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Lie group analysis and similarity solutions for the equation ∂2u∂x2 + ∂2u∂y2 + ∂2(eu)∂z2 = 0
~o;on,,nror Anc,/~m. Theory, Prmted in Grea! Bntam
.Merhods
& A~~,;co,,ons.
Vol.
13.
No.
5. pp.
489-SOS.
0362-546X189 $3.00~ .oO B 1989 Pcrgamon Press plc
1989.
LIE GROUP ANALYSIS
AND SIMILARITY SOLUTIONS a2u a2m () FOR THE EQUATION 2 + ayZ+az’= MARK
S. DREW and STEVE C. KLOSTER
School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada VSA IS6
and JACK D. GEGENBERG Department of Mathematics, University of New Brunswick, P.O. Box 4400, Fredericton, N.B., Canada, E3B 5A3 (Received 9 November
Key words and phrases: solutions.
1987; received for publication
Group analysis, nonlinear,
IS April 1988)
similarity solutions,
exact solutions,
invariant
1. INTRODUCTION
analysis of differential equations has been a carefully developed field for some time [I]. Even without prior knowledge of specific solutions of sets of nonlinear equations, the understanding of how any solutions must transform amongst themselves under a group of point transformations can lead in many cases to the determination of new solutions, the similarity solutions (see e.g. [2]) or to a fuller understanding of group orbits in the space of solutions of well-known equations [3]. In recent years, the use of the Lie group method has widened to a larger community including physicists studying new properties of some of the fundamental equations of mathematical physics (see, e.g., the excellent overview in [4]). This widened interest has been spurred by the development of a series of highly useable computer programs written in symbolic manipulation languages, such as REDUCE and MACSYMA [5-91. This development has often been used to replicate manual calculations (cf. [lo] and [ll]). (Throughout this paper, the algebraic manipulation language MACSYMA was used to develop our results.) Here, a modification of one of these programs [5] is used as an aid in applying the method to determine the group of point transformations for the partial differential equation LIE GROUP
a2u
a2u
a2@“>
axzfayzfx=
o
.
(1.1)
The solutions of this equation determine a class of Riemannian metrics in quantum gravity, derived from general relativity, for which the corresponding curvature tensors are (anti-) selfdual and which admit at least one Killing vector field [ 12-131. Included in the above class of metrics are the gravitational instantons [14 and references cited therein] which are likely to play an important role in quantum gravity [15]. The main motivation for discovering the symmetry group of equation (1.1) is in order to develop new exact similarity solutions for this equation, and hence, perhaps, new exact gravitational instanton solutions of the Euclidean version of Einstein’s equations. To date, only the 489
Xl. S. DREW et al.
490
of the particular ansatz 10, this ansatz effectively . reduces the problem to that of solving the simpler Liouville equation, in two dimensions. Here, several new solutions of equation (1.1) are determined and most do not belong to the class that reduces to solutions of the simpler equation. An analysis of the group found for equation (l.l), and also for the simpler equation found by assuming cylindrical symmetry of solutions, is presented here. The reason for consideration of the cylindrically symmetric case is that the Lie group for the full equation (1.1) turns out to split into a finite-dimensional part and an infinite Lie group determined by an arbitrary harmonic function in two variables. Specialization to the simpler cylindrically symmetric equation turns out to cast the symmetry group into a form that can automatically determine some choices for the arbitrary harmonic function of the group for the full equation, and hence guide an exploration of part of the infinite dimensional subgroup. Solutions of the simpler equation are also easier to obtain, since it has only two independent variables. The method of finding similarity solutions consists of determining invariant surfaces in the space of independent and dependent variables, and rewriting the partial differential equation in terms of the similarity variables fixing these surfaces. A similarity variable is constant along an orbit of the group of point transformations. This procedure is guaranteed to reduce by one the number of independent variables in the resulting differential equation. In the case of equation (1.1) this reduces the number of independent variables to two similarity variables. A further application of the group method to the resulting equation results in an ordinary (nonlinear) differential equation for a function with argument being another similarity variable. Solution of this equation leads directly to a solution of the full equation (1.1). For the simpler version of the equation, there are only two independent variables, so that a single application of the method leads to an ordinary differential equation. The resulting solutions are cylindrically symmetric solutions of equation (1.1). The consequences, in regard to new solutions of the Euclidean Einstein’s equations, resulting from the new similarity solutions determined here, as well as the physical meaning of the geometries determined via these new solutions, will be presented elsewhere. constant
solution
of
equation
(1.1)
and
three
related
cases
u = f(x, _v) + g(z) have been noted [12]. As shown below in Section
2. LIE
In general, we are interested variable u and N independent
GROUP
ANALYSIS
in a single kth order partial variables xi, i = 1, . . . , N
differential
equation
with dependent
(2.1) uili2 is a2u/8xi, axi,, etc. In the case of equation (1. l), one has N = 3 and of the form (2.1) the group analysis is carried out is as follows. the infinitesimal version of a one-parameter point group:
where ui is au/ki,
k = 2. For an equation Consider
n = u + &fl(U,X) + 0(&Z), pi = Xi + &ri(U, X) + O(&‘), E infinitesimal.
The generator
of this transformation
(2.2)
is written
u = tj a/au + ci a/ax,
(2.3)
Lie group
(using the summation convention Uk = q
analysis
and similarity
491
solutions
for repeated indices). The kth extension of CJis given by
a/au + & a/ax, + pi lVfS.4, + ... + pi ,.,. ir a/au, ,... ir
Here, the p’s are given in terms of the total differentiation Di = Wax, +
Ui
.
(2.4)
operator
a/au
(2.5)
via the recursive formula Pi = Di(rl) Pil...i*
-
j=l
ujDi(tj)9
= Dik(Pil...if_I)
-
9 .--, N,
Uil...it_~jDir(tj).
(2.6)
The fact that the above formulation can be handled so naturally on a computer makes this work an obvious candidate for the use of an algebraic manipulation language. In order for (2.1) to have solutions which transform into one another by means of the group, one must have u,n = 0 (2.7) whenever (2.1) holds. Therefore, the group can be found by the following steps. (a) Form the expression on the left-hand side of (2.7). (b) Simplify by applying side-condition (2.1). (c) The resulting expression is a polynomial in u and the derivatives of U. Equate separately to zero the coefficients of the polynomial. Now setting aside the condition (2. l), one is faced with solving a set of coupled linear partial differential equations in the unknowns q and C. This set is termed the set of determining equations. The set of determining equations can be quite large, and an algorithm has been developed for their solution [6]. However, in this paper it was found expedient to solve the determining equations by hand in all cases. Nevertheless, without the use of some computer help, it would have been very burdensome to develop the set of determining equations in order to solve them for the group action. In Section 3 the Lie group of point transformations for equation (1 .l) is developed by following the steps outlined above, and in Section 4 the finite versions of the infinitesimal transformations (2.2) are determined. In Sections 5 and 6 the resulting system of equations for the characteristics is integrated, and the partial differential equation is reduced to an ordinary differential equation in several special cases, with the result that new solutions are developed. In Sections 7 and 8 a similar analysis is carried out for the cylindrically symmetric version of the partial differential equation, and again new solutions are found. In Section 9 we discuss the situation occurring when limiting cases of similarity variables are used in solutions, and in Section 10 the connection with the Liouville equation is explicated. 3. LIE
GROUP
FOR
FULL
EQUATION
To determine the Lie group for the full equation (1.1) we apply the second extension U, of the group operator, since equation (1.1) is second order. When applied to (1. l), (2.7) with k = 2 results in a polynomial, in u and its derivatives, which is simplified when (1.1) is applied as a side-condition. Now setting aside the condition (1.1) and equating the coefficients in the polynomial to zero separately, one has a set of 34 coupled linear partial differential equations in rl and &, i = 1, 2, 3, the determining equations (omitted here).
M. S. DREW et al.
492
The so&ion
of the determining
equations
is as follows
(with (x, y, z) = (Xi, i = 1,2, 31):
q=-2p
(3.1) (2
=
(a
<3 =
where o and (Y;are constants
+ Lox2
cTx3 +
+
a2
cY3
but /? is an arbitrary
harmonic
function
in two dimensions:
a2p a2p s+s=o. I
(3.2)
2
The meaning of these transformations is apparent: the 01~represent three independent lations; the transformation characterized by o is a uniform dilation in three dimensions; point transformation given by p is a dilation in the two-dimensional x,-x, subspace panied by a translation in the u-direction. The choice for the harmonic function p is entirely open, so that /3 contributes an subgroup to the full Lie group for equation (1. I). In Sections 7 and 8 below the analysis cylindrically symmetric version of (1.1) will automatically lead to a special choice for p, the present let us take the particular solution jI = constant. With this choice, one is faced with solving equations, the characteristic equations.
&I +
a!,
=
(a
+
4. FINITE
The differential
equations
/l)x,
set of ordinary
the simultaneous
b3 +
infinite for the but for
(3.3)
dx,
(CT + jl)x,
transand the accom-
012 =
fJx3
+
a3
du =Tj$
differential
(3.4)
TRANSFORMATIONS
corresponding
to (3.1) are
da -2P
z=
(4.1) -
da
=
!Lox da
where a is the arc length
along
a group
(a
+ P)x2
3
+(y
+
(Y2
3,
orbit in Xi - u space.
Lie group
analysis
and similarity
493
solutions
The finite transformations of solutions u are given by the solution of equations (4.1). Since no equation involves more than one independent variable, they can be trivially solved: X, = xi e(0+i3)0+ at u X2 =
x2e(“+8)a + cx2a (4.2)
a = u - 2pa. Equations (1.1).
(4.2) constitute
the most general group of point transformations
5. INTEGRATION
OF
CHARACTERISTIC
for equation
EQUATIONS
To integrate the system (3.4), first solve the first equation, calling the constant of integration [, say. This is the first similarity variable; now replace x1 in favour of < in subsequent calculations. Integration of the x2-x3 pair yields a second similarity variable, 4. The last integration gives the general form for u in terms of x3, and in terms of an integration constant, F say, that is actually a function of { and 4, i.e. depends on the two surfaces previously selected. Since the method is guaranteed to reduce the number of independent variables, the resulting nonlinear partial differential equation for F(c, q5) (omitted here) involves only [ and C$as independent variables; it has fifteen terms and is quite complex. However, by a judicious choice of [ and 4, out of the five group constants the differential equation can be made to involve only the constants cr and 8, and splits into three parts involving just o, the product crp, and just p. This suggests beginning again to solve (3.4), but with either of o and p set separately to zero. Then the resulting partial differential equation for F([, I$) is simpler in each case, and its solution can be addressed by the same method brought to bear so far, i.e. one can find the group associated with the equation, determine a new similarity variable given in terms of [ and 4, and hence arrive at an ordinary differential equation for the resulting function. This scheme is carried out below, and results in new solutions of the original equation (1.1). 6. REDUCTION
TO
ORDINARY
DIFFERENTIAL
EQUATION
The cases p = 0 and cr = 0 are treated separately. 6.1. Case/3 = 0 To integrate (3.4), first consider the case p E 0. One is now attempting to solve the set tit du, = = ox, + CY, ox, + cY2 ox,+CYj dx,
=-
du 0’
(6.1.1)
It is convenient to redefine the xi via pi =
Xi +
(6.1.2)
CYJC7.
Then integration of the system (6.1.1) gives three constants of integration C, 4, and x, with [ = x,/x2,
4 =
rS,/q,
x
=
x3,
(6.1.3)
hf. S. DREW et al.
194
so
that
x, = c-x4, u = ni,
Substitution the number
x2
=
R, = x9
xh
(6.1.4)
4).
of these forms into the original equation of independent variables to two:
(1.1) does indeed
4” + 2g:
lead to a reduction
-i 2eFgm3
of
= 0. (6.15)
In order to solve equation (6.1 S), we apply the group method once again. There are sixteen determining equations, for a new set of c’s and an q, defining the point group for (6.1.5). In this calculation there appear two new constants of integration, A and B, and no arbitrary functions. The result is expressed as
d4
dl
-A([’
+
dF
(6.1.6)
+ B) =--
1) = ,#+l[
To use the solution of the system (6.1.6) to solve equation (6.15) it is much simpler to again split the work into separate calculations by setting in turn each of the constants to zero. (a) Case A = 0. Since now 0 appears as the denominator of the first expression in (6.1.6), the variable [ is itself a similarity variable in the c-4 space for the equation (6.1.5). The second equal sign yields (6.1.7)
F = -2 logW(01, where Z(c) arises as the integration constant, similarity variable. Substitution into (6.1.5) gives an equation differential equation: (C2 + l)Z$
which actually
depends
in one less independent
- (<2 + 1)
5
2 + 2525
- z2 -
on the previous variable,
1 = 0.
choice of
i.e. an ordinary
(6.1.8)
0 To solve this equation,
let (6.1-g)
Z = Q(i)U(P), with
(6.1.10)
P = P(C). Substitution
into (6.1.8) yields an equation
which is simplified
by the following
choices for a(c)
and P(C): a([) = qT7, (6.1.11) P(C) = tan-‘(C). Then equation
(6.1.8) becomes -!=O.
(6.1.12)
Lie group
Now, the solution
of this equation
analysis
and similarity
495
solutions
is well-known, C, CJ = cosh(C,p
(6.1.13)
+ C,),
and hence equation (6.1.8), and thus (6.1.5) and (1. l), is solved. The final form of this solution for (l.l), is exp(u)
Cf(x,
=
+ cr$
(6.1.14) + C2]’
[(x1 + d,)2 + (xz + &z)2]-cosh2[C,tan-‘~~) where ai = ~;/a.
One sees immediately
from this expression
that u is of the form (6.1.15)
*g&l, x2)1
u = 2 ]og]f(xs) for this solution.
(b) Case B = 0.Now < itself is no longer the similarity variable arising the first equality of (6.1.6) gives a new similarity variable s([, $),
from (6.1.6).
(6.1.16)
s = (C2 + l)@. The second
equality
in (6.1.6) gives (6.1.17)
F = Z(s). Substituting
into equation
(6.1.5) one again derives an ordinary 2
2 2.s(e?s
To simplify,
introduce
+
1)s
+ 2eZs2
differential
dZ + (3eZs + 2)~ = 0.
equation: (6.1.18)
V(s) = ez: 2Vs(Vs + 1)s
This equation
Instead,
is isobaric
* + V(3V.s + 2)%
- 2s g (
[ 161, and simplifies
= 0.
(6.1.19)
> via the substitution
v = Y(f),
(6.1.20)
t = log(s)
to Y(Y+ Since the independent
variable 2Y(Y+
This equation
- ;y$
I)$is missing, l)&
can in turn be simplified:
let P(Y)
+ ;y3
= 0.
(6.1.21)
= d Y/d?, so that
2P2 - 3Y2Pi-
Y3 = 0.
(6.1.22)
with the substitution Y p = (Y + l)T(Y)
(6.1.23)
496
M. S. DREW et al.
it becomes an Abel’s equation of the first kind, 2g
- T3(Y + 1) + 3T2 = 0.
(6.1.24)
This particular example of such an equation is simple to solve, because on substitution of T=
-3S(Y)/(Y
+ 1)
(6.1.25)
it becomes separable: 2$(Y+
l)-9S3-9S2-2S=O
with solution S(3S + 2) = -C,(Y+ (3s + I)2
1)
(6.1.27)
or, inverting s = &JC,(Y
+ 1) f 1 - C,(Y + 1) - 1 3C,(Y + 1) + 3
(6.1.28)
Hence one arrives at the solution for the first order equation (6.1.22): p=
Y[C,(Y
-
*JC,(Y Since P(Y)
= d Y/at,
+ 1) + 1)
+ 1) + 1 - C,(Y + 1) - 1 .
we can integrate for t = t(Y)
s = e’ = C2C,V
and hence s = s(V),
JC,(V + 1) + 1 +
L
C,(V+
(6.1.29)
c, + 1 *(“JVr)
1) + 1 - w ”
(6.1.30)
and hence yields an implicit solution for u via equation (6.1.4). This solution will not be pursued here. However, a similar solution will be examined in Section 8 below, and there it will be shown that it can be inverted in a few special cases. 6.2. Case CT= 0 In this case, we have the following equations to solve: dx, dw, du &I /Ix, + CY,= px* + cY2= 3 = q *
(6.2-l)
We shall assume that ,B is nonzero; this allows us to redefine the (Y’Sas follows: &i = ffj/B.
(6.2.2)
Since the 01’sare arbitrary, we will drop the bars in the remainder of this discussion. The effect of the substitution (6.2.2) is that we may take p = 1 in equation (6.2.1). The approach here is similar to that of Section 6.1. We find the following similarity variables: x1 + 011
C-=-_, x2
+
a2
I$
=
log(x2
+
cx2)
-
x3/cY3
1
(6.2.3)
Lie group
analysis
and similarity
solutions
491
The solution for u may be written as U = (-2/a&,
+ F(C, 6).
(6.2.4)
The function F must satisfy the following partial differential
equation:
ez’+F[$+(~~+4~$)+4]+a’[$-$+2~(!!-$&-)+([2+1)$]=0, (62.5) where cyJ has been abbreviated as 0~.The above equation can be simplified if we take I’= F+
2r#1- 2loga.
(6.2.6)
Equation (6.2.5) then becomes + (C2 + 1)s
+$-5+2+&_&)
+ 2 = 0.
(6.2.7)
We now apply the group method to solve equation (6.2.7). Solving the determining equations for the new t’s and an q, we are led to the following: (6.2.8) where A is a constant of integration.
The similarity variable for equation (6.2.8) is
s = + log([’ + 1) + A tan-‘c + 4,
(6.2.9)
and the general form of V is V = -log([2
+ 1) + G(s),
where G satisfies the following ordinary differential
(6.2.10)
equation:
(eG + A2 + l)G” + eG(G’)’ = 0.
(6.2.11)
w = eG,
(6.2.12)
w(w + A2 + 1)w” - (A2 + l)(w)2 = 0.
(6.2.13)
If we let then the equation becomes simply
Starting from the assumption A = 0 also leads to this equation, with A = 0. Notice that the above equation with A E 0 is an abbreviated version of equation (6.1.21), and it can be solved in a similar manner. The solution is s = C,[w + (.42 + l)logw]
+ c,.
(6.2.14)
This determines w implicitly as a function of s, w = w(s). The solution for tl can be expressed in terms of w(s) as follows: U = log w - log[(x, + cXJ2 + (x2 + cX2)2]+ 2 log
cY3.
(6.2.15)
498
M. S. DREW ef al.
Now, from (6.2.9) we can write s as s = + log[(x, + CY,)’+ (x2 + CY~)‘]- X,/CY, + A tan-’ so we see that this solution does not reduce to the type u = f(x,y) 7. SPECIALIZATION
TO
CYLINDRICAL
(6.2.16) + g(z).
SYMMETRY
In this case of cylindrical symmetry, equation (1.1) reduces to the following: (7.1)
where u = U(T) and x = rcos 8, y = r sin 8. Equation (7.1) has one less independent variable than (1.1). Recall that there were 34 determining equations for (1.1); for equation (7.1), there are only 16 determining equations. The solution of the set of determining equations is as follows: V,yl = -2&r
- 2Y,,, log r*
5, = (o,yl + Pcyl)~ -t Ycyl(r log r - r),
(7.2)
rz = c,,rz + o&l. There are four group parameters, crcYl,/3c,,I, ycyl, and ocYlin the cylindrical case. Now none of the parameters is an arbitrary function: instead each one is simply a constant. Comparing the solutions of the sets of determining equations for the cylindrical case, equations (7.2), and for the general case, equations (3.1), to generate the cylindrically symmetric case one could replace b in (3.1) by (7.3)
B + Pcyl - Ycyllog r.
Now, recall that in the work above, /3 was arbitrarily chosen to be a constant, as a special case of a harmonic function in 2-dimensional x,-x, space. The meaning of equations (7.3) is that when cylindrical symmetry is assumed, the resulting group is such that /3 automatically takes the form of a constant plus another harmonic function, viz. log r. This choice of harmonic function could equally well be chosen for the infinite part of the general Lie group (3.1). 8. CYLINDRICALLY
SYMMETRIC
SOLUTIONS
From equations (7.2), the characteristic equations in the cylindrically symmetric case are as follows (dropping the subscripts explicitly indicating cylindrical symmetry): dr
dz (o+P.)r+y(rlogr-r)=~=-2~-22ylogr’
du
(8.1)
In following sections, we will obtain solutions of this set of ordinary differential equations by taking certain of the parameters la, /3, y, a) to be zero.
Lie group
analysis
and similarity
399
solutions
8.1. Casey = 0 We can begin an analysis of equations (8.1) by eliminating the most complex part-that associated with the parameter y. Setting this parameter identically to zero reduces (8.1) to the simpler form dr du =-=- dz (8.1.1) (a+B)r cX+cK -2p* Integrating
the first of the above equalities, 1 -log[(a r = (a + p)
The second
equality
yields the general
it is convenient
+ /?)r] - -l$og[oz form of a solution
u = - ~log(cX
variable
f CY].
u of equation
(8.1.2) (7.1): (8.1.3)
+ IX) + F(C),
where F is a function yet to be determined. Substituting into the original partial differential differential equation,
d2F
to use a similarity
equation
(7.1),
one derives
an ordinary
F+Z(o+B)J-
(8.1.4)
dT’+e
To reduce this equation
to a simpler
form,
for Fin terms of w(c) given by
substitute
,,, ~ eF+2B+20r
(8.1.5)
9
with the result w(w + l)wV - (w’)2 - 30w2w’
+ 2a2w2 = 0.
(8.1.6)
Now this equation is precisely that seen before, in equation (6.1.21), when one writes w -+ Y, c + t, fJ -+ +, so that the method of solution has already been determined. The solution is an implicit one, and can be written e20rJm
=
c2
wxq C,(w JC,(w
+ 1) + 1 +
ci + 1 *i
(8.1.7)
.
+ 1) + 1 - Jc,+r “-)
It was mentioned in Section 6.1 that it is possible to invert an equation such as (5.1.7) in a few cases, and one can see that, while (8.1.7) is not invertable for general C,, that constant can be chosen such that equation (8.1.7) becomes a solvable cubic or quartic equation. To see this, first eliminate the root from the denominator above (for either the + 1 or - 1 exponent), producing a square in the numerator; then take the square root of both sides of (8.1.7), introducing a further + 1 or - 1. Isolate on one side of the equal sign the square root involving w, and square both sides of the resulting equation. This results in an equation, for either the + 1 or - 1 exponent in (8.1.7), (C,/C,)
ezoT~w’-~
f 2(C,/C2)“2~eot~w”2(‘-~)
Now solve by choosing special values for C,. For the choice C, = 3, the powers in equation (8.1.8) are w-‘, becomes a quartic in WI/~. For C, = 8, the powers are wm2, w-‘, w, so that (8.1.8) becomes
- C, w = 0. W-I/~,
(8.1.8)
w, so that (8.1.8)
a cubic in w.
M. S. DREW et al.
500
For the case C, = 3, the solutions of the quartic equation resulting from (8.1.8) are XG = -is
f Y,
JI; = +a *t,
(8.1.9)
where ~ _ (-8&e
20f + 3%/Q”* 2hzcy4
’
Q, = (8efie *Or+ 3Z@&Z$“2 2&Ec;/4 ’ (8.1.10)
where we call the f sign in (8.1.8) E. The reality of these solutions clearly depends on the sign and magnitude of C2 relative to cr and c; real cases exist. For the case C, = 8, the solutions of the cubic equation resulting from (8.1.8) are fie30r
+ N
w=m w=
’
- .&(l
f i&)e3”r
46N
(8.1.11) - +(l r iti)N,
where (8.1.12) Again, one can see that there are real cases. 8.2. Case y = o = 0 To consider this case, one does not simply set (i to zero in the above work; instead, one begins again from equations (8.1.1) with 0 identically zero. Now equations (8.1) reduce to the following: dr dz du -=(8.2.1) =I@* Pr ff From the first of these equations, the similarity variable is c = (l/P) logr - z/(Y,
(8.2.2)
- 2pz u = + F(C). cy
(8.2.3)
and the solution for u is given by
When equation (8.2.3) is substituted into (7.1), we obtain the following ordinary differential equation: 2 dF dC*
+
(~9~2)
. e*Bt+F
.[$+
(5+2PT]
=o.
(8.2.4)
Lie group
analysis
and similarity
501
solutions
If we let H = F + 2/l[ + 2 log(p/a),
(8.2.5)
then the previous equation becomes (8.2.6)
H” + eH[H” + (H’)*] = 0. Letting w = eH, we arrive at the equation
(8.2.7)
w(w + 1)w” - (w’)* = 0.
This equation is just the same as a special case of (6.2.13) in form, although of course the meaning of w is entirely different than the same symbol in Section 6.2. Here, the solution for w is the following: (8.2.8) c = C,(w + log w) + c,. Now, w(C) is determined implicitly by the above equation, and the solution for u is U = log w - 2 log f - 2 log(P/a).
(8.2.9)
We note that this solution, like the solution of Section 6.2, is not of the type u = f(x, y) + g(z). 8.3. Case r~ = /I = 0 Integrating the system (8.1) in this case leads to a similarity variable c = log(log r - 1) - yz/a,
(8.3.1)
# = - 2 log@ log r - r) + log[w(4)],
(8.3.2)
and a general form for u
where w(c) is arbitrary so far. Substituting into equation (7.1), one finds that w must satisfy the ordinary equation w(w + l)wfl - (w’)2 - ww’ + 2w2 = 0.
differential (8.3.3)
Now, this equation looks quite similar to equation (8.1.6) but it has a somewhat different structure and a less straighforward solution. Noting that the independent variable is missing, we proceed in the usual way to a first order equation dw dp (8.3.4) w(w+ l)pdw+2w2wp-p*=o, P(W) = z * With the substitution p=
+
k(w)
(8.3.5)
we have the simple equation wkk’ = -2w
+ k - 2.
(We might note the complex solution k = 2 + 2&v
for this equation.)
(8.3.6)
M. S. DREW et ol.
502
The above equation can easily be turned into an Abel’s equation of the first kind [17] by the substitution k(w) = l/c(w), but the standard method of solution for the resulting equation makes use of a transformation that cannot be made explicit. Instead, to solve (8.3.6), let k(w) = m(u),
U = log w, (8.3.7)
mm’ = -2(e” To simplify,
introduce
+ 1) + m.
t such that
a new parameter
u’(t) = f??(u), (8.3.8) u” - v’ + 2e” = - 2. Finally,
let t = log 5,
u(t) = v + 2 log 5, (8.3.9) v” + 2e’ = 0,
and we arrive at the solution C, e-” = cosh’[G(C,
- r)].
(8.3.10)
Evaluating m(v) = u’(t), using the above, one finds that it cannot be written in terms of v(t) itself; instead, (8.3.10) constitutes a parametric solution for w in terms of t since both w and [ can be written in terms of t. Firstly, one can write w in terms of t via equation (8.3.10) since r = e’ and w is given in terms of v via equations (8.3.9) and (8.3.7). Also, since
duct) = m(u) = k(w) = -w+ 1 w P(W) = K$-!$= dt
(e”+
-
I).$.!!,
(8.3.11)
we have [ given in terms of t by
dC
z=e“(O+ 1, so that [given in terms of t via integration of the right-hand tainly does not belong to the class u = f(x,y) + g(z). 8.4. Case IJ = CY= 0 In this case, the denominator of dt in equation similarity variable. Integrating, the general form of a solution is
(8.3.12) side of this equation.
(8.1) is identically
This case cer-
zero, so that z is itself a
(8.4.1)
U = - 2 log(y log r - y + 8) - 2 log r + F(z), where F is arbitrary so far. One sees immediately that u is of the form u = f(x,y) Substituting into (7.1), one is left with the ordinary differential equation d2(eF) dzZ Hence,
eF is constrained
to be any quadratic
+ 2y2 = 0. in z with leading
coefficient
- 2~‘.
+ g(z).
.
Lie group
analysis
and similarity
solutions
503
The result for the special case /3 = 0 is contained within (8.4.1): starting with /? = 0 leads to the solution d2(eF) (8.4.3) U = -2 log(rlog f - r) + F(z), -@ = - 2.
9. LIMITS
OF
SIMILARITY
VARIABLES
We have been finding solutions to the characteristic equations by selectively setting some parameters identically to zero. It is worth enquiring whether the same results could have been obtained by setting some parameters equal to zero after having found similarity variables and solutions. Recall that we derived equation (8.2.1) from (8.1.1) by taking D = 0. The similarity variable used in Section 8.1, where we had o # 0 was [,=--
log r
log(az + o) CJ ’
o+P
whereas the similarity variable used in Section 8.2 was log r 12=a--
2 Q’
(9.2)
However, it is not immediately clear how to take the limit in (9.1) as CJgoes to zero. Recall that [, arose as a constant of integration, and thus we may add a constant of our choosing to it. Let us take log r log(az + 01) - log (Y r;=-0 o+P As (T- 0, the first term approaches log r/P. The second term is a O/O form. By using 1’Hospital’s rule and taking derivatives with respect to CJ,this converges to -z/a, the second term of (9.2). Thus we see that the similarity variable c2 does arise in the limit. It is interesting to note that in fact equation (8.2.7) is the limit of equation (8.1.6). However, this limit process cannot be used to derive equation (8.2.7), because some of the intermediate steps on the way to (8.1.6) involve l/a. Instead, the derivation must start by putting o to zero in equation (8.1.1). This conclusion is validated by the observation that from the 15 term equation for F([, 4) mentioned in Section 5, with (Tand /? arbitrary, one does indeed arrive at equation (6.1.5) for the fl = 0 case simply by setting fl to zero, but one does not produce the o = 0 case, equation (6.2.5), by setting o to zero in the general equation. 10. CONCLUSION
Prior to this work, known solutions of equation Liouville equation,
a2L a2L
g+7+eL=0
ay
modified by the addition of a z-dependent term.
(1.1) were really just solutions of the
(10.1)
504
M. S. DREW et al.
For if one substitutes
the ansa~
u = f(x, Y) + g(z) into equation
(l.l),
the resulting
equation
has a solution
a2f+Cf +a&
ax
ay2
-3
=
(10.2) * only if
0
(10.3)
and a2(eg) -z where cx is a constant.
Clearly
+(Y=o,
(10.4) has the general eg = (a/2)z2
(10.4) solution
+ pz + y,
(10.5)
where /?, y are constants. In (10.3). set CY= ep (if (Y > 0; do the other two cases CY= 0, (Y < 0 separately). Then write L = f + a to arrive at the Liouville equation (10.1). Now, a general solution for equation (10.1) is known [18], and consists of the logarithm of a nonlinear combination of two arbitrary functions F, G and their derivatives, where F, G are (in general complex) functions of x f iy. Real solutions can be constructed via G = F. Here, F is an arbitrary analytic function of x + iy. The situation is comparable to that for the two-dimensional Laplace equation, where the real and imaginary parts of an arbitrary analytic F(x + iy) are both solutions of Laplace’s equation. When it comes to solving a particular boundary-value problem for Laplace’s equation, there is no standard procedure for finding the appropriate complex function, although conformal mapping may be useful; similarly, for the Liouville equation it proves to be very difficult to adapt the general solution to boundary conditions and demand real solutions [19]. Hence the real solutions developed here are of interest. The solutions developed in [12] belong to the ansafz (10.2) and f is of the form f (x, y) = -a
+ [ 1 + 6(x2 + y*)]-2,
(10.6)
where CY= e’; a, b constant. The solutions of part (a) of Section 6.1 and those of Section 8.4 are comparably simple. Particularly for the latter case, it would be interesting to determine whether these solutions, although different than previously known solutions, produce equivalent spacetime metrics and quantum gravity solutions. These enquiries will be developed elsewhere. Acknowledgement-One of the authors (MSD) is indebted to the Laboratory for Computer Communication at Simon Fraser University for machine support for algebraic computing.
Research
REFERENCES 1. LIE S., Theorie der Transformafiongruppen, 2nd edition. Chelsea, New York (1970). 2. BLUMANG. W. & COLEJ. D.. Similarify Methods for Differential Equations. Springer, New York (1974). 3. WULFMANC. E. & WYBOURNEB. G., The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator, J. Phys. A9, 507 (1976). 4. OVSIANNIKOV L. V.. Group Analysis of Differential Equations. Academic, New York (1982). 5. SCHWARZF., A REDUCE Package for determining Lie symmetries of ordinary and partial differential equations, Camp. Phys. Commun. 27, 179 (1982). 6. SCHWARZF., Automatically determining symmetries of partial differential equations, Computing 34, 91 (1985).
Lie group
analysis
and similarity
7. SCHWARZ F., A REDUCE package for determining f&r integrals equations, Comp. Phys. Commun. 39, 285 (1986). 8. ROSENCRANSS. I., Computation of higher-order fluid symmetries
505
solutions of autonomous
systems of ordinary
using MACSYMA,
differential
Comp. Phys. Commun. 38,
347 (1985). 9. ELISEEV V. P., FEDOROVA R. N. & KORNYAK V. V., A REDUCE program for determining point and contact Lie symmetries of differential equations, Comp. Phys. Commun. 36, 383 (1985). 10. AMES K. A. & AMES W. F.. On group analysis of the Von Karman equations, Nonlinear Analysis 6, 845 (1982). 1I. SCHWARZ F., Lie symmetries of the Von Karman equations, Camp. Phys. Commun. 31, 113 (1984). 12. GEGENBERG J. D. & DAS A., Stationary Riemannian space-times with self-dual curvature, Gen. Ref. Grav. 16, 817 (1984). 13. BOYER C. P. & FINLEYY III J. D., Killing vectors in self-dual, Euclidean Einstein space, J. Mar/t. Phys. 23, 1126 (1981). 14. EGUCHI T., GILKEY P. B. & HANSON A. J.. Gravitation, gauge theories, and differential geometry, Phys. Rep. 66, 213 (1980). 15. HAWKING S. W., Euclidean quantum gravity, in Recenf Developments in Gravitation (Edited by S. DESER). Plenum, New York (1979). 16. MURPHY G. M., Ordinary Differential Equations and Their Solutions. Van Nostrand, Princeton, NJ (1960). 17. KAMKE E., Differentialgleichungen, Vol. 1. Chelsea, New York (1959). 18. BATMAN H., Partial Differenfial Equations of Mathematical Physics. Cambridge University Press, London (1964). 19. TRACY E. R., CHIN C. H. & CHEN H. H., Real periodic solutions of the Liouvilleequation, Physica23D, 91 (1986).
.Merhods
& A~~,;co,,ons.
Vol.
13.
No.
5. pp.
489-SOS.
0362-546X189 $3.00~ .oO B 1989 Pcrgamon Press plc
1989.
LIE GROUP ANALYSIS
AND SIMILARITY SOLUTIONS a2u a2m () FOR THE EQUATION 2 + ayZ+az’= MARK
S. DREW and STEVE C. KLOSTER
School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada VSA IS6
and JACK D. GEGENBERG Department of Mathematics, University of New Brunswick, P.O. Box 4400, Fredericton, N.B., Canada, E3B 5A3 (Received 9 November
Key words and phrases: solutions.
1987; received for publication
Group analysis, nonlinear,
IS April 1988)
similarity solutions,
exact solutions,
invariant
1. INTRODUCTION
analysis of differential equations has been a carefully developed field for some time [I]. Even without prior knowledge of specific solutions of sets of nonlinear equations, the understanding of how any solutions must transform amongst themselves under a group of point transformations can lead in many cases to the determination of new solutions, the similarity solutions (see e.g. [2]) or to a fuller understanding of group orbits in the space of solutions of well-known equations [3]. In recent years, the use of the Lie group method has widened to a larger community including physicists studying new properties of some of the fundamental equations of mathematical physics (see, e.g., the excellent overview in [4]). This widened interest has been spurred by the development of a series of highly useable computer programs written in symbolic manipulation languages, such as REDUCE and MACSYMA [5-91. This development has often been used to replicate manual calculations (cf. [lo] and [ll]). (Throughout this paper, the algebraic manipulation language MACSYMA was used to develop our results.) Here, a modification of one of these programs [5] is used as an aid in applying the method to determine the group of point transformations for the partial differential equation LIE GROUP
a2u
a2u
a2@“>
axzfayzfx=
o
.
(1.1)
The solutions of this equation determine a class of Riemannian metrics in quantum gravity, derived from general relativity, for which the corresponding curvature tensors are (anti-) selfdual and which admit at least one Killing vector field [ 12-131. Included in the above class of metrics are the gravitational instantons [14 and references cited therein] which are likely to play an important role in quantum gravity [15]. The main motivation for discovering the symmetry group of equation (1.1) is in order to develop new exact similarity solutions for this equation, and hence, perhaps, new exact gravitational instanton solutions of the Euclidean version of Einstein’s equations. To date, only the 489
Xl. S. DREW et al.
490
of the particular ansatz 10, this ansatz effectively . reduces the problem to that of solving the simpler Liouville equation, in two dimensions. Here, several new solutions of equation (1.1) are determined and most do not belong to the class that reduces to solutions of the simpler equation. An analysis of the group found for equation (l.l), and also for the simpler equation found by assuming cylindrical symmetry of solutions, is presented here. The reason for consideration of the cylindrically symmetric case is that the Lie group for the full equation (1.1) turns out to split into a finite-dimensional part and an infinite Lie group determined by an arbitrary harmonic function in two variables. Specialization to the simpler cylindrically symmetric equation turns out to cast the symmetry group into a form that can automatically determine some choices for the arbitrary harmonic function of the group for the full equation, and hence guide an exploration of part of the infinite dimensional subgroup. Solutions of the simpler equation are also easier to obtain, since it has only two independent variables. The method of finding similarity solutions consists of determining invariant surfaces in the space of independent and dependent variables, and rewriting the partial differential equation in terms of the similarity variables fixing these surfaces. A similarity variable is constant along an orbit of the group of point transformations. This procedure is guaranteed to reduce by one the number of independent variables in the resulting differential equation. In the case of equation (1.1) this reduces the number of independent variables to two similarity variables. A further application of the group method to the resulting equation results in an ordinary (nonlinear) differential equation for a function with argument being another similarity variable. Solution of this equation leads directly to a solution of the full equation (1.1). For the simpler version of the equation, there are only two independent variables, so that a single application of the method leads to an ordinary differential equation. The resulting solutions are cylindrically symmetric solutions of equation (1.1). The consequences, in regard to new solutions of the Euclidean Einstein’s equations, resulting from the new similarity solutions determined here, as well as the physical meaning of the geometries determined via these new solutions, will be presented elsewhere. constant
solution
of
equation
(1.1)
and
three
related
cases
u = f(x, _v) + g(z) have been noted [12]. As shown below in Section
2. LIE
In general, we are interested variable u and N independent
GROUP
ANALYSIS
in a single kth order partial variables xi, i = 1, . . . , N
differential
equation
with dependent
(2.1) uili2 is a2u/8xi, axi,, etc. In the case of equation (1. l), one has N = 3 and of the form (2.1) the group analysis is carried out is as follows. the infinitesimal version of a one-parameter point group:
where ui is au/ki,
k = 2. For an equation Consider
n = u + &fl(U,X) + 0(&Z), pi = Xi + &ri(U, X) + O(&‘), E infinitesimal.
The generator
of this transformation
(2.2)
is written
u = tj a/au + ci a/ax,
(2.3)
Lie group
(using the summation convention Uk = q
analysis
and similarity
491
solutions
for repeated indices). The kth extension of CJis given by
a/au + & a/ax, + pi lVfS.4, + ... + pi ,.,. ir a/au, ,... ir
Here, the p’s are given in terms of the total differentiation Di = Wax, +
Ui
.
(2.4)
operator
a/au
(2.5)
via the recursive formula Pi = Di(rl) Pil...i*
-
j=l
ujDi(tj)9
= Dik(Pil...if_I)
-
9 .--, N,
Uil...it_~jDir(tj).
(2.6)
The fact that the above formulation can be handled so naturally on a computer makes this work an obvious candidate for the use of an algebraic manipulation language. In order for (2.1) to have solutions which transform into one another by means of the group, one must have u,n = 0 (2.7) whenever (2.1) holds. Therefore, the group can be found by the following steps. (a) Form the expression on the left-hand side of (2.7). (b) Simplify by applying side-condition (2.1). (c) The resulting expression is a polynomial in u and the derivatives of U. Equate separately to zero the coefficients of the polynomial. Now setting aside the condition (2. l), one is faced with solving a set of coupled linear partial differential equations in the unknowns q and C. This set is termed the set of determining equations. The set of determining equations can be quite large, and an algorithm has been developed for their solution [6]. However, in this paper it was found expedient to solve the determining equations by hand in all cases. Nevertheless, without the use of some computer help, it would have been very burdensome to develop the set of determining equations in order to solve them for the group action. In Section 3 the Lie group of point transformations for equation (1 .l) is developed by following the steps outlined above, and in Section 4 the finite versions of the infinitesimal transformations (2.2) are determined. In Sections 5 and 6 the resulting system of equations for the characteristics is integrated, and the partial differential equation is reduced to an ordinary differential equation in several special cases, with the result that new solutions are developed. In Sections 7 and 8 a similar analysis is carried out for the cylindrically symmetric version of the partial differential equation, and again new solutions are found. In Section 9 we discuss the situation occurring when limiting cases of similarity variables are used in solutions, and in Section 10 the connection with the Liouville equation is explicated. 3. LIE
GROUP
FOR
FULL
EQUATION
To determine the Lie group for the full equation (1.1) we apply the second extension U, of the group operator, since equation (1.1) is second order. When applied to (1. l), (2.7) with k = 2 results in a polynomial, in u and its derivatives, which is simplified when (1.1) is applied as a side-condition. Now setting aside the condition (1.1) and equating the coefficients in the polynomial to zero separately, one has a set of 34 coupled linear partial differential equations in rl and &, i = 1, 2, 3, the determining equations (omitted here).
M. S. DREW et al.
492
The so&ion
of the determining
equations
is as follows
(with (x, y, z) = (Xi, i = 1,2, 31):
q=-2p
(3.1) (2
=
(a
<3 =
where o and (Y;are constants
+ Lox2
cTx3 +
+
a2
cY3
but /? is an arbitrary
harmonic
function
in two dimensions:
a2p a2p s+s=o. I
(3.2)
2
The meaning of these transformations is apparent: the 01~represent three independent lations; the transformation characterized by o is a uniform dilation in three dimensions; point transformation given by p is a dilation in the two-dimensional x,-x, subspace panied by a translation in the u-direction. The choice for the harmonic function p is entirely open, so that /3 contributes an subgroup to the full Lie group for equation (1. I). In Sections 7 and 8 below the analysis cylindrically symmetric version of (1.1) will automatically lead to a special choice for p, the present let us take the particular solution jI = constant. With this choice, one is faced with solving equations, the characteristic equations.
&I +
a!,
=
(a
+
4. FINITE
The differential
equations
/l)x,
set of ordinary
the simultaneous
b3 +
infinite for the but for
(3.3)
dx,
(CT + jl)x,
transand the accom-
012 =
fJx3
+
a3
du =Tj$
differential
(3.4)
TRANSFORMATIONS
corresponding
to (3.1) are
da -2P
z=
(4.1) -
da
=
!Lox da
where a is the arc length
along
a group
(a
+ P)x2
3
+(y
+
(Y2
3,
orbit in Xi - u space.
Lie group
analysis
and similarity
493
solutions
The finite transformations of solutions u are given by the solution of equations (4.1). Since no equation involves more than one independent variable, they can be trivially solved: X, = xi e(0+i3)0+ at u X2 =
x2e(“+8)a + cx2a (4.2)
a = u - 2pa. Equations (1.1).
(4.2) constitute
the most general group of point transformations
5. INTEGRATION
OF
CHARACTERISTIC
for equation
EQUATIONS
To integrate the system (3.4), first solve the first equation, calling the constant of integration [, say. This is the first similarity variable; now replace x1 in favour of < in subsequent calculations. Integration of the x2-x3 pair yields a second similarity variable, 4. The last integration gives the general form for u in terms of x3, and in terms of an integration constant, F say, that is actually a function of { and 4, i.e. depends on the two surfaces previously selected. Since the method is guaranteed to reduce the number of independent variables, the resulting nonlinear partial differential equation for F(c, q5) (omitted here) involves only [ and C$as independent variables; it has fifteen terms and is quite complex. However, by a judicious choice of [ and 4, out of the five group constants the differential equation can be made to involve only the constants cr and 8, and splits into three parts involving just o, the product crp, and just p. This suggests beginning again to solve (3.4), but with either of o and p set separately to zero. Then the resulting partial differential equation for F([, I$) is simpler in each case, and its solution can be addressed by the same method brought to bear so far, i.e. one can find the group associated with the equation, determine a new similarity variable given in terms of [ and 4, and hence arrive at an ordinary differential equation for the resulting function. This scheme is carried out below, and results in new solutions of the original equation (1.1). 6. REDUCTION
TO
ORDINARY
DIFFERENTIAL
EQUATION
The cases p = 0 and cr = 0 are treated separately. 6.1. Case/3 = 0 To integrate (3.4), first consider the case p E 0. One is now attempting to solve the set tit du, = = ox, + CY, ox, + cY2 ox,+CYj dx,
=-
du 0’
(6.1.1)
It is convenient to redefine the xi via pi =
Xi +
(6.1.2)
CYJC7.
Then integration of the system (6.1.1) gives three constants of integration C, 4, and x, with [ = x,/x2,
4 =
rS,/q,
x
=
x3,
(6.1.3)
hf. S. DREW et al.
194
so
that
x, = c-x4, u = ni,
Substitution the number
x2
=
R, = x9
xh
(6.1.4)
4).
of these forms into the original equation of independent variables to two:
(1.1) does indeed
4” + 2g:
lead to a reduction
-i 2eFgm3
of
= 0. (6.15)
In order to solve equation (6.1 S), we apply the group method once again. There are sixteen determining equations, for a new set of c’s and an q, defining the point group for (6.1.5). In this calculation there appear two new constants of integration, A and B, and no arbitrary functions. The result is expressed as
d4
dl
-A([’
+
dF
(6.1.6)
+ B) =--
1) = ,#+l[
To use the solution of the system (6.1.6) to solve equation (6.15) it is much simpler to again split the work into separate calculations by setting in turn each of the constants to zero. (a) Case A = 0. Since now 0 appears as the denominator of the first expression in (6.1.6), the variable [ is itself a similarity variable in the c-4 space for the equation (6.1.5). The second equal sign yields (6.1.7)
F = -2 logW(01, where Z(c) arises as the integration constant, similarity variable. Substitution into (6.1.5) gives an equation differential equation: (C2 + l)Z$
which actually
depends
in one less independent
- (<2 + 1)
5
2 + 2525
- z2 -
on the previous variable,
1 = 0.
choice of
i.e. an ordinary
(6.1.8)
0 To solve this equation,
let (6.1-g)
Z = Q(i)U(P), with
(6.1.10)
P = P(C). Substitution
into (6.1.8) yields an equation
which is simplified
by the following
choices for a(c)
and P(C): a([) = qT7, (6.1.11) P(C) = tan-‘(C). Then equation
(6.1.8) becomes -!=O.
(6.1.12)
Lie group
Now, the solution
of this equation
analysis
and similarity
495
solutions
is well-known, C, CJ = cosh(C,p
(6.1.13)
+ C,),
and hence equation (6.1.8), and thus (6.1.5) and (1. l), is solved. The final form of this solution for (l.l), is exp(u)
Cf(x,
=
+ cr$
(6.1.14) + C2]’
[(x1 + d,)2 + (xz + &z)2]-cosh2[C,tan-‘~~) where ai = ~;/a.
One sees immediately
from this expression
that u is of the form (6.1.15)
*g&l, x2)1
u = 2 ]og]f(xs) for this solution.
(b) Case B = 0.Now < itself is no longer the similarity variable arising the first equality of (6.1.6) gives a new similarity variable s([, $),
from (6.1.6).
(6.1.16)
s = (C2 + l)@. The second
equality
in (6.1.6) gives (6.1.17)
F = Z(s). Substituting
into equation
(6.1.5) one again derives an ordinary 2
2 2.s(e?s
To simplify,
introduce
+
1)s
+ 2eZs2
differential
dZ + (3eZs + 2)~ = 0.
equation: (6.1.18)
V(s) = ez: 2Vs(Vs + 1)s
This equation
Instead,
is isobaric
* + V(3V.s + 2)%
- 2s g (
[ 161, and simplifies
= 0.
(6.1.19)
> via the substitution
v = Y(f),
(6.1.20)
t = log(s)
to Y(Y+ Since the independent
variable 2Y(Y+
This equation
- ;y$
I)$is missing, l)&
can in turn be simplified:
let P(Y)
+ ;y3
= 0.
(6.1.21)
= d Y/d?, so that
2P2 - 3Y2Pi-
Y3 = 0.
(6.1.22)
with the substitution Y p = (Y + l)T(Y)
(6.1.23)
496
M. S. DREW et al.
it becomes an Abel’s equation of the first kind, 2g
- T3(Y + 1) + 3T2 = 0.
(6.1.24)
This particular example of such an equation is simple to solve, because on substitution of T=
-3S(Y)/(Y
+ 1)
(6.1.25)
it becomes separable: 2$(Y+
l)-9S3-9S2-2S=O
with solution S(3S + 2) = -C,(Y+ (3s + I)2
1)
(6.1.27)
or, inverting s = &JC,(Y
+ 1) f 1 - C,(Y + 1) - 1 3C,(Y + 1) + 3
(6.1.28)
Hence one arrives at the solution for the first order equation (6.1.22): p=
Y[C,(Y
-
*JC,(Y Since P(Y)
= d Y/at,
+ 1) + 1)
+ 1) + 1 - C,(Y + 1) - 1 .
we can integrate for t = t(Y)
s = e’ = C2C,V
and hence s = s(V),
JC,(V + 1) + 1 +
L
C,(V+
(6.1.29)
c, + 1 *(“JVr)
1) + 1 - w ”
(6.1.30)
and hence yields an implicit solution for u via equation (6.1.4). This solution will not be pursued here. However, a similar solution will be examined in Section 8 below, and there it will be shown that it can be inverted in a few special cases. 6.2. Case CT= 0 In this case, we have the following equations to solve: dx, dw, du &I /Ix, + CY,= px* + cY2= 3 = q *
(6.2-l)
We shall assume that ,B is nonzero; this allows us to redefine the (Y’Sas follows: &i = ffj/B.
(6.2.2)
Since the 01’sare arbitrary, we will drop the bars in the remainder of this discussion. The effect of the substitution (6.2.2) is that we may take p = 1 in equation (6.2.1). The approach here is similar to that of Section 6.1. We find the following similarity variables: x1 + 011
C-=-_, x2
+
a2
I$
=
log(x2
+
cx2)
-
x3/cY3
1
(6.2.3)
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491
The solution for u may be written as U = (-2/a&,
+ F(C, 6).
(6.2.4)
The function F must satisfy the following partial differential
equation:
ez’+F[$+(~~+4~$)+4]+a’[$-$+2~(!!-$&-)+([2+1)$]=0, (62.5) where cyJ has been abbreviated as 0~.The above equation can be simplified if we take I’= F+
2r#1- 2loga.
(6.2.6)
Equation (6.2.5) then becomes + (C2 + 1)s
+$-5+2+&_&)
+ 2 = 0.
(6.2.7)
We now apply the group method to solve equation (6.2.7). Solving the determining equations for the new t’s and an q, we are led to the following: (6.2.8) where A is a constant of integration.
The similarity variable for equation (6.2.8) is
s = + log([’ + 1) + A tan-‘c + 4,
(6.2.9)
and the general form of V is V = -log([2
+ 1) + G(s),
where G satisfies the following ordinary differential
(6.2.10)
equation:
(eG + A2 + l)G” + eG(G’)’ = 0.
(6.2.11)
w = eG,
(6.2.12)
w(w + A2 + 1)w” - (A2 + l)(w)2 = 0.
(6.2.13)
If we let then the equation becomes simply
Starting from the assumption A = 0 also leads to this equation, with A = 0. Notice that the above equation with A E 0 is an abbreviated version of equation (6.1.21), and it can be solved in a similar manner. The solution is s = C,[w + (.42 + l)logw]
+ c,.
(6.2.14)
This determines w implicitly as a function of s, w = w(s). The solution for tl can be expressed in terms of w(s) as follows: U = log w - log[(x, + cXJ2 + (x2 + cX2)2]+ 2 log
cY3.
(6.2.15)
498
M. S. DREW ef al.
Now, from (6.2.9) we can write s as s = + log[(x, + CY,)’+ (x2 + CY~)‘]- X,/CY, + A tan-’ so we see that this solution does not reduce to the type u = f(x,y) 7. SPECIALIZATION
TO
CYLINDRICAL
(6.2.16) + g(z).
SYMMETRY
In this case of cylindrical symmetry, equation (1.1) reduces to the following: (7.1)
where u = U(T) and x = rcos 8, y = r sin 8. Equation (7.1) has one less independent variable than (1.1). Recall that there were 34 determining equations for (1.1); for equation (7.1), there are only 16 determining equations. The solution of the set of determining equations is as follows: V,yl = -2&r
- 2Y,,, log r*
5, = (o,yl + Pcyl)~ -t Ycyl(r log r - r),
(7.2)
rz = c,,rz + o&l. There are four group parameters, crcYl,/3c,,I, ycyl, and ocYlin the cylindrical case. Now none of the parameters is an arbitrary function: instead each one is simply a constant. Comparing the solutions of the sets of determining equations for the cylindrical case, equations (7.2), and for the general case, equations (3.1), to generate the cylindrically symmetric case one could replace b in (3.1) by (7.3)
B + Pcyl - Ycyllog r.
Now, recall that in the work above, /3 was arbitrarily chosen to be a constant, as a special case of a harmonic function in 2-dimensional x,-x, space. The meaning of equations (7.3) is that when cylindrical symmetry is assumed, the resulting group is such that /3 automatically takes the form of a constant plus another harmonic function, viz. log r. This choice of harmonic function could equally well be chosen for the infinite part of the general Lie group (3.1). 8. CYLINDRICALLY
SYMMETRIC
SOLUTIONS
From equations (7.2), the characteristic equations in the cylindrically symmetric case are as follows (dropping the subscripts explicitly indicating cylindrical symmetry): dr
dz (o+P.)r+y(rlogr-r)=~=-2~-22ylogr’
du
(8.1)
In following sections, we will obtain solutions of this set of ordinary differential equations by taking certain of the parameters la, /3, y, a) to be zero.
Lie group
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399
solutions
8.1. Casey = 0 We can begin an analysis of equations (8.1) by eliminating the most complex part-that associated with the parameter y. Setting this parameter identically to zero reduces (8.1) to the simpler form dr du =-=- dz (8.1.1) (a+B)r cX+cK -2p* Integrating
the first of the above equalities, 1 -log[(a r = (a + p)
The second
equality
yields the general
it is convenient
+ /?)r] - -l$og[oz form of a solution
u = - ~log(cX
variable
f CY].
u of equation
(8.1.2) (7.1): (8.1.3)
+ IX) + F(C),
where F is a function yet to be determined. Substituting into the original partial differential differential equation,
d2F
to use a similarity
equation
(7.1),
one derives
an ordinary
F+Z(o+B)J-
(8.1.4)
dT’+e
To reduce this equation
to a simpler
form,
for Fin terms of w(c) given by
substitute
,,, ~ eF+2B+20r
(8.1.5)
9
with the result w(w + l)wV - (w’)2 - 30w2w’
+ 2a2w2 = 0.
(8.1.6)
Now this equation is precisely that seen before, in equation (6.1.21), when one writes w -+ Y, c + t, fJ -+ +, so that the method of solution has already been determined. The solution is an implicit one, and can be written e20rJm
=
c2
wxq C,(w JC,(w
+ 1) + 1 +
ci + 1 *i
(8.1.7)
.
+ 1) + 1 - Jc,+r “-)
It was mentioned in Section 6.1 that it is possible to invert an equation such as (5.1.7) in a few cases, and one can see that, while (8.1.7) is not invertable for general C,, that constant can be chosen such that equation (8.1.7) becomes a solvable cubic or quartic equation. To see this, first eliminate the root from the denominator above (for either the + 1 or - 1 exponent), producing a square in the numerator; then take the square root of both sides of (8.1.7), introducing a further + 1 or - 1. Isolate on one side of the equal sign the square root involving w, and square both sides of the resulting equation. This results in an equation, for either the + 1 or - 1 exponent in (8.1.7), (C,/C,)
ezoT~w’-~
f 2(C,/C2)“2~eot~w”2(‘-~)
Now solve by choosing special values for C,. For the choice C, = 3, the powers in equation (8.1.8) are w-‘, becomes a quartic in WI/~. For C, = 8, the powers are wm2, w-‘, w, so that (8.1.8) becomes
- C, w = 0. W-I/~,
(8.1.8)
w, so that (8.1.8)
a cubic in w.
M. S. DREW et al.
500
For the case C, = 3, the solutions of the quartic equation resulting from (8.1.8) are XG = -is
f Y,
JI; = +a *t,
(8.1.9)
where ~ _ (-8&e
20f + 3%/Q”* 2hzcy4
’
Q, = (8efie *Or+ 3Z@&Z$“2 2&Ec;/4 ’ (8.1.10)
where we call the f sign in (8.1.8) E. The reality of these solutions clearly depends on the sign and magnitude of C2 relative to cr and c; real cases exist. For the case C, = 8, the solutions of the cubic equation resulting from (8.1.8) are fie30r
+ N
w=m w=
’
- .&(l
f i&)e3”r
46N
(8.1.11) - +(l r iti)N,
where (8.1.12) Again, one can see that there are real cases. 8.2. Case y = o = 0 To consider this case, one does not simply set (i to zero in the above work; instead, one begins again from equations (8.1.1) with 0 identically zero. Now equations (8.1) reduce to the following: dr dz du -=(8.2.1) =I@* Pr ff From the first of these equations, the similarity variable is c = (l/P) logr - z/(Y,
(8.2.2)
- 2pz u = + F(C). cy
(8.2.3)
and the solution for u is given by
When equation (8.2.3) is substituted into (7.1), we obtain the following ordinary differential equation: 2 dF dC*
+
(~9~2)
. e*Bt+F
.[$+
(5+2PT]
=o.
(8.2.4)
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501
solutions
If we let H = F + 2/l[ + 2 log(p/a),
(8.2.5)
then the previous equation becomes (8.2.6)
H” + eH[H” + (H’)*] = 0. Letting w = eH, we arrive at the equation
(8.2.7)
w(w + 1)w” - (w’)* = 0.
This equation is just the same as a special case of (6.2.13) in form, although of course the meaning of w is entirely different than the same symbol in Section 6.2. Here, the solution for w is the following: (8.2.8) c = C,(w + log w) + c,. Now, w(C) is determined implicitly by the above equation, and the solution for u is U = log w - 2 log f - 2 log(P/a).
(8.2.9)
We note that this solution, like the solution of Section 6.2, is not of the type u = f(x, y) + g(z). 8.3. Case r~ = /I = 0 Integrating the system (8.1) in this case leads to a similarity variable c = log(log r - 1) - yz/a,
(8.3.1)
# = - 2 log@ log r - r) + log[w(4)],
(8.3.2)
and a general form for u
where w(c) is arbitrary so far. Substituting into equation (7.1), one finds that w must satisfy the ordinary equation w(w + l)wfl - (w’)2 - ww’ + 2w2 = 0.
differential (8.3.3)
Now, this equation looks quite similar to equation (8.1.6) but it has a somewhat different structure and a less straighforward solution. Noting that the independent variable is missing, we proceed in the usual way to a first order equation dw dp (8.3.4) w(w+ l)pdw+2w2wp-p*=o, P(W) = z * With the substitution p=
+
k(w)
(8.3.5)
we have the simple equation wkk’ = -2w
+ k - 2.
(We might note the complex solution k = 2 + 2&v
for this equation.)
(8.3.6)
M. S. DREW et ol.
502
The above equation can easily be turned into an Abel’s equation of the first kind [17] by the substitution k(w) = l/c(w), but the standard method of solution for the resulting equation makes use of a transformation that cannot be made explicit. Instead, to solve (8.3.6), let k(w) = m(u),
U = log w, (8.3.7)
mm’ = -2(e” To simplify,
introduce
+ 1) + m.
t such that
a new parameter
u’(t) = f??(u), (8.3.8) u” - v’ + 2e” = - 2. Finally,
let t = log 5,
u(t) = v + 2 log 5, (8.3.9) v” + 2e’ = 0,
and we arrive at the solution C, e-” = cosh’[G(C,
- r)].
(8.3.10)
Evaluating m(v) = u’(t), using the above, one finds that it cannot be written in terms of v(t) itself; instead, (8.3.10) constitutes a parametric solution for w in terms of t since both w and [ can be written in terms of t. Firstly, one can write w in terms of t via equation (8.3.10) since r = e’ and w is given in terms of v via equations (8.3.9) and (8.3.7). Also, since
duct) = m(u) = k(w) = -w+ 1 w P(W) = K$-!$= dt
(e”+
-
I).$.!!,
(8.3.11)
we have [ given in terms of t by
dC
z=e“(O+ 1, so that [given in terms of t via integration of the right-hand tainly does not belong to the class u = f(x,y) + g(z). 8.4. Case IJ = CY= 0 In this case, the denominator of dt in equation similarity variable. Integrating, the general form of a solution is
(8.3.12) side of this equation.
(8.1) is identically
This case cer-
zero, so that z is itself a
(8.4.1)
U = - 2 log(y log r - y + 8) - 2 log r + F(z), where F is arbitrary so far. One sees immediately that u is of the form u = f(x,y) Substituting into (7.1), one is left with the ordinary differential equation d2(eF) dzZ Hence,
eF is constrained
to be any quadratic
+ 2y2 = 0. in z with leading
coefficient
- 2~‘.
+ g(z).
.
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and similarity
solutions
503
The result for the special case /3 = 0 is contained within (8.4.1): starting with /? = 0 leads to the solution d2(eF) (8.4.3) U = -2 log(rlog f - r) + F(z), -@ = - 2.
9. LIMITS
OF
SIMILARITY
VARIABLES
We have been finding solutions to the characteristic equations by selectively setting some parameters identically to zero. It is worth enquiring whether the same results could have been obtained by setting some parameters equal to zero after having found similarity variables and solutions. Recall that we derived equation (8.2.1) from (8.1.1) by taking D = 0. The similarity variable used in Section 8.1, where we had o # 0 was [,=--
log r
log(az + o) CJ ’
o+P
whereas the similarity variable used in Section 8.2 was log r 12=a--
2 Q’
(9.2)
However, it is not immediately clear how to take the limit in (9.1) as CJgoes to zero. Recall that [, arose as a constant of integration, and thus we may add a constant of our choosing to it. Let us take log r log(az + 01) - log (Y r;=-0 o+P As (T- 0, the first term approaches log r/P. The second term is a O/O form. By using 1’Hospital’s rule and taking derivatives with respect to CJ,this converges to -z/a, the second term of (9.2). Thus we see that the similarity variable c2 does arise in the limit. It is interesting to note that in fact equation (8.2.7) is the limit of equation (8.1.6). However, this limit process cannot be used to derive equation (8.2.7), because some of the intermediate steps on the way to (8.1.6) involve l/a. Instead, the derivation must start by putting o to zero in equation (8.1.1). This conclusion is validated by the observation that from the 15 term equation for F([, 4) mentioned in Section 5, with (Tand /? arbitrary, one does indeed arrive at equation (6.1.5) for the fl = 0 case simply by setting fl to zero, but one does not produce the o = 0 case, equation (6.2.5), by setting o to zero in the general equation. 10. CONCLUSION
Prior to this work, known solutions of equation Liouville equation,
a2L a2L
g+7+eL=0
ay
modified by the addition of a z-dependent term.
(1.1) were really just solutions of the
(10.1)
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M. S. DREW et al.
For if one substitutes
the ansa~
u = f(x, Y) + g(z) into equation
(l.l),
the resulting
equation
has a solution
a2f+Cf +a&
ax
ay2
-3
=
(10.2) * only if
0
(10.3)
and a2(eg) -z where cx is a constant.
Clearly
+(Y=o,
(10.4) has the general eg = (a/2)z2
(10.4) solution
+ pz + y,
(10.5)
where /?, y are constants. In (10.3). set CY= ep (if (Y > 0; do the other two cases CY= 0, (Y < 0 separately). Then write L = f + a to arrive at the Liouville equation (10.1). Now, a general solution for equation (10.1) is known [18], and consists of the logarithm of a nonlinear combination of two arbitrary functions F, G and their derivatives, where F, G are (in general complex) functions of x f iy. Real solutions can be constructed via G = F. Here, F is an arbitrary analytic function of x + iy. The situation is comparable to that for the two-dimensional Laplace equation, where the real and imaginary parts of an arbitrary analytic F(x + iy) are both solutions of Laplace’s equation. When it comes to solving a particular boundary-value problem for Laplace’s equation, there is no standard procedure for finding the appropriate complex function, although conformal mapping may be useful; similarly, for the Liouville equation it proves to be very difficult to adapt the general solution to boundary conditions and demand real solutions [19]. Hence the real solutions developed here are of interest. The solutions developed in [12] belong to the ansafz (10.2) and f is of the form f (x, y) = -a
+ [ 1 + 6(x2 + y*)]-2,
(10.6)
where CY= e’; a, b constant. The solutions of part (a) of Section 6.1 and those of Section 8.4 are comparably simple. Particularly for the latter case, it would be interesting to determine whether these solutions, although different than previously known solutions, produce equivalent spacetime metrics and quantum gravity solutions. These enquiries will be developed elsewhere. Acknowledgement-One of the authors (MSD) is indebted to the Laboratory for Computer Communication at Simon Fraser University for machine support for algebraic computing.
Research
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analysis
and similarity
7. SCHWARZ F., A REDUCE package for determining f&r integrals equations, Comp. Phys. Commun. 39, 285 (1986). 8. ROSENCRANSS. I., Computation of higher-order fluid symmetries
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solutions of autonomous
systems of ordinary
using MACSYMA,
differential
Comp. Phys. Commun. 38,
347 (1985). 9. ELISEEV V. P., FEDOROVA R. N. & KORNYAK V. V., A REDUCE program for determining point and contact Lie symmetries of differential equations, Comp. Phys. Commun. 36, 383 (1985). 10. AMES K. A. & AMES W. F.. On group analysis of the Von Karman equations, Nonlinear Analysis 6, 845 (1982). 1I. SCHWARZ F., Lie symmetries of the Von Karman equations, Camp. Phys. Commun. 31, 113 (1984). 12. GEGENBERG J. D. & DAS A., Stationary Riemannian space-times with self-dual curvature, Gen. Ref. Grav. 16, 817 (1984). 13. BOYER C. P. & FINLEYY III J. D., Killing vectors in self-dual, Euclidean Einstein space, J. Mar/t. Phys. 23, 1126 (1981). 14. EGUCHI T., GILKEY P. B. & HANSON A. J.. Gravitation, gauge theories, and differential geometry, Phys. Rep. 66, 213 (1980). 15. HAWKING S. W., Euclidean quantum gravity, in Recenf Developments in Gravitation (Edited by S. DESER). Plenum, New York (1979). 16. MURPHY G. M., Ordinary Differential Equations and Their Solutions. Van Nostrand, Princeton, NJ (1960). 17. KAMKE E., Differentialgleichungen, Vol. 1. Chelsea, New York (1959). 18. BATMAN H., Partial Differenfial Equations of Mathematical Physics. Cambridge University Press, London (1964). 19. TRACY E. R., CHIN C. H. & CHEN H. H., Real periodic solutions of the Liouvilleequation, Physica23D, 91 (1986).