- Email: [email protected]
Symmetry analysis of differential equations with Mathematica
bates. Comput. ~~e~f~ng Vol. 25, No. S/9, pp. 2547, 1997 Co~rig~t~l997 ElsevierScienceLtd Printedin Great Britain. All rightsreserved 08957177197 $17.00+ 0.00
Pergamon
PII: s0895-7177(97)00056-3
Symmetry
Analysis of Different ial Equations with ~~th~~at~~a G. BAU~ANN Abteilung fiir Mathemat~he Physik, Universitiit Ulm Albert-Einstein-A&e 11, D-89069 Ulm, Germany [email protected]
Abstract-we will discuss three different methods for finding symmetry solutions based on the Frkchet derivative common to each procedure. The methods discussed are Lie’s standardprocedure of symmetry analysis, the nonclassical method, and the derivation of potential symmetries. A ferromagnet in a strong external field represented by a nonlinear telegraph equation serves as an example describing the application of all three methods. The symmetry methods discussed are realized in a Mathematics package called MathLie performing all of the required calculations. Keywords-Lie,
Symmetry, Computer algebra, ~~th~~t~~,
Group theory.
1. INTRODUCTION The subject of this paper is systems of differential equations (DEQ). We discuss the invariance of the dependent and independent variables of the systems under transformation. The central point of our examinations is a special type of transformation, the so-called Lie symmetry. A Lie or a point symmetry depends on continuous parameters mapping every solution of the equations to another solution of the same equations. Before we give a definition for a Lie symmetry, we will define the more general term of any kind of symmetry for differential equations. DEFINITION:
SYMMETRY. We define a symmetry of any given di~erenti~ equation es a tmnsformation whj~h maps any caution of the d~~erentj~ equatjon to another ~~utjon.
Such a general definition of a symmetry allows a great variety of tr~sformations under which the differential equation can be invariant. One of the many possibilities of such transformations was introduced by Lie in the form of point transformations. In later sections of this paper, we will discuss other types of transformations like potential and nonclassical transformations. These transformations are also possibilities contained in the definition of a symmetry. Let us now specify the content of Lie or point symmetries. DEFINITION: LIESYMMETRY.
A Lie (point) symmetry is characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all jndependent and dependent va&&es. It z&o possesses the properties of our first definition. Evidently, the Lie symmetries of DEQs form a group: composition of any two s~metri~ yields a symmetry, there is an identity tr~sformation, and for any symmetry there is an inverse transformation. The composition of symmetries is obviously associative. The theory of Lie symmetry groups of DEQs was developed by Sophus Lie in the late nineteenth century and is today a well-studied topic. Such Lie groups are invertible point transformations of both the dependent and independent variables of the DEQs and may also depend on continuous parameters. In his work, Lie pointed out that this type of symmetry group is of great importance to understand and
25
26
G. B~UMANN
construct solutions of DE&s. Lie demonstrated that the many techniques for finding solutions of differential equations can be unified and extended by considering symmetry groups. Today, we know several applications of Lie groups in the theory of differential equations. The most important ones are summarized below: (i) (ii) (iii) (iv) (v) (vi) (vii)
reduction of the order for ordinary differential equations, mapping solutions to other solutions, reduction of the number of independent variables of partial differential equations, construction of invariant solutions, construction of invariant solutions to boundary value problems, construction of conser~tion laws, detection of linearizing tra~formations of DE&s.
For the many other applications of Lie group symmetries, refer to [l-3]. To use any of the above-mentioned methods for a given set of DEQs we must first be able to find the symmetries of the equations. The first step for finding point symmetries is to make a general change of all variables and then enforce the new variables to satisfy the same set of DEQs. This approach leads to complicated nonlinear systems of DEQs for the functions used in the transformations. Lie proved that such a procedure is not necessary. He established an efficient method based on an infinitesimal formulation of the problem of finding the symmetry group of a set of DEQs. This formulation replaces these highly complicated and in most cases intractable nonlinear equations with tractable linear overdetermin~ systems of partial di~erential equations. The solution of these so-called infinit~im~ determining equations can be used to determine symmetry transformations. The Lie point symmetries used here are point transformations in the space of dependent and independent variables of a system of DEQs. Under a symmetry transformation, the image of a point traces out a path which is analogous to the trajectory of a particle. The collection of all trajectories can be viewed as a fluid flow. The tangents to this flow represent its velocity vector field and may be regarded as a differential operator. This differential operator is interpreted by the rate of change following the fluid. Finding symmetries for a given set of differential equations involves setting up and solving an associated system of linear homogeneous partial differential equations called determining equations. We will discuss how determining equations arise from symmet~ properties and illustrate this development by outlining the derivation of such equations for several examples. The concepts and principles of symmetries play an important role in mathematics and physics. It appears that cases of exact solutions of differential equations are based on the use of symmetries of these equations with respect to certain transformation. Symmetry analysis is one of the systematic and accurate ways to obtain solutions of differential equations. An alternative method especially for solving nonlinear partial differential equations is the inverse scattering method. This method is only applicable for completely integrable equations of motion, whereas a symmetry analysis is also useful in the study of nonintegrable equations of motion. A major obstacle in the application of symmetry analysis is that a large number of tedious calculations is usually involved. The purpose of the following sections is to present the theoretical background for the calculations of Lie point symmetries. Based on thii theory we have developed a program in ~~~~e~a~~ca doing all the necessary calculations within no time. Similar programs written in REDUCE, MACSYMA, AXIOM, and MAPLE are available on large scale computers [4--81. Recent reviews of such programs are given in [5,6,9].
2. LIE’S PROCEDURE In this section, we outline the general procedure for determining point symmetries for any system of equations. Let us consider the general case of a nonlinear system of differential equations for an arbitrary number q of unknown functions ua which may depend on p independent vari-
Symmetry
Analysis
27
ables xi. We simply denote these sets of variables by a= (r~‘, u2,. . . , uq) and 5 = (1i,22,. respectively. The general case is given by a system of m nonlinear differential equations Ai (z,@))
(1)
i=l,Z,...,m
= 0,
. , zp),
of order Ic. The term ~l(~) is understood as kth derivative of u with respect to 2. We note that m, Ic, p, and Q are arbitrary positive integers. Consider further a one-parameter e-Lie group of transformations 5’ = E(z, 21;c), u’ = @(X121;c) (2) under which (1) must be invariant. Invariance of (1) under the transformation (2) means that any solution u = e(z) of (1) maps into another solution u = @(s; E) of (1). Let 2~= 8(x:) be a soliution of (1). If we replace the dependent and independent variables u and x with ‘u and z f = E, equations (1) become Ai (z’, V(~)) = 0,
i = 1,2 ,..., m.
(3)
Then TJ= e(z’) are solutions of (3). This implies that if (1) and (3) have a unique solution, then 8
Hence 8 satisfies the one-parameter
(2’) = * (x,0(x); c) .
(4
functional equation 8 (Z(2; E)) = (a(%,6; E).
(51
Expanding equations (2) around the identity E = 0, we can generate the following infinitesimal transformations:
Z~=x~+EJi(x,U)+O(~Z), u ‘~==*4EiQla(2,U)+O(E2),
i=1,2,...,p,
(6)
o=
(7)
1,2 ,*..,Q*
The functions & and & are the infinitesimals of the transformations for the independent and dependent variables, respectively. In order to find the unknown infinitesimals
where 8 represents a linear combination of the vector fields generating .C, which in turn is based on the characteristic quantities & and I& of the transformation (6),(7). The algorithm used in the Mathematiccs program for finding the infinitesimals & and & is described below. Please note that the infinites~~s depend on this simple form only with respect to independent and dependent variables. A prolongation of the dependencies to derivatives extends the Lie symmetries to socalled generalized Lie symmetries. Transformations (6),(7) together with the transformations for the first, second, . . . derivatives of the @‘s are called first, second, . . . prolongations. Using these various extensions, the infinitesimal criterion for the invariance of (1) under the group (2) can be derived by P’(“)~A~,=~ = 0, (91
28
G. BAUMANN
where the kth prolongation
of the vector field v’ is given by
f--J): (Z,dk))6.
pr(“)v’=v’+
a=1
The second
summation
The kth prolongation
extends coefficients
00)
.I
to all multi-indices
J = (jl, . . . , jl) with 1 5 jl < p, 1 5 1 5 k.
c$; are given by
+;I(x7 u(k)
(11)
where ug = g and ~3,~ = 2. Thus, the system of differential equations (1) is invariant under the transformation of a one-parameter group with the infinitesimal generator (8) if the
procedure
for deriving
the determining
equations
for
the infinitesimals
3. INVARIANCE
BASED
ON FRtiCHET
DERIVATIVES
The F’rechet derivative can be considered to be a generalization of the total derivative. In this section, we will use this kind of derivative for formulating an efficient procedure for the calculation of the invariance condition used in the derivation of the determining equations. The F’rechet derivative as a variation of the dependent variables is defined by
WQ)
= $(.+,Q)i
.
(12)
E=O
In actuality the above equation means that the dependent variables are being replaced by their variations in the support function P. The dependent variables are replaced by the variables themselves and by test functions Q, which are multiplied by c. After our substitutions, we differentiate with respect to E and then set E = 0. This relation defined for an r-dimensional support function P and for a q-dimensional test function can be implemented in Mathematics very efficiently by using the pattern matching mechanism of Mathematics. Before discussing the implementation in Mathematics, let us look at the equivalence of the invariance condition in both formulations. To calculate the determining equations, we need the prolongation of a vector field ‘i?Q applied to the system of differential equations A. If we assume that the related characteristics Q depend on the dependent variables and their derivatives, we can write down a relation connecting the prolongation of a differential system with the Frechet derivative of the system in an ‘evolutionary representation.’ The term ‘evolutionary representation’ means that we only consider infinitesimal transformations which are independent of the independent variables. The connection between the prolongation and the F’rechet derivative is given by pr’k’fi~(A) = DA(Q). (13) This relation
follows from the definition
of the prolongation
in an evolutionary
representation
(14)
Symmetry Anaiysis
29
with &* = Qa(~(k)) depending only on the derivatives of the dependent variables k = 0, 1, . . . , see [2]. If, in addition, we use the definition of the Frechet derivative given by equation (12), we are able to reproduce immediately equation (13). For the calculation of point symmetries, we need a representation of the characteristics showing at least one linear dependence on the first derivatives of the dependent variables. that we can represent the characteristics by
This way of representing the char~teristics
We assume
is a special case of the general form Qa = Qa (~(~1).
With this definition, the kth prolongation coefficient (11) takes the form
$;c= 0.1&a +
2 Q-&
(16)
i=l
Substituting relation (16) into (10) and rearranging terms in the sums, we get
(17) A comparison of the terms in braced curled brackets with the definition of the complete derivative
shows that we can replace equation (17) with
where we used the expressions (14) to represent the Ph prolongation in an evolutionary form. If we now use the relation between the prolongation and the Frechet derivative (13), we can express the prolongation of a vector field by the Frechet derivative as
pdk47= D*(Q) + f: &Di.
(20)
i=l The above calculation demonstrates the equivalence of the standard formulation and the use of fiechet derivatives. We are now able to use this relation to reformulate the invariance condition (9) to derive the determining equations. The invariance condition (9) is replaced by the expression
At first glance, this expression appears to be very complicated from a calculational point of view. Indeed, using this formula in a manual calculation would be very cumbersome. In a pencil calculation, we would first have to replace all dependent variables and their derivatives with the variation of the dependent variables which must be extended up to kth order in the derivatives. After their substitutions, we would have to differentiate the expression with respect to (5and then set e = 0. These steps are very laborious by hand. However, all the steps can be handled quickly
30
G. BAUMANN
with Mathematics. Mathematics’s powerful matching procedures carry out these calculations with ease. The implementation steps are the same as the steps just described. The advantage of calculating the prolongation of a given system of differential equations does not only lie in its speed, but also in our flexibility the calculation
allowing
an extension
The above steps to derive the determining called MathLie. partial
differential
the independent
equations and dependent
symmetries of the system We note that equation symmetry
method.
relation (21). demonstrating
symmetries
equations
of the Mathematics
The result
to choose expressions
to generalized
for the infinitesimals variables.
& and &,
is that
the second
4. NONCLASSICAL
of linear
homogeneous
in which x and u are vectors determining
to the invariance side condition
In the following section, we will discuss its connection with equation (21).
in
in a Mathematics package
Lie is a system
These are the so-called
A. (21) looks very similar
The difference
are incorporated
function
for the characteristics
[lO,ll].
condition
equations
of
for the
of the nonclassical
Qa is not equal to zero in
the nonclassical
symmetry
method
by
METHOD
The nonclassical method was introduced by Bluman and Cole [12] upon studying the diffusion equation. A group theoretical explanation of this method is given in [13,14]. The procedure was extended in [13] to “weak symmetries” and “side conditions.” The vector fields of the nonclassical method do not need to form a Lie algebra. Hence, there can be a wider class of similarity solutions than in the classical case. However, oftentimes the results of both classical and nonclassical methods are equivalent. In such cases, the nonclassical method fails to produce new solutions. An essential observation by Bluman and Cole was that an invariant solution u(x) of (1) does not only solve the original system but also its surface condition given by
&a=%
o!=1,2
)...,
(22)
q.
The characteristics Qa are defined by relation (15). The nonclassical Lie method of symmetry reduction to the extended system A’(x,u
(k) ) = 0,
i = 1,2 ,...,
&a=% The invariance
criterion
for this system
DA(&)
method
applies the classical
m,
(23)
a= 1,2 ,...) q.
(24)
reads
+2
i=l
J&A
A=0 Qcx=o
= 0.
The added surface condition can be interpreted as a side condition or as a conditional equation introducing new dependencies on the derivatives of u. These relations have to be taken into account during the calculations. Basically there are two ways to use the side conditions in the calculation of the invariance criterion in (25). The algorithm proposed by Levi and Winternitz [14] eliminates the additional relation among the derivatives of u by substituting (22) into (25). On the other hand, the algorithm used by Clarkson and Mansfield [15] replaces the surface condition (24) in the original equations (23). After executing the replacements in both procedures, the classical symmetry reduction of Lie is used to derive the determining equations. Common to both algorithms is that the resulting determining equations for the infinitesimals ti and +* are an overdetermined nonlinear system of PDEs. Because of the additional relations among the derivatives, the set of solutions may but need not be larger than the one obtained by the classical method. These solutions yield neither a vector space nor a Lie-algebra. Using the
31
Symmetry Analysis
nonclassical method, the old solutions of (1) cannot be transformed to new ones. It is, however, possible to derive similarity solutions. The nonclassical method has been applied to various PDEs. New classes of solutions which cannot be obtained
by the classical
method have been found for the heat equation
Boussinesq equation [14], Burger’s equation [16], and the Fizhugh-Nagumo
(121, the
equation [17].
Since the calculation is very time-consuming, we have developed a Mathematics program which allows the derivation of the nonlinear determining equations for nonclassical symmetries. This alteration of Lie’s method is incorporated in the package MathLie. This package also gives the solutions of the nonlinear determining equations for some simple cases. Up to now we have discussed local symmetries of a given equation. we will describe the derivation of nonlocal symmetries.
In the following section,
We will show that the Frechet derivative
is also a useful tool to determine this type of symmetry.
5. POTENTIAL
SYMMETRIES
If we want to describe nonlocal symmetries, it is convenient to introduce new variables V(X) which are related to the old variables u(z) by additional equations. The original system has to be derivable from these equations. In other words, if (u(z), u(z)) satisfy the extended equations, u(x) also has to be a solution of the system A = 0. An auxiliary system with new variables can be introduced if at least one PDE of A can be written as a conservation law. Suppose one PDE of A, without loss of generality Am = 0, can be expressed as a conservation law [18] 2
Difi
(x, r&k-i))
= 0.
i=l
The system A can now be written in the form AV XJL(~) = 0, ( >
2 Difi
~=l,...,m-1,
(x,t&k-l))= 0.
(26) (27)
i=l According to (27), it is possible to introduce n - 1 auxiliary variables to form a new system of PDEs or a potential system Q(x, utk), u(l))
f’ (XJJ fl (V
(k-1)
_ 1 ) -t&
tk-l) ) = (-l)l
(28) [WE+1+ Vf;:] )
l
w-9 (30)
A” (x, ~(~1) = 0,
Y = 1,. . . ,m - 1.
(31)
The systems A and Q are closely related to each other. A symmetry of the system q is a symmetry of the original system A and vice versa. However, the same symmetry may assume a different purpose in the two systems. A point symmetry of !I’ could yield a nonlocal symmetry transformation in A. Such symmetries of Q are called potential symmetries of A (see [1,18,19]). A point symmetry of Q with the vector field
G. BAUMANN
32
is a potential symmetry of A if $(z, U, V) and [( z, u, V) essentially depend on the new auxiliary variables V(Z). Otherwise #’ projects onto a point symmetry of A corresponding to the vector field
The main problem in determining potential symmetries is to find useful potential systems which allow potential symmetries. In the following we will discuss how this problem can be solved. In the derivation of a potential system, we have to consider two cases. The first case is that the equations are already in a potential representation, and the second case is that a transformation to a potential form exists. The first case consists of the conserved form of equation (1) as discussed in the representations (26) and (27). In the second case, we have to take into account that some PDEs can be multiplied by factors X transforming the original equations into a conserved form [18,1!3]. This second case is important for deriving a potential system. Let us suppose the existence of a set of integrating factors X(2,?.&)= (X’(z, u), . -. , P(x,u)) for which at least one Xfi does not vanish. Then we can write down the following equation connecting the original equation with the conserved representation:
In such a case, system A can be replaced with the system a given by Ai = 0,
,...) p-l,p+l,...,m,
i=1,2
&&fi=o,
(34)
i=l
which we call an auxiliary system. The factors X have to be chosen carefully because both solutions of A = 0 and solutions of the system Ai = 0,
i=1,2
,...) /A-l,p+l,...,
m,
X’l(z, ?A)= 0
satisfy the modified system zi;. Thus, not all possible integrating equation XP(z,u) = 0
factors X are useful.
If the
has no solutions for any U, system 3 yields a useful potential system. In this case, we know that the point symmetries of the corresponding potential system are symmetries of the original system, too. A necessary condition for the factors X is
A=0
=0.
(35)
This condition systematically opens a way for finding integrating factors X (see [2,19]). The adjoint Prechet derivative ‘0; is the differential operator which is defined by the relation
s n
V’DAWdX
=
I
n
WDiVdx
(36)
Symmetry Analysis
for any
domain
fl c Wn and
’ * * 7Wm(z))
W’(4, we will easily see that
any smooth
with compact
V(x)
functions
support
33
=
in R (see [2,18]).
(V’(z),.
. . ,P(x)),
W(s)
By doing
a partial
integration,
=
formula (37)
is a matrix
representation
of the adjoint
Frechet
derivative
VO;;.
The procedure for finding potential systems can also be applied to an already known potential system. By applying this method step-by-step, it is possible to construct an entire chain of potential systems. Finally, the symmetry analysis can be applied to all those systems. The
solutions
functions
of the determining
condition
(35) for the integrating
that have to satisfy a PDE or a system
new potential
systems.
They only indicate
that
of PDEs.
factors
may yield
free
These factors are of no use for finding
the original
system
can be linearized.
In order
to use the information given by (35), an algorithm was proposed by Bluman [1,20] to linearize a scalar PDE or a system of PDEs by invertible mappings. With the aid of potential symmetries, new solutions which cannot be derived from local symmetries can be found for the original system A. The new solutions can be determined by calculating the finite transformations corresponding to a potential symmetry to construct new solutions from old ones or by computing the invariant solution. Another useful tool for finding new solutions of potential systems is given by the possibility of linearization by no invertible mappings [21]. S ometimes, the original system cannot be linearized by invertible mapping. In contrast to such a result, a potential system could admit an invertible mapping which in turn leads to a linearization of the potential system.
6. SYMMETRIES
OF A NONLINEAR
TELEGRAPH
EQUATION
The equation (38) represents the physical model of a nonlinear electrical system derivable from Maxwell’s equations. We assume in this derivation that the dielectric permeability E is a constant, and the magnetic permeability p depends on the field H. Following [22], we have E(E) = ~1 and p(H) which describes a ferromagnet in a strong external magnetic field. The point symmetry group of equation (38) is represented by the infinitesimals 51 = c3 + c4z,
generating separation
52 = Cl + C2h
space and time translations ansatz in the form
where ICIand k2 are integration
41 = (c4 - c2> ‘u,
as well as a scaling r&t)
constants.
= &,
1
= pi/H2,
invariant
solution
(3%
and a particular
2
This symmetry
group contains
no infinite-dimensional
subgroup since we are dealing with a nonlinear equation. There is thus no possibility to linearize equation (38) by invertible mappings. The symmetry analysis of the nonclassical method delivers another interesting case with infinitesimals 51 = 6%
52 = 1,
41 = 0,
(40)
where 6, 52, and ~$1 are again related to the variables z, t, and u. Here we have a remarkable situation in that the solutions of the invariant surface condition uu3:
+
ut = 0
(41)
G. BAUMANN
34
also solve the nonlinear telegraph equations (38). The problem of solving the PDE (38) has been simplified in order to solve a quasi-linear PDE of first order. PDEs of this type can be solved by the method of characteristics. The solution zt(s,t) of (41) obtained by this method is defined by the implicit equation z = G(u) + ut (42) for any arbitrary G(u). Of course, equation (42) cannot be solved explicitly for arbitrary G(Q). Explicit solutions u(z, t) can only be obtained if the defining equation (42) can be expressed as a polynomial in u of maximal degree 4, i.e., G(u) = &-,+ Sru + ts2u2 + asus + 54~‘. The alternate cases are
and G(U) = So +&u + &up
1 1 3 -- 2’ ‘zl 2’
withp=
For example, the solution of the equation z = 60 + 61~ + tu + &u2-a given by the expression u(z,t) =
&2 (-61
- t f
J(Sl
polynomial of degree 2-is
+ t)2 - 41i3(61 - 2,)
.
Calculating the potential systems of equation (38) gives us four nontrivial integrating factors X1, X2, As, and As. The integrating factors AZ, As, and X4 are not important for our examination of potential symmetries. Potential symmetries exist for Xr = 1. This case is connected with the potential system ?J; -
Ut U2
Ue = 0,
vi
=
(43)
0,
where u1 and ZLare the field variables. In addition to symmetries projecting onto the point symmetries of our original equation, the symmetry analysis of the system (43) shows a potential symmetry generated by the infinitesimals
ml,
$1 = -221211,
c5 = ; +td,
$9 = -2 log(u).
For this case, it is not possible to calculate explicitly the finite transformations of this infinitesimal transformation. The last two equations (46) of the corresponding system of ODES
fee(c) =tu -
d,
d ---U(E) = -2u& de
$6)
= f + td,
-$2(E)
= -2log(u),
can be decoupled and a solution can be found. The solutions of the remaining equations (45,46) cannot be computed due to the large number of dependent and independent variables involved. This is a typical problem when calculating potential symmetries. The discrete symmetry group of (38) a 1so contains another infinite-dimensional subgroup with the infinitesimals Cl = f1 (wl)
1
G = f2 (wl)
I
cpl = 0,
#2 = 0
(47)
representing a potential symmetry which allows a linearization of (38) by noninvertible mappings. The free functions fr(u, v’> and f2(u, v’) have to satisfy the linear system f,’ - f$ = 0,
-f,‘l + u”f, = 0.
(48)
Symmetry
Analysis
35
It should be noted that the nonlinear telegraph equation (38) cannot be linearized by an invertible mapping. Yet, the symmetries (47) indicate that it is possible to linearize the potential Equation (48) delivers the target system. system (43) by an invertible point transformation. Consequently, the original equation (38) is indirectly linearized since the solutions of system (43) (u, vl) are also solutions of equation (38) if ZJ~is ignored. Following the method proposed by Bluman [l], the potential system (43) is linearized by a hodograph tr~sformation Zi = zb, Z2 =vl
and
W1 =z 7 w= = t
(49)
with new independent variables 31, ~2 and new dependent variables w’(zr , ~2) and w2(zi, .x2), The transformed equations
-w& + wz2= 0,
-wl z2 +
zfwf,= 0
are a linear homogeneous system of PDEs. There are various techniques for solving equations (50). A simple way to solve this system is by introducing a potential function f(~r, zz), which solves the scalar PDE &,ZZ - fi,Z, = 0 (51) of second order where the dependent variables are given by gradients of the potential: w1 = fi,, w2 = fi,. A common way to solve PDE (51) is by making a separation ansatz in the following form: f(zl, ~2) = J’(aP(~2). After some manipulations and inverting equations (49), we find the solution in an implicit form x2 - u + t2u2 = 0 , whence .(x,,,=&*$Jm.
(52)
This type of solution cannot be derived by other methods. Another way to solve equation (51) is by using Lie’s symmet~ procedure. The result of the analysis is a four-Dimensions symmetry group represented by the infinitesim~s
El =
c4z1
+
2C2ZlZ2,
b
=
c3 + 2c2 log (z2)
,
$1 = sf
+
C2Z2f
+ g (.a,
a)
.
(53)
The free function g(zi, zz) reflects linearity and needs to satisfy the original equation (51). The group constant cl connects to the homogeneity of equation (51) while ca generates translations in time. These group constants are of no use for finding invariant solutions. The invariant solution corresponding to the infinitesimal generator
is a special form of the separation ansatz used in the discussion above. Only the invariant solution of the generator
created by setting cz to unity and the remaining group constants to zero leads to a new class of solutions. The invariant surface condition v’f = 0 is solved by the similarity solution (54) with the similarity variable C = ~22- (iog(zr))2. The reduced equation of (51) reads zu(C) -t 16w’(C) + 16w”(~~ = 0 HmZI:B,P-8’
G. BAUMANN
36
and can be solved in terms of Bessel functions
Differentiation
of equation (54) yields the solutions of equations (50)
Wl(Q,z2>=
~(-k~J~(~)-keys),
(55)
w2(a f 22) = ~~~J~(~)+k2Y~(~))
+ &zzl)
(IcrJi ($)
+k2Yl
($)).
(56)
The solutions of the potenti~ system (43) can be obtained by resubstituting rules (49). The solutions are defined by the implicit equations &Vi 5 = 2t/(Vi)2 - (log(V))2
t
= -
1
26
l-
@Jo
( $./(v’)~
Ji; hid4
Q(Vi)2 2
- (log(~))2
the transformation
(VI)2 - (log(V)2 Q;,
- (10g(u))2) + kzYo ( &$J’)~ (klJ1 (~~~)
- (bdu))2)) +kzYl
(~J~)),,
2j/?7TGz
It is obvious that U(Z, t) and Vi(zr, t) cannot be derived in explicit form by these equations. In addition to the Lie point symmetries of (51), the nonclassical method yields no further results. In the case of the nonlinear telegraph equation (38), it is interesting to look for nonclassical symmetries of the potential system. Generally, the nonclassical method delivers no new solutions for infinitesimals unless C#Q and (62are zero. In such a case, the determining equations are satisfied identically for arbitrary &(z, t, u,d). Here, U(Z, t) and vl(z, t) have to solve the invariant surface condition &%
+ % =
0,
.$v;+vt’=o
(59)
in addition to the potential system (43). The only solutions of this extended system are ~(2, t) = k1 and v1 (2, t) = k2 for arbitrary &. So u and V1 are trivial solutions and linearly dependent. Hence, the nonclassical method for the potential system is of no use if we wish to discover new solutions for the nonlinear telegraph equation (38). 7. CONCLUSIONS All calculations presented in the previous sections are easily performed by Muthematica. The symmetry analysis in its standard Lie form and the determination of potential symmetries are completely automatic calculations found in the package MathLie. For the nonclassical method, we can use the MathLie package to derive the nonlinear determining equations. In some cases, these equations can be solved with the equation solver inside the package. In other cases, we can use the package to solve the equations interactively. The standard and potential form of the resulting deter~ning equations are linear partial differenti~ equations. For such cases, the MathLie package is capable of solving equations in a completely automatic way. The solution procedure consists of two independent steps. If necessary, a canonical form of the determining equation is created. The second independent step consists of simple integration rules for linear partial differenti~ equations. Thus, we are able to find the solutions of a large class of linear coupled partial differential equations.
Symmetry The
derivation
of the determining
We are able to attain connection
with simple definitions based on the F’rkhet
of the prolongation. derivative.
37
of the discussed
great efficiency if we use the pattern
operations
of the determining
equations
Analysis
symmetry
matching
The prolongation
Both properties
methods
capabilities
is very efficient.
of Mathematics in
is calculated
allow a fast and reliable
by simple derivation
equations.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
11. 12. 13. 14. 15. 16. 17. 18.
19. 20. 21. 22. 23.
G.W. Bluman and S. Kumei, Symmetries and Diflerential Equations, Springer, New York, (1989). P.J. Olver, Applications of Lie Groups to Diflerential Equations, Springer, Berlin, (1986). L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, (1982). F. Schwarz, Comput. Phys. Comm. 39, 285 (1986). B. Champagne, W. Hereman and P. Winternitz, CRM-1689, Universitb de Montrbal, (1990). F. Schwarz, SIAM Review 30, 450 (1988). T. Wolf, An efficient improved program LIEPDE for determining Lie-symmetries of PDEs, Preprint, (1993). G.J. Reid and A. Wittkopf, A differential algebra package in MAPLE, (1993); ftp 137.82.36.21 login: anonymous, password: your e-mail address, directory: pub/reid/standardf orm. W. Hereman, Review of symbolic software for Lie symmetry analysis, Mathl. Comput. ModelZing (this issue). G. Baumann, Application of the generalized symmetry method, In Proc. of the International Workshop on Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, (Edited by M. Torresi and N. Ibragimov), Kluwer, Dordrecht, (1993). G. Baumann, Generalized symmetries: A Mathematics program to determine Lie-Biicklund symmetries, MathSource 0204-680, Wolfram Research, Champaign, IL, (1993). G.W. Bluman and J.D. Cole, J. Math. Mech. 18, 1025 (1969). P.J. Olver and P. Rosenau, SIAM J. Appl. Math. 47, 263 (1987). D. Levi and P. Winternitz, J. Phys. A: Math. Gen. 22, 2915 (1989). P.A. Clarkson and E.L. Mansfield, Algorithms for the non classical method of symmetry reduction, Preprint, (1994). E. Pucci, J. Phys. A: Math. Gen. 25, 2631 (1992). M.C. Nucci and P.A. Clarkson, Phys. Lett. A 164, 49 (1992). G.W. Bluman, Potential symmetries and equivalent conservation laws, In Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, (Edited by N.H. Ibragimov), p. 71, Kluwer, Dordrecht , (1993). G.W. Bluman, The use of factors to discover potential systems or lineariaations, Preprint, (1993). G.W. Bluman and S. Kumei, Euro. J. Appl. Math. 1, 189 (1990). G.W. Bluman and S. Kumei, Euro. J. Appl. Math. 1, 217 (1990). I.G. Katayev, Electromagnetic Shock Waves, Iliffe Books, London, (1966). G. Baumann, Lie symmetries of differential equations, a Mathematics program to determine Lie symmetries, MathSource 0202-622, Wolfram Research, Champaign, IL, (1993).
Pergamon
PII: s0895-7177(97)00056-3
Symmetry
Analysis of Different ial Equations with ~~th~~at~~a G. BAU~ANN Abteilung fiir Mathemat~he Physik, Universitiit Ulm Albert-Einstein-A&e 11, D-89069 Ulm, Germany [email protected]
Abstract-we will discuss three different methods for finding symmetry solutions based on the Frkchet derivative common to each procedure. The methods discussed are Lie’s standardprocedure of symmetry analysis, the nonclassical method, and the derivation of potential symmetries. A ferromagnet in a strong external field represented by a nonlinear telegraph equation serves as an example describing the application of all three methods. The symmetry methods discussed are realized in a Mathematics package called MathLie performing all of the required calculations. Keywords-Lie,
Symmetry, Computer algebra, ~~th~~t~~,
Group theory.
1. INTRODUCTION The subject of this paper is systems of differential equations (DEQ). We discuss the invariance of the dependent and independent variables of the systems under transformation. The central point of our examinations is a special type of transformation, the so-called Lie symmetry. A Lie or a point symmetry depends on continuous parameters mapping every solution of the equations to another solution of the same equations. Before we give a definition for a Lie symmetry, we will define the more general term of any kind of symmetry for differential equations. DEFINITION:
SYMMETRY. We define a symmetry of any given di~erenti~ equation es a tmnsformation whj~h maps any caution of the d~~erentj~ equatjon to another ~~utjon.
Such a general definition of a symmetry allows a great variety of tr~sformations under which the differential equation can be invariant. One of the many possibilities of such transformations was introduced by Lie in the form of point transformations. In later sections of this paper, we will discuss other types of transformations like potential and nonclassical transformations. These transformations are also possibilities contained in the definition of a symmetry. Let us now specify the content of Lie or point symmetries. DEFINITION: LIESYMMETRY.
A Lie (point) symmetry is characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all jndependent and dependent va&&es. It z&o possesses the properties of our first definition. Evidently, the Lie symmetries of DEQs form a group: composition of any two s~metri~ yields a symmetry, there is an identity tr~sformation, and for any symmetry there is an inverse transformation. The composition of symmetries is obviously associative. The theory of Lie symmetry groups of DEQs was developed by Sophus Lie in the late nineteenth century and is today a well-studied topic. Such Lie groups are invertible point transformations of both the dependent and independent variables of the DEQs and may also depend on continuous parameters. In his work, Lie pointed out that this type of symmetry group is of great importance to understand and
25
26
G. B~UMANN
construct solutions of DE&s. Lie demonstrated that the many techniques for finding solutions of differential equations can be unified and extended by considering symmetry groups. Today, we know several applications of Lie groups in the theory of differential equations. The most important ones are summarized below: (i) (ii) (iii) (iv) (v) (vi) (vii)
reduction of the order for ordinary differential equations, mapping solutions to other solutions, reduction of the number of independent variables of partial differential equations, construction of invariant solutions, construction of invariant solutions to boundary value problems, construction of conser~tion laws, detection of linearizing tra~formations of DE&s.
For the many other applications of Lie group symmetries, refer to [l-3]. To use any of the above-mentioned methods for a given set of DEQs we must first be able to find the symmetries of the equations. The first step for finding point symmetries is to make a general change of all variables and then enforce the new variables to satisfy the same set of DEQs. This approach leads to complicated nonlinear systems of DEQs for the functions used in the transformations. Lie proved that such a procedure is not necessary. He established an efficient method based on an infinitesimal formulation of the problem of finding the symmetry group of a set of DEQs. This formulation replaces these highly complicated and in most cases intractable nonlinear equations with tractable linear overdetermin~ systems of partial di~erential equations. The solution of these so-called infinit~im~ determining equations can be used to determine symmetry transformations. The Lie point symmetries used here are point transformations in the space of dependent and independent variables of a system of DEQs. Under a symmetry transformation, the image of a point traces out a path which is analogous to the trajectory of a particle. The collection of all trajectories can be viewed as a fluid flow. The tangents to this flow represent its velocity vector field and may be regarded as a differential operator. This differential operator is interpreted by the rate of change following the fluid. Finding symmetries for a given set of differential equations involves setting up and solving an associated system of linear homogeneous partial differential equations called determining equations. We will discuss how determining equations arise from symmet~ properties and illustrate this development by outlining the derivation of such equations for several examples. The concepts and principles of symmetries play an important role in mathematics and physics. It appears that cases of exact solutions of differential equations are based on the use of symmetries of these equations with respect to certain transformation. Symmetry analysis is one of the systematic and accurate ways to obtain solutions of differential equations. An alternative method especially for solving nonlinear partial differential equations is the inverse scattering method. This method is only applicable for completely integrable equations of motion, whereas a symmetry analysis is also useful in the study of nonintegrable equations of motion. A major obstacle in the application of symmetry analysis is that a large number of tedious calculations is usually involved. The purpose of the following sections is to present the theoretical background for the calculations of Lie point symmetries. Based on thii theory we have developed a program in ~~~~e~a~~ca doing all the necessary calculations within no time. Similar programs written in REDUCE, MACSYMA, AXIOM, and MAPLE are available on large scale computers [4--81. Recent reviews of such programs are given in [5,6,9].
2. LIE’S PROCEDURE In this section, we outline the general procedure for determining point symmetries for any system of equations. Let us consider the general case of a nonlinear system of differential equations for an arbitrary number q of unknown functions ua which may depend on p independent vari-
Symmetry
Analysis
27
ables xi. We simply denote these sets of variables by a= (r~‘, u2,. . . , uq) and 5 = (1i,22,. respectively. The general case is given by a system of m nonlinear differential equations Ai (z,@))
(1)
i=l,Z,...,m
= 0,
. , zp),
of order Ic. The term ~l(~) is understood as kth derivative of u with respect to 2. We note that m, Ic, p, and Q are arbitrary positive integers. Consider further a one-parameter e-Lie group of transformations 5’ = E(z, 21;c), u’ = @(X121;c) (2) under which (1) must be invariant. Invariance of (1) under the transformation (2) means that any solution u = e(z) of (1) maps into another solution u = @(s; E) of (1). Let 2~= 8(x:) be a soliution of (1). If we replace the dependent and independent variables u and x with ‘u and z f = E, equations (1) become Ai (z’, V(~)) = 0,
i = 1,2 ,..., m.
(3)
Then TJ= e(z’) are solutions of (3). This implies that if (1) and (3) have a unique solution, then 8
Hence 8 satisfies the one-parameter
(2’) = * (x,0(x); c) .
(4
functional equation 8 (Z(2; E)) = (a(%,6; E).
(51
Expanding equations (2) around the identity E = 0, we can generate the following infinitesimal transformations:
Z~=x~+EJi(x,U)+O(~Z), u ‘~==*4EiQla(2,U)+O(E2),
i=1,2,...,p,
(6)
o=
(7)
1,2 ,*..,Q*
The functions & and & are the infinitesimals of the transformations for the independent and dependent variables, respectively. In order to find the unknown infinitesimals
where 8 represents a linear combination of the vector fields generating .C, which in turn is based on the characteristic quantities & and I& of the transformation (6),(7). The algorithm used in the Mathematiccs program for finding the infinitesimals & and & is described below. Please note that the infinites~~s depend on this simple form only with respect to independent and dependent variables. A prolongation of the dependencies to derivatives extends the Lie symmetries to socalled generalized Lie symmetries. Transformations (6),(7) together with the transformations for the first, second, . . . derivatives of the @‘s are called first, second, . . . prolongations. Using these various extensions, the infinitesimal criterion for the invariance of (1) under the group (2) can be derived by P’(“)~A~,=~ = 0, (91
28
G. BAUMANN
where the kth prolongation
of the vector field v’ is given by
f--J): (Z,dk))6.
pr(“)v’=v’+
a=1
The second
summation
The kth prolongation
extends coefficients
00)
.I
to all multi-indices
J = (jl, . . . , jl) with 1 5 jl < p, 1 5 1 5 k.
c$; are given by
+;I(x7 u(k)
(11)
where ug = g and ~3,~ = 2. Thus, the system of differential equations (1) is invariant under the transformation of a one-parameter group with the infinitesimal generator (8) if the
procedure
for deriving
the determining
equations
for
the infinitesimals
3. INVARIANCE
BASED
ON FRtiCHET
DERIVATIVES
The F’rechet derivative can be considered to be a generalization of the total derivative. In this section, we will use this kind of derivative for formulating an efficient procedure for the calculation of the invariance condition used in the derivation of the determining equations. The F’rechet derivative as a variation of the dependent variables is defined by
WQ)
= $(.+,Q)i
.
(12)
E=O
In actuality the above equation means that the dependent variables are being replaced by their variations in the support function P. The dependent variables are replaced by the variables themselves and by test functions Q, which are multiplied by c. After our substitutions, we differentiate with respect to E and then set E = 0. This relation defined for an r-dimensional support function P and for a q-dimensional test function can be implemented in Mathematics very efficiently by using the pattern matching mechanism of Mathematics. Before discussing the implementation in Mathematics, let us look at the equivalence of the invariance condition in both formulations. To calculate the determining equations, we need the prolongation of a vector field ‘i?Q applied to the system of differential equations A. If we assume that the related characteristics Q depend on the dependent variables and their derivatives, we can write down a relation connecting the prolongation of a differential system with the Frechet derivative of the system in an ‘evolutionary representation.’ The term ‘evolutionary representation’ means that we only consider infinitesimal transformations which are independent of the independent variables. The connection between the prolongation and the F’rechet derivative is given by pr’k’fi~(A) = DA(Q). (13) This relation
follows from the definition
of the prolongation
in an evolutionary
representation
(14)
Symmetry Anaiysis
29
with &* = Qa(~(k)) depending only on the derivatives of the dependent variables k = 0, 1, . . . , see [2]. If, in addition, we use the definition of the Frechet derivative given by equation (12), we are able to reproduce immediately equation (13). For the calculation of point symmetries, we need a representation of the characteristics showing at least one linear dependence on the first derivatives of the dependent variables. that we can represent the characteristics by
This way of representing the char~teristics
We assume
is a special case of the general form Qa = Qa (~(~1).
With this definition, the kth prolongation coefficient (11) takes the form
$;c= 0.1&a +
2 Q-&
(16)
i=l
Substituting relation (16) into (10) and rearranging terms in the sums, we get
(17) A comparison of the terms in braced curled brackets with the definition of the complete derivative
shows that we can replace equation (17) with
where we used the expressions (14) to represent the Ph prolongation in an evolutionary form. If we now use the relation between the prolongation and the Frechet derivative (13), we can express the prolongation of a vector field by the Frechet derivative as
pdk47= D*(Q) + f: &Di.
(20)
i=l The above calculation demonstrates the equivalence of the standard formulation and the use of fiechet derivatives. We are now able to use this relation to reformulate the invariance condition (9) to derive the determining equations. The invariance condition (9) is replaced by the expression
At first glance, this expression appears to be very complicated from a calculational point of view. Indeed, using this formula in a manual calculation would be very cumbersome. In a pencil calculation, we would first have to replace all dependent variables and their derivatives with the variation of the dependent variables which must be extended up to kth order in the derivatives. After their substitutions, we would have to differentiate the expression with respect to (5and then set e = 0. These steps are very laborious by hand. However, all the steps can be handled quickly
30
G. BAUMANN
with Mathematics. Mathematics’s powerful matching procedures carry out these calculations with ease. The implementation steps are the same as the steps just described. The advantage of calculating the prolongation of a given system of differential equations does not only lie in its speed, but also in our flexibility the calculation
allowing
an extension
The above steps to derive the determining called MathLie. partial
differential
the independent
equations and dependent
symmetries of the system We note that equation symmetry
method.
relation (21). demonstrating
symmetries
equations
of the Mathematics
The result
to choose expressions
to generalized
for the infinitesimals variables.
& and &,
is that
the second
4. NONCLASSICAL
of linear
homogeneous
in which x and u are vectors determining
to the invariance side condition
In the following section, we will discuss its connection with equation (21).
in
in a Mathematics package
Lie is a system
These are the so-called
A. (21) looks very similar
The difference
are incorporated
function
for the characteristics
[lO,ll].
condition
equations
of
for the
of the nonclassical
Qa is not equal to zero in
the nonclassical
symmetry
method
by
METHOD
The nonclassical method was introduced by Bluman and Cole [12] upon studying the diffusion equation. A group theoretical explanation of this method is given in [13,14]. The procedure was extended in [13] to “weak symmetries” and “side conditions.” The vector fields of the nonclassical method do not need to form a Lie algebra. Hence, there can be a wider class of similarity solutions than in the classical case. However, oftentimes the results of both classical and nonclassical methods are equivalent. In such cases, the nonclassical method fails to produce new solutions. An essential observation by Bluman and Cole was that an invariant solution u(x) of (1) does not only solve the original system but also its surface condition given by
&a=%
o!=1,2
)...,
(22)
q.
The characteristics Qa are defined by relation (15). The nonclassical Lie method of symmetry reduction to the extended system A’(x,u
(k) ) = 0,
i = 1,2 ,...,
&a=% The invariance
criterion
for this system
DA(&)
method
applies the classical
m,
(23)
a= 1,2 ,...) q.
(24)
reads
+2
i=l
J&A
A=0 Qcx=o
= 0.
The added surface condition can be interpreted as a side condition or as a conditional equation introducing new dependencies on the derivatives of u. These relations have to be taken into account during the calculations. Basically there are two ways to use the side conditions in the calculation of the invariance criterion in (25). The algorithm proposed by Levi and Winternitz [14] eliminates the additional relation among the derivatives of u by substituting (22) into (25). On the other hand, the algorithm used by Clarkson and Mansfield [15] replaces the surface condition (24) in the original equations (23). After executing the replacements in both procedures, the classical symmetry reduction of Lie is used to derive the determining equations. Common to both algorithms is that the resulting determining equations for the infinitesimals ti and +* are an overdetermined nonlinear system of PDEs. Because of the additional relations among the derivatives, the set of solutions may but need not be larger than the one obtained by the classical method. These solutions yield neither a vector space nor a Lie-algebra. Using the
31
Symmetry Analysis
nonclassical method, the old solutions of (1) cannot be transformed to new ones. It is, however, possible to derive similarity solutions. The nonclassical method has been applied to various PDEs. New classes of solutions which cannot be obtained
by the classical
method have been found for the heat equation
Boussinesq equation [14], Burger’s equation [16], and the Fizhugh-Nagumo
(121, the
equation [17].
Since the calculation is very time-consuming, we have developed a Mathematics program which allows the derivation of the nonlinear determining equations for nonclassical symmetries. This alteration of Lie’s method is incorporated in the package MathLie. This package also gives the solutions of the nonlinear determining equations for some simple cases. Up to now we have discussed local symmetries of a given equation. we will describe the derivation of nonlocal symmetries.
In the following section,
We will show that the Frechet derivative
is also a useful tool to determine this type of symmetry.
5. POTENTIAL
SYMMETRIES
If we want to describe nonlocal symmetries, it is convenient to introduce new variables V(X) which are related to the old variables u(z) by additional equations. The original system has to be derivable from these equations. In other words, if (u(z), u(z)) satisfy the extended equations, u(x) also has to be a solution of the system A = 0. An auxiliary system with new variables can be introduced if at least one PDE of A can be written as a conservation law. Suppose one PDE of A, without loss of generality Am = 0, can be expressed as a conservation law [18] 2
Difi
(x, r&k-i))
= 0.
i=l
The system A can now be written in the form AV XJL(~) = 0, ( >
2 Difi
~=l,...,m-1,
(x,t&k-l))= 0.
(26) (27)
i=l According to (27), it is possible to introduce n - 1 auxiliary variables to form a new system of PDEs or a potential system Q(x, utk), u(l))
f’ (XJJ fl (V
(k-1)
_ 1 ) -t&
tk-l) ) = (-l)l
(28) [WE+1+ Vf;:] )
l
w-9 (30)
A” (x, ~(~1) = 0,
Y = 1,. . . ,m - 1.
(31)
The systems A and Q are closely related to each other. A symmetry of the system q is a symmetry of the original system A and vice versa. However, the same symmetry may assume a different purpose in the two systems. A point symmetry of !I’ could yield a nonlocal symmetry transformation in A. Such symmetries of Q are called potential symmetries of A (see [1,18,19]). A point symmetry of Q with the vector field
G. BAUMANN
32
is a potential symmetry of A if $(z, U, V) and [( z, u, V) essentially depend on the new auxiliary variables V(Z). Otherwise #’ projects onto a point symmetry of A corresponding to the vector field
The main problem in determining potential symmetries is to find useful potential systems which allow potential symmetries. In the following we will discuss how this problem can be solved. In the derivation of a potential system, we have to consider two cases. The first case is that the equations are already in a potential representation, and the second case is that a transformation to a potential form exists. The first case consists of the conserved form of equation (1) as discussed in the representations (26) and (27). In the second case, we have to take into account that some PDEs can be multiplied by factors X transforming the original equations into a conserved form [18,1!3]. This second case is important for deriving a potential system. Let us suppose the existence of a set of integrating factors X(2,?.&)= (X’(z, u), . -. , P(x,u)) for which at least one Xfi does not vanish. Then we can write down the following equation connecting the original equation with the conserved representation:
In such a case, system A can be replaced with the system a given by Ai = 0,
,...) p-l,p+l,...,m,
i=1,2
&&fi=o,
(34)
i=l
which we call an auxiliary system. The factors X have to be chosen carefully because both solutions of A = 0 and solutions of the system Ai = 0,
i=1,2
,...) /A-l,p+l,...,
m,
X’l(z, ?A)= 0
satisfy the modified system zi;. Thus, not all possible integrating equation XP(z,u) = 0
factors X are useful.
If the
has no solutions for any U, system 3 yields a useful potential system. In this case, we know that the point symmetries of the corresponding potential system are symmetries of the original system, too. A necessary condition for the factors X is
A=0
=0.
(35)
This condition systematically opens a way for finding integrating factors X (see [2,19]). The adjoint Prechet derivative ‘0; is the differential operator which is defined by the relation
s n
V’DAWdX
=
I
n
WDiVdx
(36)
Symmetry Analysis
for any
domain
fl c Wn and
’ * * 7Wm(z))
W’(4, we will easily see that
any smooth
with compact
V(x)
functions
support
33
=
in R (see [2,18]).
(V’(z),.
. . ,P(x)),
W(s)
By doing
a partial
integration,
=
formula (37)
is a matrix
representation
of the adjoint
Frechet
derivative
VO;;.
The procedure for finding potential systems can also be applied to an already known potential system. By applying this method step-by-step, it is possible to construct an entire chain of potential systems. Finally, the symmetry analysis can be applied to all those systems. The
solutions
functions
of the determining
condition
(35) for the integrating
that have to satisfy a PDE or a system
new potential
systems.
They only indicate
that
of PDEs.
factors
may yield
free
These factors are of no use for finding
the original
system
can be linearized.
In order
to use the information given by (35), an algorithm was proposed by Bluman [1,20] to linearize a scalar PDE or a system of PDEs by invertible mappings. With the aid of potential symmetries, new solutions which cannot be derived from local symmetries can be found for the original system A. The new solutions can be determined by calculating the finite transformations corresponding to a potential symmetry to construct new solutions from old ones or by computing the invariant solution. Another useful tool for finding new solutions of potential systems is given by the possibility of linearization by no invertible mappings [21]. S ometimes, the original system cannot be linearized by invertible mapping. In contrast to such a result, a potential system could admit an invertible mapping which in turn leads to a linearization of the potential system.
6. SYMMETRIES
OF A NONLINEAR
TELEGRAPH
EQUATION
The equation (38) represents the physical model of a nonlinear electrical system derivable from Maxwell’s equations. We assume in this derivation that the dielectric permeability E is a constant, and the magnetic permeability p depends on the field H. Following [22], we have E(E) = ~1 and p(H) which describes a ferromagnet in a strong external magnetic field. The point symmetry group of equation (38) is represented by the infinitesimals 51 = c3 + c4z,
generating separation
52 = Cl + C2h
space and time translations ansatz in the form
where ICIand k2 are integration
41 = (c4 - c2> ‘u,
as well as a scaling r&t)
constants.
= &,
1
= pi/H2,
invariant
solution
(3%
and a particular
2
This symmetry
group contains
no infinite-dimensional
subgroup since we are dealing with a nonlinear equation. There is thus no possibility to linearize equation (38) by invertible mappings. The symmetry analysis of the nonclassical method delivers another interesting case with infinitesimals 51 = 6%
52 = 1,
41 = 0,
(40)
where 6, 52, and ~$1 are again related to the variables z, t, and u. Here we have a remarkable situation in that the solutions of the invariant surface condition uu3:
+
ut = 0
(41)
G. BAUMANN
34
also solve the nonlinear telegraph equations (38). The problem of solving the PDE (38) has been simplified in order to solve a quasi-linear PDE of first order. PDEs of this type can be solved by the method of characteristics. The solution zt(s,t) of (41) obtained by this method is defined by the implicit equation z = G(u) + ut (42) for any arbitrary G(u). Of course, equation (42) cannot be solved explicitly for arbitrary G(Q). Explicit solutions u(z, t) can only be obtained if the defining equation (42) can be expressed as a polynomial in u of maximal degree 4, i.e., G(u) = &-,+ Sru + ts2u2 + asus + 54~‘. The alternate cases are
and G(U) = So +&u + &up
1 1 3 -- 2’ ‘zl 2’
withp=
For example, the solution of the equation z = 60 + 61~ + tu + &u2-a given by the expression u(z,t) =
&2 (-61
- t f
J(Sl
polynomial of degree 2-is
+ t)2 - 41i3(61 - 2,)
.
Calculating the potential systems of equation (38) gives us four nontrivial integrating factors X1, X2, As, and As. The integrating factors AZ, As, and X4 are not important for our examination of potential symmetries. Potential symmetries exist for Xr = 1. This case is connected with the potential system ?J; -
Ut U2
Ue = 0,
vi
=
(43)
0,
where u1 and ZLare the field variables. In addition to symmetries projecting onto the point symmetries of our original equation, the symmetry analysis of the system (43) shows a potential symmetry generated by the infinitesimals
ml,
$1 = -221211,
c5 = ; +td,
$9 = -2 log(u).
For this case, it is not possible to calculate explicitly the finite transformations of this infinitesimal transformation. The last two equations (46) of the corresponding system of ODES
fee(c) =tu -
d,
d ---U(E) = -2u& de
$6)
= f + td,
-$2(E)
= -2log(u),
can be decoupled and a solution can be found. The solutions of the remaining equations (45,46) cannot be computed due to the large number of dependent and independent variables involved. This is a typical problem when calculating potential symmetries. The discrete symmetry group of (38) a 1so contains another infinite-dimensional subgroup with the infinitesimals Cl = f1 (wl)
1
G = f2 (wl)
I
cpl = 0,
#2 = 0
(47)
representing a potential symmetry which allows a linearization of (38) by noninvertible mappings. The free functions fr(u, v’> and f2(u, v’) have to satisfy the linear system f,’ - f$ = 0,
-f,‘l + u”f, = 0.
(48)
Symmetry
Analysis
35
It should be noted that the nonlinear telegraph equation (38) cannot be linearized by an invertible mapping. Yet, the symmetries (47) indicate that it is possible to linearize the potential Equation (48) delivers the target system. system (43) by an invertible point transformation. Consequently, the original equation (38) is indirectly linearized since the solutions of system (43) (u, vl) are also solutions of equation (38) if ZJ~is ignored. Following the method proposed by Bluman [l], the potential system (43) is linearized by a hodograph tr~sformation Zi = zb, Z2 =vl
and
W1 =z 7 w= = t
(49)
with new independent variables 31, ~2 and new dependent variables w’(zr , ~2) and w2(zi, .x2), The transformed equations
-w& + wz2= 0,
-wl z2 +
zfwf,= 0
are a linear homogeneous system of PDEs. There are various techniques for solving equations (50). A simple way to solve this system is by introducing a potential function f(~r, zz), which solves the scalar PDE &,ZZ - fi,Z, = 0 (51) of second order where the dependent variables are given by gradients of the potential: w1 = fi,, w2 = fi,. A common way to solve PDE (51) is by making a separation ansatz in the following form: f(zl, ~2) = J’(aP(~2). After some manipulations and inverting equations (49), we find the solution in an implicit form x2 - u + t2u2 = 0 , whence .(x,,,=&*$Jm.
(52)
This type of solution cannot be derived by other methods. Another way to solve equation (51) is by using Lie’s symmet~ procedure. The result of the analysis is a four-Dimensions symmetry group represented by the infinitesim~s
El =
c4z1
+
2C2ZlZ2,
b
=
c3 + 2c2 log (z2)
,
$1 = sf
+
C2Z2f
+ g (.a,
a)
.
(53)
The free function g(zi, zz) reflects linearity and needs to satisfy the original equation (51). The group constant cl connects to the homogeneity of equation (51) while ca generates translations in time. These group constants are of no use for finding invariant solutions. The invariant solution corresponding to the infinitesimal generator
is a special form of the separation ansatz used in the discussion above. Only the invariant solution of the generator
created by setting cz to unity and the remaining group constants to zero leads to a new class of solutions. The invariant surface condition v’f = 0 is solved by the similarity solution (54) with the similarity variable C = ~22- (iog(zr))2. The reduced equation of (51) reads zu(C) -t 16w’(C) + 16w”(~~ = 0 HmZI:B,P-8’
G. BAUMANN
36
and can be solved in terms of Bessel functions
Differentiation
of equation (54) yields the solutions of equations (50)
Wl(Q,z2>=
~(-k~J~(~)-keys),
(55)
w2(a f 22) = ~~~J~(~)+k2Y~(~))
+ &zzl)
(IcrJi ($)
+k2Yl
($)).
(56)
The solutions of the potenti~ system (43) can be obtained by resubstituting rules (49). The solutions are defined by the implicit equations &Vi 5 = 2t/(Vi)2 - (log(V))2
t
= -
1
26
l-
@Jo
( $./(v’)~
Ji; hid4
Q(Vi)2 2
- (log(~))2
the transformation
(VI)2 - (log(V)2 Q;,
- (10g(u))2) + kzYo ( &$J’)~ (klJ1 (~~~)
- (bdu))2)) +kzYl
(~J~)),,
2j/?7TGz
It is obvious that U(Z, t) and Vi(zr, t) cannot be derived in explicit form by these equations. In addition to the Lie point symmetries of (51), the nonclassical method yields no further results. In the case of the nonlinear telegraph equation (38), it is interesting to look for nonclassical symmetries of the potential system. Generally, the nonclassical method delivers no new solutions for infinitesimals unless C#Q and (62are zero. In such a case, the determining equations are satisfied identically for arbitrary &(z, t, u,d). Here, U(Z, t) and vl(z, t) have to solve the invariant surface condition &%
+ % =
0,
.$v;+vt’=o
(59)
in addition to the potential system (43). The only solutions of this extended system are ~(2, t) = k1 and v1 (2, t) = k2 for arbitrary &. So u and V1 are trivial solutions and linearly dependent. Hence, the nonclassical method for the potential system is of no use if we wish to discover new solutions for the nonlinear telegraph equation (38). 7. CONCLUSIONS All calculations presented in the previous sections are easily performed by Muthematica. The symmetry analysis in its standard Lie form and the determination of potential symmetries are completely automatic calculations found in the package MathLie. For the nonclassical method, we can use the MathLie package to derive the nonlinear determining equations. In some cases, these equations can be solved with the equation solver inside the package. In other cases, we can use the package to solve the equations interactively. The standard and potential form of the resulting deter~ning equations are linear partial differenti~ equations. For such cases, the MathLie package is capable of solving equations in a completely automatic way. The solution procedure consists of two independent steps. If necessary, a canonical form of the determining equation is created. The second independent step consists of simple integration rules for linear partial differenti~ equations. Thus, we are able to find the solutions of a large class of linear coupled partial differential equations.
Symmetry The
derivation
of the determining
We are able to attain connection
with simple definitions based on the F’rkhet
of the prolongation. derivative.
37
of the discussed
great efficiency if we use the pattern
operations
of the determining
equations
Analysis
symmetry
matching
The prolongation
Both properties
methods
capabilities
is very efficient.
of Mathematics in
is calculated
allow a fast and reliable
by simple derivation
equations.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
11. 12. 13. 14. 15. 16. 17. 18.
19. 20. 21. 22. 23.
G.W. Bluman and S. Kumei, Symmetries and Diflerential Equations, Springer, New York, (1989). P.J. Olver, Applications of Lie Groups to Diflerential Equations, Springer, Berlin, (1986). L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, (1982). F. Schwarz, Comput. Phys. Comm. 39, 285 (1986). B. Champagne, W. Hereman and P. Winternitz, CRM-1689, Universitb de Montrbal, (1990). F. Schwarz, SIAM Review 30, 450 (1988). T. Wolf, An efficient improved program LIEPDE for determining Lie-symmetries of PDEs, Preprint, (1993). G.J. Reid and A. Wittkopf, A differential algebra package in MAPLE, (1993); ftp 137.82.36.21 login: anonymous, password: your e-mail address, directory: pub/reid/standardf orm. W. Hereman, Review of symbolic software for Lie symmetry analysis, Mathl. Comput. ModelZing (this issue). G. Baumann, Application of the generalized symmetry method, In Proc. of the International Workshop on Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, (Edited by M. Torresi and N. Ibragimov), Kluwer, Dordrecht, (1993). G. Baumann, Generalized symmetries: A Mathematics program to determine Lie-Biicklund symmetries, MathSource 0204-680, Wolfram Research, Champaign, IL, (1993). G.W. Bluman and J.D. Cole, J. Math. Mech. 18, 1025 (1969). P.J. Olver and P. Rosenau, SIAM J. Appl. Math. 47, 263 (1987). D. Levi and P. Winternitz, J. Phys. A: Math. Gen. 22, 2915 (1989). P.A. Clarkson and E.L. Mansfield, Algorithms for the non classical method of symmetry reduction, Preprint, (1994). E. Pucci, J. Phys. A: Math. Gen. 25, 2631 (1992). M.C. Nucci and P.A. Clarkson, Phys. Lett. A 164, 49 (1992). G.W. Bluman, Potential symmetries and equivalent conservation laws, In Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, (Edited by N.H. Ibragimov), p. 71, Kluwer, Dordrecht , (1993). G.W. Bluman, The use of factors to discover potential systems or lineariaations, Preprint, (1993). G.W. Bluman and S. Kumei, Euro. J. Appl. Math. 1, 189 (1990). G.W. Bluman and S. Kumei, Euro. J. Appl. Math. 1, 217 (1990). I.G. Katayev, Electromagnetic Shock Waves, Iliffe Books, London, (1966). G. Baumann, Lie symmetries of differential equations, a Mathematics program to determine Lie symmetries, MathSource 0202-622, Wolfram Research, Champaign, IL, (1993).