲 dt, t 2
共33兲
where R 1 and R 2 are the Riemann functions for U y y ⫺U xx ⫹ 1 共 y 兲 U⫽0 and U y y ⫺U xx ⫺ 2 共 x 兲 U⫽0. Looking through the results from the previous section, 共20兲 and 共25兲 can both be applied to 共33兲. First let m 1 (1⫺m 1 )⫽m 3 (1⫺m 3 )⫽0 and a→ 1 in 共20兲 to obtain U ⫺U ⫺ 21
冋
册
m 2 共 1⫺m 2 兲 m 4 共 1⫺m 4 兲 ⫹ U⫽0, sin2 1 cos2 1
共34兲
which has the Riemann function R 4 共 , , 0 , 0 兲 ⫽F 3 共 m 2 ,m 4 ,1⫺m 2 ,1⫺m 4 ,1,z 10 ,z 12兲 ,
共35兲
where z 10 and z 12 are defined in 共22兲 and 共23兲. Analogously we can obtain U ⫺U ⫹ 22
冋
册
m 1 共 1⫺m 1 兲 m 3 共 1⫺m 3 兲 ⫺ U⫽0 sinh2 2 cosh2 2
共36兲
from the PDE 共25兲. The Riemann function for 共36兲 is R 5 共 , , 0 , 0 兲 ⫽F 3 共 m 1 ,m 3 ,1⫺m 1 ,1⫺m 3 ,1,z 13 ,z 16兲 ,
共37兲
where z 13 and z 16 were defined in 共27兲 and 共28兲. Combining 共34兲 and 共36兲 in 共33兲 we find that the Riemann function for the equation
冋 冉
U ⫺U ⫹ 22
冊 冉
m 1 共 1⫺m 1 兲 m 3 共 1⫺m 3 兲 m 2 共 1⫺m 2 兲 m 4 共 1⫺m 4 兲 ⫺ ⫺ 21 ⫹ 2 2 sinh 2 cosh 2 sin2 1 cos2 1
冊册
U⫽0 共38兲
is given by R 共 , , 0 , 0 兲 ⫽F 3 共 m 2 ,m 4 ,1⫺m 2 ,1⫺m 4 ,1,z 33 ,z 34兲 ⫹
冕
⫻
F 共 m ,m ,1⫺m 1 ,1⫺m 3 ,1,v 1 共 t 兲 , v 2 共 t 兲兲 dt, t 3 1 3
⫺0
⫺0
F 3 共 m 2 ,m 4 ,1⫺m 2 ,1⫺m 4 ,1,u 1 共 t 兲 ,u 2 共 t 兲兲 共39兲
3000
J. Math. Phys., Vol. 45, No. 8, August 2004
P. J. Zeitsch
where cos 1 共 ⫺ 0 兲 ⫺cos 1 t , 2 sin 1 sin 1 0
u 2共 t 兲 ⫽
cosh 2 t⫺cosh 2 共 ⫺ 0 兲 , 2 sinh 2 sinh 2 0
v 2共 t 兲 ⫽
u 1共 t 兲 ⫽ v 1共 t 兲 ⫽
cos 1 共 ⫺ 0 兲 ⫺cos 1 t 2 cos 1 cos 1 0
cosh 2 共 ⫺ 0 兲 ⫺cosh 2 t . 2 cosh 2 cosh 2 0
The ratio 1 / 2 is essential in 共38兲. It is possible to transform away either 1 or 2 via a change of variables, but not both. It is useful to write the equation as 共38兲 though, which at first glance incorporates six parameters, as the equation is then symmetric. Effectively a five parameter Riemann function has been obtained. Equation 共38兲 contains one more essential parameter than Chaundy’s equation 共1兲. There is strong evidence to suggest that 共38兲 is not isomorphic to Chaundy’s equation 共1兲. There are several reasons for this. First 共38兲 possesses no nontrivial first-order symmetries and repeating the calculations of Sec. II shows that it has no second-order symmetry operators either. So combining 共34兲 and 共36兲 in the addition formula destroys the symmetries found in Sec. II. Of course this does not rule out the possibility of a contact, or other type of transformation, between 共1兲 and 共38兲 but the author believes that this is unlikely. Secondly, during the course of analyzing Chaundy’s equation the unusual fact that the independent variables z 1 ,...,z 4 of 共1兲 are linearly dependent was discovered. In fact a few calculations will show that 1 1 1 1 ⫹ ⫹ ⫹ ⫽2, z1 z2 z3 z4 where z 1 ,...,z 4 are given by 共3兲 and 共4兲. If a parallel calculation is performed with the independent variables of 共38兲, then this linear relationship is lost. Effectively 1 1 1 1 ⫹ ⫹ ⫹ ⫽2. z 9 z 11 z 13 z 16 The loss of such a property adds further weight to the conjecture that the two equations are not isomorphic. ACKNOWLEDGMENTS
The author is grateful to Edward D. Fackerell and Christopher M. Cosgrove for many enlightening discussions. T. W. Chaundy, Q. J. Math. 9, 234 共1938兲. P. Appell and J. Kampe´ de Fe´riet, Fonctions Hyperge´ome´triques Et Hypersphe´riques, Polynomes D’Hermite 共GauthierVillars, Paris, 1926兲. 3 E. T. Copson, Arch. Ration. Mech. Anal. 1, 324 共1958兲. 4 E. G. Kalnins and W. Miller, Jr., J. Math. Phys. 17, 369 共1976兲. 5 W. Miller, Jr., SIAM J. Math. Anal. 4, 314 共1973兲. 6 W. Miller, Jr., Symmetry and Separation of Variables 共Addison–Wesley, London, 1977兲. 7 G. W. Bluman, ‘‘Construction of solutions to partial differential equations by the use of transformation groups,’’ Ph.D. thesis, California Institute Of Technology, 1967. 8 E. A. Daggit, J. Math. Anal. Appl. 29, 91 共1970兲. 9 N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Applications in Engineering and Physical Sciences 共CRC Press, Boca Raton, FL, 1995兲. 10 K. Iwasaki, SIAM J. Math. Anal. 19, 903 共1988兲. 11 M. N. Olevski, Dokl. Akad. Nauk SSSR 87, 337 共1952兲. 12 P. Henrici, Z. Angew. Math. Phys. 8, 169 共1957兲, particularly the table on p. 180. 13 E. G. Kalnins and W. Miller, Jr., SIAM J. Math. Anal. 9, 12 共1978兲. 14 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 共Dover, New York, 1965兲. 15 H. Cohn, Duke Math. J. 14, 297 共1947兲. 16 J. S. Papadakis and D. H. Wood, J. Diff. Eqns. 24, 397 共1977兲. 1 2
