Symmetry Analysis of Partial Differential Equations (PDEs)

1. Introduction

The objective of MathLie is to efficiently combine computer algebra with solution strategies for differential equations. The advantage of MathLie is to obtain analytic solutions of partial as well as ordinary differential equations in split seconds. MathLie revives a unique theory by Sophus Lie and offers a large number of strategies to solve differential equations.

MathLie is a Mathematica program which allows to determine the symmetries of ordinary as well as  partial differential equations. The name MathLie is coined of the first part of Mathematica and in honor of Sophus Lie (1842-1899).

MathLie is available for Macs, PCs/Windows, and UNIX systems running Mathematica  Version 3.0 or later. The program can be used in an interactive and in a notebook version of Mathematica. MathLie allows the completly automatic derivation of   symmetries without any interaction by the user. The package MathLie is  capable of calculating point symmetries, non-classical symmetries, potential symmetries, approximate symmetries, and generalized symmetries of a given system of equations. The point symmetries, the potential symmetries, and the approximate symmetries are determined by the program without any interaction. The non-classical symmetries of a differential equation can be derived by an interactive procedure using functions offered in the MathLie package.

MathLie  allows the determination of symmetries for nearly all differential equations; linear, nonlinear, ordinary, partial, containing free functions, etc. The symmetries calculated by MathLie are

Point symmetries

Non-classical symmetries

Potential symmetries

Approximate symmetries

Generalized symmetries

and …in the future

Once the symmetries are known, MathLie is capable of checking the invariance of the differential equation by using the given symmetries. MathLie also allows the determination of the algebraic properties of the related Lie algebra. The package is capable of deriving  differential representations of the Lie algebra, calculating the commutator table,  determining the structure constants, deriving the metric of the algebra, calculating  finite group transformations, etc. Graphical representations of  vector fields and of  group transformations producing new solutions from known ones can also be created.

The method of symmetry analysis established by Sophus Lie is an algorithmic but time-consuming procedure to find solutions for differential equations (DEs). Today, this difficulty is solved by using computer algebra systems (CAS) such as Macsyma, Maple, Mathematica, or Reduce. Unfortunately, these standard systems do not support symmetry methods. Consequently, it is necessary to extend CAS to point symmetries, generalized symmetries, potential symmetries, and approximate symmetries.

The packages used in MathLie are a centered hierarchy of basic functions. The inner most levels contain data objects and basic functions used in outer levels.

The package LieBasic defines low level functions and data objects needed by other packages. The packages Matrix and Generator provide corresponding objects as well as appropriate functions for manipulating the basic functions. Consequently, all higher level packages, in particular LieAlgebra, are able to handle all objects. The structure is designed in such a way that it can be extended to any other objects, e.g. spinors, etc. The package DetEqus calculates the determining equations with options for point symmetries as well as generalized symmetries, potential symmetries, non-classical symmetries and approximate symmetries. Finally, the packages LieSolveODE and  LieSolvePDE provide functions for integrating determining equations, determining similarity reductions and solutions of differential equations. The inner packages of MathLie make up a complete Lie tool.

For computer systems with graphical user interface like X or Windows, there is an additional Graphical User Interface (GUI) providing standard mathematical notations. The master package MathLie loads the package GUI and any other package as soon as a built-in function is called.

We are currently working on extensions for generalized symmetries, first and higher order ODEs, algebraic structures and optimal systems of sub-algebras.

This talk is organized as follows:  In section two we review Lie's method using the terminology of today.  In section three, we discuss some capabilities of the package MathLie and present some examples of how to use MathLie to find symmetries. Section four contains some concluding remarks.

2. Lie's Theory Used in MathLie

Lie pointed out in his work that the symmetry of any differential equation is defined as follows:

Definition: Lie Symmetry

A Lie (point) symmetry is characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all independent and dependent variables.●

The Lie symmetries of differential equations (DEQs) naturally form a group: since the composition of any two symmetries is also a symmetry, there is an identity transformation; the composition of symmetries is obviously associative and any symmetry has an inverse. Such groups are called Lie groups and are invertible point transformations of both the dependent and independent variables of the DEQs. The DEQs may depend on continuous parameters. Lie pointed out that this group is of great importance in understanding and constructing solutions of DEQs. Lie demonstrated that many techniques for finding solutions can be unified and extended by considering symmetry groups. Today we know several applications of Lie groups in the theory of differential equations (cf. Ibragimov [1985], Bluman and Kumei  [1989], Olver [1986], Ovsiannikov [1982], Ibragimov [1994-1996], Baumann [2000]).

To use the symmetry groups in any application we first need to find the symmetries of the equations. A first approach to find point symmetries of such systems 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 demonstrated that such a procedure is unnecessary. He established an efficient method based on an infinitesimal formulation of the problem of finding the symmetry group of a set of DEQs replacing these highly complicated and in most cases intractable nonlinear equations by tractable linear overdetermined systems of partial differential equations. The solution of these so-called infinitesimal determining equations can be used to determine symmetry transformation.

Let us consider the general case of a nonlinear system of differential equations for an arbitrary number of unknown functions which may depend on   independent variables . We denote these sets of variables simply by and , respectively. The general case is given by a system of nonlinear differential equations

of order .  The term is understood as th derivative of   with respect to .  We note that , , and are arbitrary, positive integers.  Consider further a one-parameter ε-Lie group of transformations

under which (1) must be invariant. The star on the variables and denote the new variables. Invariance of (1) under the action of (2) and (3) means that any solution of (1) maps into some other solution of (1).  Let be a solution of (1).  If we replace the dependent and independent variables and by and , equations (1) become

Then are solutions of (4).  This implies that if (1) and (4) have a unique solution, then

.

Hence Θ satisfies the one-parameter functional equation

.

Expanding equations (2) and (3) around the identity , we can generate the  following infinitesimal transformations

functions and are the infinitesimals of the transformations for the independent and dependent variables, respectively.  In order to find the unknown infinitesimals and , we need to extend respectively to prolong the transformation group to include the properties of the derivatives. Meaning that we need to consider equations (1) and (4) as one and the same equation.  It is an infinitesimal approach which considers the Lie algebra L corresponding to the Lie group . In the algebraic description of the invariance the infinitesimal transformations (7) and (8) are associated with the vectorfield

representing a linear combination of the basic vector fields generating L which in turn is based on the characteristic quantities and of the transformations (7) and (8). The algorithm used in MathLie for finding the infinitesimals and is contained in the relations (1), (4), (7), and (8). We emphasize that the infinitesimals in this simple form only depend on independent and dependent variables. A prolongation of the dependencies to derivatives extends the Lie symmetries to so-called generalized Lie symmetries. Transformations (7,8), 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 (2,3) is derivable from the relation

.

Equation (10) represents the invariance of the differential equations under the transformation of the independent and dependent variables. To satisfy the invariance condition and the group properties it is necessary to expand the right hand side of equation (10) around the identity of (2) and (3) resulting to

.

Thus the invariance of (10) requires that the first expansion coefficients of with respect to must vanish. This allows us to introduce the notation

denoting the th prolongation of the vector field applied to the equation . Since the differential equation vanishes by itself, we can formulate a sufficient and necessary invariance condition by

.

The meaning of (13) from a calculations point of view with CAS is that the invariance condition is given by

.

In fact, equation (14) is used by MathLie to calculate the invariance condition (13) and derive the determining equations for the infinitesimals and . Contrary to the textbook procedures where the kth prolongation of the vector field is given by

,

we do not evaluate the recursive definition of the prolongation in (15) but use (14). The circumvention of (15) has the advantage that we do not need to calculate the expansion coefficients . Thus we profit in time and memory in a CAS calculation. In (15) the second summation extends over all multi indices with , . The kth expansion coefficients of the prolongation are given by

where and . Thus, the system of differential equations (1) is invariant under the transformation of a one-parameter group with the infinitesimal generator (9) if the 's and 's are determined from equation (14).

Since from (7) and (8) it is obvious that the infinitesimals and depend only on the independent and dependent variables and . Thus we can derive from (14) a set of determining equations for the infinitesimals by equating the coefficients of any product of derivatives of equal to zero. The result of this extraction problem is a linear system of coupled PDEs for the infinitesimals and . The extraction of the determining equations is also done by MathLie.

Knowing the determining equations, we can apply the solution procedures for linear coupled PDEs. Before we apply the solution steps, we calculate a general canonical form (standard form) of the determining equations by means of the Janet-Riquier theory. This pre-solution step allows us to simplify the determining equations and eliminate redundant information. After the derivation of the canonical form, we automatically solve the determining equations to obtain the admitted symmetries of the equation (1). The symmetries found determine the group properties of and are useful in deriving the final solution.

Before we construct the final solution let us discuss Lie's remarks on invariants. He called a function   an invariant of a one-parameter Lie group of transformation if the condition

is satisfied. This condition always results into a first-order PDE no matter of how large are the numbers of dependent and independent variables. Equation (17) is solvable by applying the method of characteristics. The reduction procedure itself is based on the following theorem.

Theorem: Invariant Representation

Let the equation be invariant under a one-parameter group G and let the infinitesimals , , , and be nonvanishing functions on the solution surface of the equation. In this case the surface can be represented by equations of the form

where , …, define a basis of invariants of the group G.

The steps discussed in the theorem are implemented in the MathLie function LieReduction[]. From the outlined procedure it is obvious that the reduction steps work as long as we can solve the invariant surface condition (17).  The procedure generally fails if we cannot solve (17) or we are unable to explicitly represent the invariants . In computer algebra systems such as Mathematica  this happens if the invariants contain transcendental functions.  MathLie fails exactly under these conditions because Mathematica is in general unable to solve transcendental equations.

The knowledge of the invariants allows us to simplify the original equation (1) in such a way that we can reduce the PDE to an ODE or in case of a ODE carry out quadratures. If the quadratures are successful, we end up with a solution. The whole procedure is summarized in the following diagram.

On the right side of the flow chart the corresponding functions of MathLie are given for each step. At this stage of the discussion we are prepared to apply the theory to some examples.

3. Application of the Theory

In this section we discuss the application of the theoretical formulas discussed to appropriately calculate Lie point symmetries. We apply the theoretical notions in terms of MathLie functions. The main tools discussed here are the prolongation operator, the derivation of the determining equations, the reduction of the equation, and the solution of the equation.

3.1 Calculation of Prolongations

Calculation of the prolongation in MathLie can easily be carried out by using equation (14). The theoretical concept of the prolongation is realized in MathLie by the function Prolongation[] or symbolically . This function expects for input four different quantities. The first argument of the function contains the equations , the second and third arguments specify the dependent and independent variables. The fourth slot contains the possible parameters of the equation. The input of parameters is optional and can be omitted if the equation contains no parameters. If the equation contains parameters they must be specified in the fourth slot. The application of Prolongation[] to  the heavenly equation demonstrates the use. The heavenly equation is given by

This equation is used in Einstein's theory of general relativity in the discussion of self-dual, Euclidean, Einstein spaces with one rotational Killing vector (heavens). It is important that for such a space the equation is determined by a single scalar field obeying the above nonlinear partial differential equation. The scalar field of the heavenly equation depends on three coordinates , , and . The application of this equation is not only restricted to Einstein's theory but appears in a variety of physical areas, ranging from the theory of Hamiltonian systems to general relativity.

Heavenly spaces have been studied by several authors in different frame works. The main aspects of these investigations concern i) the problem of the classification of the Riemann metric admitting at least one Killing vector field; ii) the discovery of extended conformal symmetries, and iii) the search of exact solutions of physical significance, such as self-dual gravitational instantons, whose roll is important in quantum gravity.

The prolongation of the heavenly equation follows by applying the function Prolongation[] to the equation

The result of the calculation represents the second order prolongation of the heavenly equation. The result created by the prolongation operator is identical with the result derived by the function Prolongation[]. The function behind the operator automatically detects the number of variables involved and creating the representation of the infinitesimals.

The result of this calculation in Mathematica notation was converted to a more readable form in index notation by the function LTF[]. This function reduces the standard Mathematica output to a shorter representation by deleting the arguments of any derivative and using the variables of differentiation as index. The expressions free of any derivatives remain unchanged.

As we know from the theoretical considerations, the prolongation of a differential equation is the basis for the derivation of determining equations.  In the following section, we will discuss the corresponding  function of MathLie which is instrumental in the derivation of determining equations.

3.2 Derivation of Determining Equations

Determining equations for infinitesimals are the result of invariance condition (14). The package MathLie provides a function allowing to derive determining equations for a given system of differential equations. The name of the function is  DeterminingEquations[] or in symbolic notation . This function needs five input arguments. The first argument contains the left hand sides of the equation . The second and third arguments are lists for the dependent and independent variables. The fourth list contains terms for which the equations Δ=0 is solved. The solutions with respect to these terms are used as side conditions in the invariance relation (14). If the equations under examination contain parameters, we can feed in these symbols in the last list. This list can be suppressed if no parameters are contained in the PDE. Function DeterminingEquations[] uses function Prolongation[] to calculate the kth prolongation of the equations. After the calculation of the prolongation the side conditions are applied to the result. This step reduces the redundant information in the manifold of the equation. Upon application of the side conditions, the determining equations are extracted as coefficients of the derivatives of the dependent variables. Since the infinitesimals themselves are independent of derivatives, we find the determining equations as a set of coupled PDEs.

The application of function DeterminingEquations[] is demonstrated by the heavenly equation. The determining equations of the infinitesimals for the heavenly equation follow from

The result consists of 16 determining equations. We transformed the Mathematica  expressions again to a more mathematical notation by using the function LTF[]. The 16 equations contain the unknown functions , and . Function DeterminingEquations[] automatically implants these names for the infinitesimals. The unknown functions , , and depend on the independent variables and on the dependent variable . The symmetries of the heavenly equation are determined by this set of equations.

Taking a closer look at these equations, we realize that they are linear but coupled. However, the main observation is that they are linear. Linearity is a general feature of the determining equations for point symmetries. This feature is of great advantage in the solution step of the determining equations. Another general property of the determining equations is that this set of equations is always overdetermined. This means that in general there exist more equations than unknown functions. This fact helps a lot in the derivation of the solution.

So far we have been able to calculate the determining equations for a given system of partial differential equations. The question arises how to solve these equations.

3.3 Solution of the Determining Equations

A function supporting the determination of the point symmetries in MathLie is Infinitesimals[]. The short hand notation of this function is  . Applying this function to the original equation MathLie automatically calculates the symmetries. The function Infinitesimals[] needs as input the equation of motion, the dependent and independent variables, and the parameters occurring in the equation.

The result for the heavenly equation consists of a two dimensional group of symmetries embedded in an infinite dimensional group represented by the arbitrary function . The finite subgroups are characterized by the group constants and representing translations and scaling symmetries. The arbitrary function has to satisfy the second order linear Poisson equation in and coordinates. Knowing the symmetries of the heavenly equation, we are able to reduce the original PDE to an ODE or at least to a PDE depending on a reduced set of independent variables.

3.4 Reduction and Solution of the Original Equation

For any reduction of the original PDE, we have to select a subgroup from the total symmetry  group.  In case of the heavenly equation, we can select different subgroups to find reductions. As a first example let us choose the arbitrary function in such a way that the Poisson equation is satisfied. We also assume that the group parameters and vanish.

First reduction  for

Assume the free function is given by the product y satisfying the Poisson equation. Then the subgroup for vanishing and is selected by the MathLie function SubGroupForReductions[]

The first similarity reduction of the heavenly equation is calculated by applying the MathLie function LieReduction[] to the original equation. The function LieReduction[] generates a similarity solution of the heavenly equation by determining the invariants of the subgroup.

z - τ == 0

The resulting similarity representation of the solution consist of the three invariants , , and . and are considered as the new independent variables and depends on these variables. Additionally has to satisfy a second order nonlinear PDE which determines the function . Thus we reduced a PDE in three independent variables , , and to a PDE in two variables and . At this stage we face the problem to solve a nonlinear PDE in two variables. The solution of the first reduced heavenly equation is derivable if we apply Lie's method a second time. Before caring out the calculations we rename the variables to prevent confusion in a second reduction. We choose

The point symmetries of the first reduced heavenly equation follow by applying the MathLie function Infinitesimals[] to the equation

The result of the calculation is a four dimensional discrete symmetry group. The symmetries are denoted by the group constants , …, . Among these symmetries are translation and scaling symmetries. We are now in a position to choose the group constants in different ways. Each selection will reduce the first reduction of the heavenly equation a second time. Before we start with the reduction let us examine some algebraic properties. First we calculate the commutator table and in the next step we decide which subalgebras are solvable.

The related commutator table is created by the function LieCommutatorTable[] and CTF[].

[,]
	0		0	0
		0	0	0
	0	0	0	0
	0	0	0	0

We can check the solvability of an algebra with the function SolvableAlgebrasOfOrderN[]. This function delivers those group constants which will result in solvable algebras. First let us calculate all subalgebras of order two which are solvable.

The result shows that there are six combinations of the group constants which will result into solvable algebras of order two. The next line calculates all algebras of order three

We find four solvable subalgebras of order three. The last check is concerned with the question is the original algebra solvable at all.

In fact, we find that the four dimensional algebra is solvable. So we know at this point that the reductions should be solvable too. Our next task is to find the solutions.

Reduction 1

The subgroup and is selected from the above group by

The second similarity reduction of the heavenly equation follows by applying the MathLie function LieReduction[] to the first reduction of the heavenly equation

Again the result represents a similarity solution. The invariants are now and . ω is called similarity variable and is known as similarity function. The similarity function has to satisfy a nonlinear ODE. This ODE reads

The solution of the reduced heavenly equation is gained by DSolve[]

The result represents the second similarity function in terms of the ProductLog[]. Where the product log function gives the solution for in . The function is also known as the Lambert function. The function can be viewed as a generalization of a logarithm. It can be used to represent solutions to a variety of transcendental equations. The solution of the first similarity reduction of the heavenly equation follows immediately from this solution by inserting the above result into the similarity representation of the second reduction. The solution with respect to the similarity function provides the solution of the first reduction

Inserting this representation of the solution into the first similarity solution and solving with respect to delivers the solution in original variables.

The solution is simplified if we specify the parameters and and the integration constants and .

The graphical representation of the solution follows by

We observe that the solution has a singularity for and .

Reduction 2

A second kind of solution follows by choosing the subgroup with and the remaining group constant equal to zero.

The corresponding similarity reduction follows by

τ - ω == 0

The similarity solution is given in the second line above. The similarity variable is just . The similarity function has to satisfy a nonlinear second order ODE which reads after elimination of common factors

The solution of this ODE follows from

representing a simple logarithm. The constants and are constants of integration. Inverting the similarity transformations of the second

and first reduction, we end up with the solution in original variables.

Specifying the integration constants and the parameter we find

The graphical representation of this solution is

We observe that the solution is singular for and decreases in a characteristic way for .

So far the construction of the final solution was based on the Mathematica function DSolve[]. The following similarity reduction demonstrates how Lie's theory can be used to find solutions of an ODE. This method is especially useful if DSolve[] fails to find a solution.

Reduction 3

The symmetry group we examine next is generated with and .

The subgroup represents a scaling of the second similarity variable and a translation with respect to the similarity function. The related reduction is determined by

-zeta1 + ζ == 0

The similarity function has to satisfy again a nonlinear second order ODE

Lie's theory is not only valid for PDEs but also for ODEs. The first step in an analysis is the determination of the symmetries by means of the MathLie function Infinitesimals[]

The result of the analysis is a two dimensional finite symmetry group. From this result it follows that the reduced ODE is solvable by quadratures. A theorem by Lie states that an nth order ODE possessing a n-dimensional symmetry group is always solvable by quadratures. The next step in the solution process is to find the quadratures. Another theorem by Lie stating that a known symmetry group can be used to simplify the representation of an ODE. A simplification occurs if we introduce so called canonical variables. Canonical variables are derivable from the point symmetries by solving a set of first order PDE possessing the infinitesimals as coefficients. In MathLie the function CanonicalVariables[] carries out this task. Specifying the symmetries

the canonical variables and are expressed in terms of the old variables and . These variables can be used to simplify the ODE. The MathLie function CanonicalRepresentation[] carries out this calculation. The calculation consists of a replacement of the old variables by the canonical variables, an ordering and a simplification.

The result compared with the original ODE is much simpler. Since the canonical representation of the ODE is free of the independent variable , we are able to apply a quadrature. First multiply with and take an integration over . The second step consists of a separation of variables and an integration. However, we try to solve the equation by DSolve[]

Since the canonical representation of the ODE is much simpler than the original ODE DSolve[] is now capable to find a solution. If we invert the canonical transformations and the similarity transformations step by step we gain the solution in original variables.

Specifying the constants of integration and the parameter κ of the heavenly equation in an appropriate way we find

The plot of this solution shows that a singularity due to the Sec[] function occurs.

Second reduction for

A second subgroup of the heavenly equation is gained if we incorporate the part of the finite group with in addition to the choice . The function in fact solves the Poisson equation. The related subgroup follows from

The first similarity reduction for this subgroup follows from

x - zeta2 == 0

Again the similarity function has to satisfy a nonlinear second order PDE

whose symmetries are

Thus the PDE of the first reduction allows a finite group of dimension two. Subgroups of this group determine the solution structure of the heavenly equation.

Reduction 1

A similarity reduction incorporating scaling and translation is selected by and .

The corresponding similarity reduction follows with

-zeta1 + ζ == 0

The remaining ODE for simplifies to the nonlinear ODE

Applying the MathLie function SecondOrderIntegrate[] to the second order ODE, we find a first integral for the unknown .

The solution of the ODE follows by integrating the first order ODE a second time.

Inverting the similarity transformations step by step, we end up with the solution

Specifying the constants of integration

The exponential representation of the solution is

A plot shows that the solution possesses the behavior of a soliton in the direction. We also observe that the dependence in the solution completely disappears.

Third reduction for

Another kind of solution follows if we select the subgroup where the continuous part vanishes and .

The first similarity reduction of the heavenly equation follows by

x - zeta1 == 0

y - zeta2 == 0

which uses and as similarity variables and as similarity solution. The PDE has to satisfy is a nonlinear second order equation of Liouville type.

The symmetries of this equation are given by

an infinite dimensional symmetry group determined by which must solve the Poisson equation.

Reduction 1

We already know that the Poisson equation is solved by harmonic functions. It is also known that first and second degree polynomials solve this equation. For simplicity let us assume that .

The second stage similarity reduction for this kind of subgroup is then given by

where the ratio represents the similarity variable and gives the similarity solution. The similarity variable has to satisfy the nonlinear second order ODE

The infinitesimals of this ODE are

To solve the ODE, we introduce canonical variables which simplify the representation of the equation

Applying the canonical transformations to the ODE, we find

the Liouville equation in one dimension. The solution of the Liouville equation is derived by DSolve[]

Inversion of the canonical transformation

and the similarity transformations

results to the solution

Specifying the constants of integration , , and , we can simplify the representation to

A graphical representation of the solution is given below

Reduction 2

Since the Poisson equation allows harmonic solutions , we can use this to represent a second kind of subgroup

The reduction for this specific subgroup is given by

The resulting ODE reads

The symmetries of the ODE are

From the two dimensional finite group we select the subgroup with and and calculate the canonical variables

Applying the canonical variables to the ODE, we find

DSolve[] is able to solve the canonical equation

The inversion of the canonical transformation provides us with the solution to be

The inversion of the similarity transformations gives

the solution for the special subgroup

A simplified representation of this solution is

The exponential representation of this expression follows by

The plot of the solution for different values of demonstrates that the number of singularities increases with

Reduction 3

If we choose , we select from the infinite dimensional group the subgroup

The corresponding similarity reduction reads

-zeta1 + ζ == 0

The remaining ODE is given by

Solving the ODE we find

Inserting the result into the similarity representations

we gain the final solution to be

Selecting numerical values for the parameters simplifies the solution to

It is obvious that for this kind of reduction the dependence completely disappears.

The plot of the solution demonstrates the behavior.

Fourth reduction for

Let us assume for the second kind of reduction that the finite part of the symmetry group is involved by . The free function satisfying the Poisson equation is given by . The related subgroup follows by

This subgroup allows the similarity reduction of the heavenly equation

The first stage similarity representation of the solution depends on a nonlinear second order PDE

allowing the symmetries

For this choice of initial symmetries the similarity reduction allows a two dimensional finite symmetry group.

Reduction 1

From the above discrete group we select the subgroup with .

The reduction for this subgroup follows by

The ODE we need to solve to get the solution is

The symmetries of this ODE are

The one dimensional group signals that the second order ODE is not solvable in an explicit form. To demonstrate this we calculate the canonical representation of the ODE by

The solution by DSolve[] demonstrates that an explicit solution is impossible.

The result for this subgroup is that we only know the implicit representation of the solution.

The First Atomic Explosion

4. Referencs

P.J. Olver, Applications of Lie groups to differential equations, Springer, Berlin, 1986.

N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics,  D.Reidel Publishing Company, Dortrecht, 1985.

N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1-3, CRC-Press, Boca Raton, 1994, 1995, 1996.

G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989.

L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

G. Baumann, Symmetry Analysis of Differential Equations with Mathematica, TELOS/Springer, New York, 2000.

J.G.Gegenberg and A. Das,Stationary Riemann Space-Times with Self-Dual Curvature, Gen. Rel. Grav., 16, 817-829, 1984.

J.F. Plebanski, Some Solutions of Complex Einstein Equations, J. Math. Phys., 16, 2395-2402, 1975.

E. Alfinito, G. Soliani, and L. Solombrino, The Symmetry Structure of the Heavenly Equation, hep-th/9604085 16 Apr 1996.

C.P. Boyer and J.D. Finley, Killing Vectors in Self-Dual, Euclidian Einstein Spaces, J. Math. Phys. 23, 1126-1130, (1989).

M.S. Drew,  S.C. Kloster,  and J.D. Gegenberg, Lie Group Analysis and Similarity Solution for the Equation , Nonlinear Analysis, Theory, Methods, and Applications, 13, 489-505,  (1989).

Created by Mathematica  (September 15, 2003)