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MathLie a program of doing symmetry analysis
Mathematics and Computers in Simulation 48 (1998) 205±223
MathLie a program of doing symmetry analysis G. Baumann* Department of Mathematical Physics, University of Ulm, D-89069 Ulm, Germany Received 20 April 1998; accepted 27 August 1998
Abstract We discuss the capabilities of MathLie in supporting the symmetry analysis of differential equations. MathLie is a computer algebra program written in Mathematica 3.0 and capable to carry out different types of symmetry calculations for ordinary as well as for partial differential equations. MathLie determines point symmetries, non-classical symmetries, potential symmetries, approximate symmetries, and generalized symmetries for partial differential equations. Three examples, the Harry±Dyme, the turbulent burst equations, and the Zabolotskaya±Khoklov equation demonstrate the application of MathLie in connection with point symmetries. We also present similarity solutions of these equations. # 1998 IMACS/Elsevier Science B.V. Keywords: MathLie; Symmetry analysis; Harry±Dyme equation; Zabolotskaya±Khoklov equation; Point symmetries
1. Introduction A central problem in physics, mathematics and engineering is to find solutions of a given system of differential equations. The type of equations may be linear or nonlinear. The generic case of practical problems which handle ordinary as well as partial differential equations are nonlinear models. Let us summarize these equations by the notation i x; u k 0;
i 1; 2; . . . ; m;
(1)
where x is a p-dimensional vector of independent variables and u(k) denotes the derivatives up to order k 0; 1; . . . of a q-dimensional vector of dependent variables u. The central question for such a general system of nonlinear partial or ordinary differential equations is: can we find a universal procedure which gives us solutions for this system of equations? We are not looking for the general solution but for one solution. That this is no trivial task has been known for a long time. Since the question was ÐÐÐÐ * E-mail: [email protected] 0378-4754/98/$ ± see front matter # 1998 IMACS/Elsevier Science B.V. All rights reserved PII: S 0 3 7 8 - 4 7 5 4 ( 9 8 ) 0 0 1 4 3 - 8
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raised by Sophus Lie at the end of the last century great effort has been made in solving this question. However, there still remain problems awaiting their solutions. One of the main difficulties of Lie's theory is the tremendous amount of work necessary to derive a solution of a given differential equation. The work of algebraic manipulation increases if the differential equation depends not only on one but on several independent variables. The calculations increase even more if we study a system of equations. It may well happen that we have to handle hundreds of equations to find a single solution for a system of equations. In the past this large amount of work was a huge barrier for using Lie's theory. Today we are fortunate to be able to handle the problem of algebraic manipulations of a general number of equations. Using computer algebra systems like Mathematica, Maple, Macsyma, Axiom, to name the more powerful systems, we are able to manage the laborious work in an efficient manner. An overview of available programs is given by Hereman [1,2]. The present paper discusses new capabilities that have not been implemented in the older programs to derive symmetries and reductions completely automatically. Additional features of the Mathematica based program MathLie are the automatic calculations of non-classical, potential and approximate symmetries. The package MathLie also finds solutions for ordinary differential equations of arbitrary order. In this paper, we restrict our discussion to point symmetries. The other topics are discussed elsewhere [3]. The paper is organized as follows. In Section 2, we discuss the theoretical background of Lie's theory. Section 3 introduces the MathLie functions corresponding to the terms introduced in Section 2. Section 4 discusses the reduction of differential equations and demonstrates the application of MathLie to three examples. In Section 5, we summarize the results. 2. Theoretical background The theory under discussion is the symmetry theory of Lie. This theory is useful for solving partial as well as ordinary differential equations in a systematic way. The question Lie had raised more than hundred years ago was how to solve differential equations systematically. Up to the present day this question of deriving solution for a given differential equation has been of topical interest for physicists and mathematicians alike. Lie found a solution to this problem by introducing a method which allows the examination of symmetry transformations of equations. Lie developed a method which is capable of handling a large number of equations. The application of this method depends neither on the type of the equation nor on the number of variables involved in the equations. Lie's method is a general procedure appropriate for any type of differential equation. However, perusing the literature on Lie's method we observe that his method has rarely been applied compared to the wealth of differential equations used in practical and theoretical problems. The reason for the widespread rejection of Lie's method during the past one hundred years by the community of mathematicians and physicists is that his method demands a huge number of algebraic calculations in order to extract the symmetries of a differential equation. Even for simple equations the algebraic amount of calculations is large as compared with other methods. If someone is genial enough to guess a solution for a particular problem, he probably does not have a deeper insight into the solution structure of the equation. However, if he or she is interested in a complete solution of the symmetry problem, we offer the ability to obtain the information needed for an equation by using a symbolic calculation in MathLie. This tool allows us to completely solve the symmetry problem either in a non-
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interactive or in an interactive way. Before we start to apply MathLie to a problem let us first recall the theoretical background of the program. In his work Lie pointed out that the symmetry of any differential equation is defined as follows: Definition (Lie symmetry). A Lie (point) symmetry is defined to be a symmetry characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all independent and dependent variables. To explain the terms infinitesimal transformation and invariance in more detail, we consider the general case of a nonlinear system of differential equations for an arbitrary number q of unknown functions uj which may depend on p independent variables xi. We denote these sets of variables by u u1 ; u2 ; . . . ; uq and x x1 ; x2 ; . . . ; xp , respectively. The general case is given by a system of m nonlinear differential equations i x; u k 0;
i 1; 2; . . . ; m
(2)
of order k. The term u(k) is understood as kth derivative of u with respect to x. We note that m, k, p and q are arbitrary, positive integers. Consider further a one-parameter -Lie group of transformations x x; u;
(3)
u x; u;
(4)
under which (2) must be invariant. The asterisk on the variables x and u denotes the new variables. Invariance of (2) under the action of (3) and (4) means that any solution u x of (2) maps into some other solution w x; of (2). Let u x be a solution of (2). If we replace the dependent and independent variables u and x by w and x , Eq. (2) become i x ; w k 0;
i 1; 2; . . . ; m
(5)
Then w x are solutions of (5). This implies that if (2) and (5) have a unique solution, then x x; x;
(6)
Hence satisfies the one-parameter functional equation x; x; ;
(7)
Expanding Eqs. (3) and (4) around the identity 0, we generate the following infinitesimal transformations: xi xi i x; u O 2 ; uj uj j x; u O 2 ;
i 1; 2; . . . ; p j 1; 2; . . . ; q
(8) (9)
Functions i and j are the infinitesimals of the transformations for the independent and dependent variables, respectively. In order to find the unknown infinitesimals i and j, we have to prolong the
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transformation group to include the transformation properties of the derivatives. The extension is an infinitesimal approach generating the Lie algebra L corresponding to the Lie group G. The infinitesimal transformation (8), (9) can be put into the form of v
p X i1
X @ @ i x; u j x; u @xi j1 @uj q
(10)
where v represents a linear combination of the vector fields generating L which in turn is based on the characteristic quantities i and j of the transformation (8), (9). The algorithm used in MathLie for finding the infinitesimals i and j is described below. We emphasize that the infinitesimals in this simple form only depend on independent and dependent variables. Transformations (8), (9), together with the transformations for the first, second, . . . derivatives of the uj's, are called first, second, . . . prolongations. Using these various extensions, the infinitesimal criterion for the invariance of (2) under the group (3), (4) can be derived by pr k vj0 0
(11)
where the kth prolongation of the vector field v is given as an expansion in the infinitesimals pr k v v
q X X j1
J
Jj x; u k
@
(12)
@ujJ
The second summation extends over all multi-indices J j1 ; . . . ; jl with 1 jl p; 1 l k. The kth prolongation coefficients Jj are given by the recursion ! p p X X Jj x; u k DJ j ÿ i uji ij ujJ;i (13) i1
i1
where uji @uj =@xi and ujJ;i @ujJ =@xi . Thus, the system of differential equations, Eq. (2), is invariant under the transformation of a one-parameter group with the infinitesimal generator (10) if the i's and j's are determined from Eq. (11). So far we discussed the standard procedure to derive the determining equations for the infinitesimals i and j. This procedure is widely used in the literature [4±7]. From an efficiency point of view of the calculation the procedure described above is very time and memory consuming. The main slowing-down step of the procedure is the recursive calculation of the expansion coefficients in Eq. (12). We discussed a more efficient way to derive the determining equations in [8]. This procedure is based on the FreÂchet derivative. Both methods for calculating the prolongation are available in MathLie. However, the standard calculation of point symmetries use the FreÂchet approach. From an algorithmic point of view the steps to calculate the invariance condition (11) are: 1. Calculate the prolongation of the system of differential equations up to kth order by pr k v 0
(14)
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2. Use the equations themselves to eliminate redundant information of the prolongation pr k vj0 0
(15)
3. Extract the determining equations from the prolongation by setting the coefficients of the derivatives in dependent variables equal to zero. 4. Solve the resulting determining equations. Steps 1±3 are standard steps available in nearly every computer algebra program supporting a symmetry analysis [2]. However, the fourth step is commonly not available in computer algebra programs. Step 4 distinguishes MathLie from other programs. The fourth step in MathLie deals with the solution of the determining equations which uses the theory by Janet [9] and Riquier [10] (JR) as well as differential Groebner bases. The procedure to solve the determining equations is a combination of the two methods. A detailed discussion of how the methods work and how the different procedures are implemented is given in [3]. We note that MathLie solved fairly 95% of hundreds of different examples completely automatically. In principal the solution of these equations is possible because they consist of an overdetermined system of linear partial differential equations. 3. Functions of MathLie MathLie provides a specific function for the first, third, and fourth step. The corresponding functions are Prolongation[], DeterminingEquations[], and PDESolve[]. These three functions among others are used in the general function Infinitesimals[] allowing the automatic calculation of point symmetries. We also designed a function LTF[] (Lie traditional form) to make the results of the above functions more readable. The following example demonstrates how these functions are applied to calculate the mathematical expressions discussed in Section 2. Let us demonstrate the application of Lie's theory and its MathLie counterparts by means of the Harry±Dyme equation (HD). The Harry±Dyme equation, originally a pure mathematical object, is discussed today in connection with physical applications [11]. The HD equation is one of many examples allowing a finite group and a reduction with one simple solution. We have chosen the HD equation as a non-trivial example showing all the properties to derive a similarity solution. The HD equation reads ut ÿ u3 ux;x;x 0
(16)
which in Mathematica has the representation
Square brackets denote the arguments of a 11-dimensional field u u x; t. The function LTF[] transforms the lengthy expressions of Mathematica into a readable index notation, displayed in the second line above. Let us first calculate the prolongation of the HD equation by applying Prolongation[]
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to (16). The function Prolongation[] needs to know the equation of motion, the dependent variable, and the independent variables. The variables are enclosed in curly brackets denoting lists in Mathematica.
The result is an expression containing all the terms of the third-order prolongation with respect to the spatial variable x. The expression is represented in index notation increasing the readability of the result. The second and third steps in the invariance algorithm is the insertion of the equation itself into the prolongation and the extraction of the determining equations for the infinitesimals. This step is realized in MathLie by the function DeterminingEquations[]. Next we apply this function to the HD equation. DeterminingEquations[] needs as input the left-hand side of the equation, the dependent and independent variables, an expression for which the equation of motion is solvable, and a list of parameters, here .
The result is a system of eight linear coupled partial differential equations for the infinitesimals 1, 2, and 1. These infinitesimals correspond to the independent variables x and t, and u, respectively.
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The derived equations determine the infinitesimal transformation properties of the HD equation. By applying the function PDESolve[] to these equations, we obtain the solution. PDESolve[] consists of many subroutines checking the structure of the determining equations and applying transformations to simplify them. Applying PDESolve[] to the determining equations causes the internal subroutines to reduce the number of unsolved determining equations successively. The JR method as well as differential Groebner basis in combination with integration strategies allow the successive solution of these equations. PDESolve[] needs as input the determining equations and the dependent and independent variables.
The output for the HD equation represents a five-dimensional finite transformation group characterized by the group constants k1; . . . ; k5. The infinitesimals above are isomorphic to the results given by Ablowitz and Clarkson [12]. The group parameters determine the transformation properties of the HD equation. For example k1 and k4 stand for the invariance with respect to translations in x and t. k2 and k5 are responsible for scaling transformations, etc. The above result for the infinitesimals is also derived in a single step by using the function Infinitesimals[]. The function Infinitesimals[] needs as inputs the equation of motion, dependent and independent variables, and a list of parameters occurring in the equation.
The resulting infinitesimals are isomorphic to the transformation derived by the three interactive steps above. A renumbering of the group constants shows the equivalence. Up to this point it was straightforward to derive information from the symmetry properties of the HD equation. It is equally easy to calculate infinitesimals for other equations of motion. However, Lie's theory consists not only of the derivation of symmetries but also of the application to solve the original equations. Thus, the next step in the solution procedure is to reduce the original equation of motion by means of a symmetry transformation. 4. Reduction of a differential equation A similarity reduction of a differential equation is closely connected with the invariance of the equation. In this section we discuss the invariance condition of a partial differential equation (PDE) to
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reduce the original equation to an ordinary differential equation (ODE) or to a PDE in less independent coordinates. The first case occurs for an equation with two independent variables. The second case occurs if a PDE contains more than two independent variables. Then a PDE is reduced to a PDE in fewer independent variables. However, in both cases the reduction procedure generates a similarity representation of the original equation. This reduction procedure works independently of the type of the PDE, linear or nonlinear. The procedure is also independent of the order of the PDE and does not need information on any boundary conditions. The benefit of the reduction is a simpler equation allowing an analytic or numeric solution. The reduced equation is known as a similarity solution. Examples of similarity solutions are often found in mechanics, hydrodynamics, in the general theory of relativity, etc. We will show that the calculation of a similarity reduction is an algorithmic procedure delivering large number of solutions frequently discussed in the literature as ansaÈtze. The main idea behind a similarity reduction is the term invariance. Invariance is discussed in this section. To simplify the mathematical representation, we restrict the examination to cases with two independent variables. A generalization to more independent variables is straightforward. Let us consider a general PDE with two independent variables and one dependent variable in the form x; t; u; ux ; ut ; ux;x ; . . . 0
(17)
which is invariant under the one-parameter Lie group of transformations x X x; t; u;
(18)
t T x; t; u;
(19)
u U x; t; u;
(20)
Let us further assume that u x; t is a solution of Eq. (17). Inserting this solution into the transformations (18)±(20), we get x X x; t; x; t;
(21)
t T x; t; x; t;
(22)
u U x; t; x; t;
(23)
stating that u* is also a solution of the transformed PDE. The use of these facts allows to define the invariance in the following way. Definition (Invariance). A PDE is invariant under a one-parameter Lie group transformation if u U x; t; x; t; satisfies the transformed PDE whenever u x; t is a solution of the original PDE. If we additionally want to solve this problem uniquely, we end up by the functional equation X x; t; x; t; ; T x; t; x; t; U x; t; x; t;
(24)
The solution of this functional equation can be derived by introducing the infinitesimal
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transformations of (21)±(23) X x 1 x; t; u O 2 ;
T t 2 x; t; u O 2 ;
U u 1 x; t; u O 2 (25)
Inserting the infinitesimal representation of the transformations into the functional Eq. (24), we get the simplification x 1 x; t; ; t 2 x; t; 1 x; t;
(26)
Where represents the group parameter. Expression (26) contains on the left-hand side the solution we are looking for. In fact the above expression also contains in a functional form. However, the functional representation depends on the infinitesimal parameter allowing a Taylor expansion around the identity 0. If we additionally subtract the right-hand side from the left-hand side and extract the terms of lowest order in , we get the result x 1 t 2 ÿ 1 0
(27)
This first-order PDE is called the invariant surface condition. The problem we face now is to solve this first-order partial differential equation. The essential point is that the solution of a first-order partial differential equation is represented by a family of surfaces u ÿ x; t const. If F x; t; 0 defines a surface satisfying the first-order partial differential equation, then this surface is an invariant of the one-parameter Lie group transformation. This is obvious from the condition vF x; t; 0
(28)
where v is the vector field of the transformation given by v 1 @x 2 @t 1 @
(29)
The result is that the vector field v applied to the surface F x; t; delivers the determining equation for the surface. These facts are stated in the following theorem. Theorem (Invariance condition). The function F x; t; is an invariant of a one-parameter Lie group transformation if the condition vF 0
(30)
is satisfied. This condition always results into a first-order partial differential equation independent of the number of dependent and independent variables. The equation is solvable by applying the method of characteristics. We demonstrate the application of the theorem by the following example. Let us apply the theorem to three arbitrary infinitesimals
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where c is a real parameter. The invariance condition (27) in Mathematica reads
The solution of this PDE follows by
representing the general solution of the first-order partial differential equation. The arbitrary function C[1] depends on two invariants given by I1 t=x2 and I2 u=xc . These two invariants are the basis for the reduction. The two invariants allow us to reduce the original equation to an ordinary differential equation. The reduction procedure itself is based on the theorem. Theorem (Invariant representation). Let the equation 0 be invariant under a one-parameter group G and let the infinitesimals i ; j ; i 1; 2; . . . ; p, and j 1; 2; . . . ; q be non-vanishing functions on the solution surface F of the equation. In this case the surface F can be represented by equations of the form k I1 x; u; . . . ; Ipÿ1 x; u 0;
k 1; 2; . . . ; q
(31)
where I1 ; . . . ; Ipÿ1 define a basis of invariants of the group G The steps discussed in the above theorems are incorporated in the MathLie function LieReduction[]. From the theoretical discussion above it is obvious that the reduction procedure works as long as we can solve the invariance surface condition and the expressions of the invariants. MathLie fails if the invariants contain transcendental functions because Mathematica is not able to solve transcendental equations. The following examples demonstrate applications of the function LieReduction[]. The first example discusses the reduction of the HD equation in connection with a simple solution. Example 2 demonstrates the application of LieReduction[] to a system of equations. In the third example we examine a PDE with three independent variables. Example 1. The reduction process for the HD equation starts with selecting a subgroup from the total symmetry group. Let us select the infinitesimals in a way that only translations are considered in the reduction; i.e. k3 and k1 1 while the remaining group parameters vanish. This subgroup is represented by
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The selected subgroup represents a transformation allowing only translations in the independent variables. The real parameter is an arbitrary constant describing a reciprocal velocity. Reduction of the HD equation follows by applying function LieReduction[] to the HD equation in connection with the information on the dependent and independent variables and the selected subgroup for these variables.
The result is a similarity representation of the solution in terms of the similarity variable and the similarity function F1 F1 . This function has to satisfy a third-order ODE. A solution of the HD equation is known if we can solve the third-order ODE. However, this third-order ODE is neither solved by Mathematica nor by Lie's method. The reason why Lie's method fails to provide a solution is due to the lack of a sufficient number of symmetries for the third-order ODE. We can check the leak of symmetries by carrying out a symmetry analysis of the equation by
The infinitesimals contain two group constants which means that only a reduction of the order of the equation is possible. However, another subgroup of the symmetries for the HD equation follows by selecting k3 1; k4 , and the other constants equal to zero
The reduction for this subgroup follows by
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The similarity representation of the solution shows that a solution of the HD equation follows from the solution of a first-order ODE.
The solution is given by DSolve[] in a pure function representation denoted by &. The similarity solution is a constant C[1]. The solution in original variables reads
which is a solution of the HD equation. Example 2. The second example discusses the application of MathLie to a system of equations. It shows that the application of the functions Infinitesimals[] and LieReduction[] is as simple as that of a single equation. The system we examine is a 11-dimensional model describing a turbulent burst in a fluid [13]. The equations of motion for the turbulent energy and the turbulent energy dissipation b are
The symmetries of the coupled system of PDEs are determined by
The result from 21 determining equations for the infinitesimals is a four-dimensional symmetry group. Let us select the scaling group characterized by k4 and k2 to reduce the original equations.
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Reduction of the turbulent vortex field equations follows by
We observe that the similarity representation consists of the similarity variable tzÿ2= 3 and two similarity functions F1 and F2 satisfying a coupled nonlinear system of second-order ODEs. These ODEs determine the solution of the original equation. We will not dwell on the solution of this system. The example demonstrates that MathLie is capable to handle systems of equations. Function LieReduction[] works even if simple transcendental functions in the similarity solution are present. However, a reduction fails if the similarity representation contains complicated transcendental functions. In this case we have to carry out the reduction by hand. As to the number of equations in the
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system, MathLie has no limitation. However, it may happen that the memory of the computer is too small to carry out calculations for large systems. Example 3. The third example applies the theory of reduction to a 21-dimensional equation to a single PDE. From the theorems and the discussion above, we expect that the 21-dimensional PDE first is reduced to a 11-dimensional PDE and in a second step simplifies to a single ODE. The equation under discussion is the 21-dimensional Zabolotskaya±Khoklov equation (ZK) [14] 6u2x ux;t 6uux;x ÿ uy;y ÿ ux;x;x 0:
(32)
Eq. (32) is used to describe shallow water waves. The equation is written in Mathematica by
The infinitesimals for ZK equation are determined by
We realize that the ZK equation allows an infinite symmetry group depending on two arbitrary functions F1 and F2. A subgroup embedded in the infinite group is the symmetry group of translations. The following calculations are based on this kind of primary symmetry. The first reduction of the ZK model allowing translation F1 1, F2 1, and k1 c thus follows by
The resulting similarity representation consists of four parts: the two similarity variables 1 t ÿ cx, 2 y ÿ x, the similarity representation u F1 1 ; 2 , and a third-order nonlinear PDE for the similarity function F1. To solve the once reduced PDE for F1, we again apply Lie's symmetry method a
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second time to the once reduced PDE. The symmetries of the first reduction follow with
The result is a finite three-dimensional symmetry group. This group can be used again to reduce the once reduced PDE a second time and thus reduces the ZK equation to an ODE. Before carrying out this second reduction, we rename variables in the once reduced PDE to clarify the notation
The second reduction of the equation eqv is carried out by selecting the subgroup of translations from the secondary symmetry group. The simplified infinitesimals for this case are given by
The corresponding reduction follows from
The result is a third-order nonlinear ODE. The examination of the symmetries for the ODE results in a two-dimensional discrete symmetry group depending on group parameters c and v of the preceding
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reductions
The derived result indicates that the third-order ODE is reducible to a second-order ODE which in fact is possible by an integration with respect to zeta1.
In principle the resulting second-order ODE can be treated again by Lie's procedure. However, here we prefer using DSolve[] an internal function of Mathematica in order to solve the problem. Function DSolve[] can solve the ODE resulting from the first integral by
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The result is a complicated expression containing special functions of the Airy type. However, the solution is simplified if we set the integration constant c1 equal to zero.
The solution for this special case reads
containing the Tan[] as a special function. At this stage of the calculation we derived a solution for a
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special subgroup embedded into the infinite symmetries of the original equation. This special solution helps us to represent the solution in original variables. To derive the representation of the solution in x, y, t, and u, we have to invert the similarity transformation carried out above. Due to limitations of space, we restrict the discussion to the second solution sol2 which in original variables read
The complicated symbolic solution simplifies in x; y; t space if we choose a special set of parameters Thus the simplified solution reads
The Tanh[] is one of many other solutions of the ZK-equation representing a propagating step in (x,y)-space. The ZK equation is an example that Lie's method of symmetry analysis is capable to derive analytic solutions for a model in 21 dimensions. The steps of the symmetry analysis are supported by MathLie shortening the calculation time and increasing the reliability. 5. Conclusion In this paper, we have carried out a comprehensive discussion of Lie's classical theory in connection with MathLie. We briefly discussed the theoretical background and demonstrated the application of
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MathLie functions to partial differential equations. The functions of MathLie are designed in a way to allow a symmetry analysis to be carried out step by step or completely automatically. Each theoretical step has a counterpart in MathLie. We demonstrated that two functions Infinitesimals[] and LieReductin[] are sufficient to solve an equation. These two functions are used to determine the infinitesimal transformations and the similarity reduction of the original equations. The symmetries and the reductions are automatically calculated by MathLie. The user only had to decide which subgroup of the symmetry transformation should be used in the reduction step. We restricted the presentation to point symmetries because these symmetries are widely used in practical applications. However, MathLie is capable to support another five different types of symmetry analysis based on non-classical, potential, approximate, and generalized symmetries. A discussion of these additional symmetry methods are given in [3]. References [1] W. Hereman, Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath. Bull. 2 (1994) 45±82. [2] W. Hereman, Symbolic software for Lie symmetry analysis, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3: New Trends in Theoretical Developments and Computational Methods, chapter 13, CRC Press, Boca Raton, 1996, pp. 367±413. [3] G. Baumann, Symmetry Analysis of Differential Equations Using Mathematica, TELOS/Springer, New York, 1998. [4] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. [5] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985. [6] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. [7] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. [8] G. Baumann, Symmetry Analysis of Differential Equations with Mathematica, Math. Comput. Modelling 25 (1997) 25± 37. [9] M. Janet, Sur les systeÁmes d'eÂriveeÂs partielles, J. Math. Pure Appl. 3 (1920) 65±151. [10] Ch. Riquier, Les systeÁmes d'eÂquations aux deÂriveÂes partielles, Gauthier-Villars, Paris, 1910. [11] L.P. Kadanoff, Exact solutions for the Saffman±Taylor problem with surface tension, Phys. Rev. Lett. 65 (1990) 2986± 2988. [12] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics, vol. 149, Cambridge University Press, Cambridge, 1991. [13] G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996. [14] V.I. Karpman, V.Yu. Belashov, Dynamics of two-dimensional solutions in weakly dispersive media, Phys. Lett. 154 (1991) 131±139.
MathLie a program of doing symmetry analysis G. Baumann* Department of Mathematical Physics, University of Ulm, D-89069 Ulm, Germany Received 20 April 1998; accepted 27 August 1998
Abstract We discuss the capabilities of MathLie in supporting the symmetry analysis of differential equations. MathLie is a computer algebra program written in Mathematica 3.0 and capable to carry out different types of symmetry calculations for ordinary as well as for partial differential equations. MathLie determines point symmetries, non-classical symmetries, potential symmetries, approximate symmetries, and generalized symmetries for partial differential equations. Three examples, the Harry±Dyme, the turbulent burst equations, and the Zabolotskaya±Khoklov equation demonstrate the application of MathLie in connection with point symmetries. We also present similarity solutions of these equations. # 1998 IMACS/Elsevier Science B.V. Keywords: MathLie; Symmetry analysis; Harry±Dyme equation; Zabolotskaya±Khoklov equation; Point symmetries
1. Introduction A central problem in physics, mathematics and engineering is to find solutions of a given system of differential equations. The type of equations may be linear or nonlinear. The generic case of practical problems which handle ordinary as well as partial differential equations are nonlinear models. Let us summarize these equations by the notation i x; u k 0;
i 1; 2; . . . ; m;
(1)
where x is a p-dimensional vector of independent variables and u(k) denotes the derivatives up to order k 0; 1; . . . of a q-dimensional vector of dependent variables u. The central question for such a general system of nonlinear partial or ordinary differential equations is: can we find a universal procedure which gives us solutions for this system of equations? We are not looking for the general solution but for one solution. That this is no trivial task has been known for a long time. Since the question was ÐÐÐÐ * E-mail: [email protected] 0378-4754/98/$ ± see front matter # 1998 IMACS/Elsevier Science B.V. All rights reserved PII: S 0 3 7 8 - 4 7 5 4 ( 9 8 ) 0 0 1 4 3 - 8
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raised by Sophus Lie at the end of the last century great effort has been made in solving this question. However, there still remain problems awaiting their solutions. One of the main difficulties of Lie's theory is the tremendous amount of work necessary to derive a solution of a given differential equation. The work of algebraic manipulation increases if the differential equation depends not only on one but on several independent variables. The calculations increase even more if we study a system of equations. It may well happen that we have to handle hundreds of equations to find a single solution for a system of equations. In the past this large amount of work was a huge barrier for using Lie's theory. Today we are fortunate to be able to handle the problem of algebraic manipulations of a general number of equations. Using computer algebra systems like Mathematica, Maple, Macsyma, Axiom, to name the more powerful systems, we are able to manage the laborious work in an efficient manner. An overview of available programs is given by Hereman [1,2]. The present paper discusses new capabilities that have not been implemented in the older programs to derive symmetries and reductions completely automatically. Additional features of the Mathematica based program MathLie are the automatic calculations of non-classical, potential and approximate symmetries. The package MathLie also finds solutions for ordinary differential equations of arbitrary order. In this paper, we restrict our discussion to point symmetries. The other topics are discussed elsewhere [3]. The paper is organized as follows. In Section 2, we discuss the theoretical background of Lie's theory. Section 3 introduces the MathLie functions corresponding to the terms introduced in Section 2. Section 4 discusses the reduction of differential equations and demonstrates the application of MathLie to three examples. In Section 5, we summarize the results. 2. Theoretical background The theory under discussion is the symmetry theory of Lie. This theory is useful for solving partial as well as ordinary differential equations in a systematic way. The question Lie had raised more than hundred years ago was how to solve differential equations systematically. Up to the present day this question of deriving solution for a given differential equation has been of topical interest for physicists and mathematicians alike. Lie found a solution to this problem by introducing a method which allows the examination of symmetry transformations of equations. Lie developed a method which is capable of handling a large number of equations. The application of this method depends neither on the type of the equation nor on the number of variables involved in the equations. Lie's method is a general procedure appropriate for any type of differential equation. However, perusing the literature on Lie's method we observe that his method has rarely been applied compared to the wealth of differential equations used in practical and theoretical problems. The reason for the widespread rejection of Lie's method during the past one hundred years by the community of mathematicians and physicists is that his method demands a huge number of algebraic calculations in order to extract the symmetries of a differential equation. Even for simple equations the algebraic amount of calculations is large as compared with other methods. If someone is genial enough to guess a solution for a particular problem, he probably does not have a deeper insight into the solution structure of the equation. However, if he or she is interested in a complete solution of the symmetry problem, we offer the ability to obtain the information needed for an equation by using a symbolic calculation in MathLie. This tool allows us to completely solve the symmetry problem either in a non-
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interactive or in an interactive way. Before we start to apply MathLie to a problem let us first recall the theoretical background of the program. In his work Lie pointed out that the symmetry of any differential equation is defined as follows: Definition (Lie symmetry). A Lie (point) symmetry is defined to be a symmetry characterized by an infinitesimal transformation which leaves the given differential equation invariant under the transformation of all independent and dependent variables. To explain the terms infinitesimal transformation and invariance in more detail, we consider the general case of a nonlinear system of differential equations for an arbitrary number q of unknown functions uj which may depend on p independent variables xi. We denote these sets of variables by u u1 ; u2 ; . . . ; uq and x x1 ; x2 ; . . . ; xp , respectively. The general case is given by a system of m nonlinear differential equations i x; u k 0;
i 1; 2; . . . ; m
(2)
of order k. The term u(k) is understood as kth derivative of u with respect to x. We note that m, k, p and q are arbitrary, positive integers. Consider further a one-parameter -Lie group of transformations x x; u;
(3)
u x; u;
(4)
under which (2) must be invariant. The asterisk on the variables x and u denotes the new variables. Invariance of (2) under the action of (3) and (4) means that any solution u x of (2) maps into some other solution w x; of (2). Let u x be a solution of (2). If we replace the dependent and independent variables u and x by w and x , Eq. (2) become i x ; w k 0;
i 1; 2; . . . ; m
(5)
Then w x are solutions of (5). This implies that if (2) and (5) have a unique solution, then x x; x;
(6)
Hence satisfies the one-parameter functional equation x; x; ;
(7)
Expanding Eqs. (3) and (4) around the identity 0, we generate the following infinitesimal transformations: xi xi i x; u O 2 ; uj uj j x; u O 2 ;
i 1; 2; . . . ; p j 1; 2; . . . ; q
(8) (9)
Functions i and j are the infinitesimals of the transformations for the independent and dependent variables, respectively. In order to find the unknown infinitesimals i and j, we have to prolong the
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transformation group to include the transformation properties of the derivatives. The extension is an infinitesimal approach generating the Lie algebra L corresponding to the Lie group G. The infinitesimal transformation (8), (9) can be put into the form of v
p X i1
X @ @ i x; u j x; u @xi j1 @uj q
(10)
where v represents a linear combination of the vector fields generating L which in turn is based on the characteristic quantities i and j of the transformation (8), (9). The algorithm used in MathLie for finding the infinitesimals i and j is described below. We emphasize that the infinitesimals in this simple form only depend on independent and dependent variables. Transformations (8), (9), together with the transformations for the first, second, . . . derivatives of the uj's, are called first, second, . . . prolongations. Using these various extensions, the infinitesimal criterion for the invariance of (2) under the group (3), (4) can be derived by pr k vj0 0
(11)
where the kth prolongation of the vector field v is given as an expansion in the infinitesimals pr k v v
q X X j1
J
Jj x; u k
@
(12)
@ujJ
The second summation extends over all multi-indices J j1 ; . . . ; jl with 1 jl p; 1 l k. The kth prolongation coefficients Jj are given by the recursion ! p p X X Jj x; u k DJ j ÿ i uji ij ujJ;i (13) i1
i1
where uji @uj =@xi and ujJ;i @ujJ =@xi . Thus, the system of differential equations, Eq. (2), is invariant under the transformation of a one-parameter group with the infinitesimal generator (10) if the i's and j's are determined from Eq. (11). So far we discussed the standard procedure to derive the determining equations for the infinitesimals i and j. This procedure is widely used in the literature [4±7]. From an efficiency point of view of the calculation the procedure described above is very time and memory consuming. The main slowing-down step of the procedure is the recursive calculation of the expansion coefficients in Eq. (12). We discussed a more efficient way to derive the determining equations in [8]. This procedure is based on the FreÂchet derivative. Both methods for calculating the prolongation are available in MathLie. However, the standard calculation of point symmetries use the FreÂchet approach. From an algorithmic point of view the steps to calculate the invariance condition (11) are: 1. Calculate the prolongation of the system of differential equations up to kth order by pr k v 0
(14)
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2. Use the equations themselves to eliminate redundant information of the prolongation pr k vj0 0
(15)
3. Extract the determining equations from the prolongation by setting the coefficients of the derivatives in dependent variables equal to zero. 4. Solve the resulting determining equations. Steps 1±3 are standard steps available in nearly every computer algebra program supporting a symmetry analysis [2]. However, the fourth step is commonly not available in computer algebra programs. Step 4 distinguishes MathLie from other programs. The fourth step in MathLie deals with the solution of the determining equations which uses the theory by Janet [9] and Riquier [10] (JR) as well as differential Groebner bases. The procedure to solve the determining equations is a combination of the two methods. A detailed discussion of how the methods work and how the different procedures are implemented is given in [3]. We note that MathLie solved fairly 95% of hundreds of different examples completely automatically. In principal the solution of these equations is possible because they consist of an overdetermined system of linear partial differential equations. 3. Functions of MathLie MathLie provides a specific function for the first, third, and fourth step. The corresponding functions are Prolongation[], DeterminingEquations[], and PDESolve[]. These three functions among others are used in the general function Infinitesimals[] allowing the automatic calculation of point symmetries. We also designed a function LTF[] (Lie traditional form) to make the results of the above functions more readable. The following example demonstrates how these functions are applied to calculate the mathematical expressions discussed in Section 2. Let us demonstrate the application of Lie's theory and its MathLie counterparts by means of the Harry±Dyme equation (HD). The Harry±Dyme equation, originally a pure mathematical object, is discussed today in connection with physical applications [11]. The HD equation is one of many examples allowing a finite group and a reduction with one simple solution. We have chosen the HD equation as a non-trivial example showing all the properties to derive a similarity solution. The HD equation reads ut ÿ u3 ux;x;x 0
(16)
which in Mathematica has the representation
Square brackets denote the arguments of a 11-dimensional field u u x; t. The function LTF[] transforms the lengthy expressions of Mathematica into a readable index notation, displayed in the second line above. Let us first calculate the prolongation of the HD equation by applying Prolongation[]
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to (16). The function Prolongation[] needs to know the equation of motion, the dependent variable, and the independent variables. The variables are enclosed in curly brackets denoting lists in Mathematica.
The result is an expression containing all the terms of the third-order prolongation with respect to the spatial variable x. The expression is represented in index notation increasing the readability of the result. The second and third steps in the invariance algorithm is the insertion of the equation itself into the prolongation and the extraction of the determining equations for the infinitesimals. This step is realized in MathLie by the function DeterminingEquations[]. Next we apply this function to the HD equation. DeterminingEquations[] needs as input the left-hand side of the equation, the dependent and independent variables, an expression for which the equation of motion is solvable, and a list of parameters, here .
The result is a system of eight linear coupled partial differential equations for the infinitesimals 1, 2, and 1. These infinitesimals correspond to the independent variables x and t, and u, respectively.
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The derived equations determine the infinitesimal transformation properties of the HD equation. By applying the function PDESolve[] to these equations, we obtain the solution. PDESolve[] consists of many subroutines checking the structure of the determining equations and applying transformations to simplify them. Applying PDESolve[] to the determining equations causes the internal subroutines to reduce the number of unsolved determining equations successively. The JR method as well as differential Groebner basis in combination with integration strategies allow the successive solution of these equations. PDESolve[] needs as input the determining equations and the dependent and independent variables.
The output for the HD equation represents a five-dimensional finite transformation group characterized by the group constants k1; . . . ; k5. The infinitesimals above are isomorphic to the results given by Ablowitz and Clarkson [12]. The group parameters determine the transformation properties of the HD equation. For example k1 and k4 stand for the invariance with respect to translations in x and t. k2 and k5 are responsible for scaling transformations, etc. The above result for the infinitesimals is also derived in a single step by using the function Infinitesimals[]. The function Infinitesimals[] needs as inputs the equation of motion, dependent and independent variables, and a list of parameters occurring in the equation.
The resulting infinitesimals are isomorphic to the transformation derived by the three interactive steps above. A renumbering of the group constants shows the equivalence. Up to this point it was straightforward to derive information from the symmetry properties of the HD equation. It is equally easy to calculate infinitesimals for other equations of motion. However, Lie's theory consists not only of the derivation of symmetries but also of the application to solve the original equations. Thus, the next step in the solution procedure is to reduce the original equation of motion by means of a symmetry transformation. 4. Reduction of a differential equation A similarity reduction of a differential equation is closely connected with the invariance of the equation. In this section we discuss the invariance condition of a partial differential equation (PDE) to
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reduce the original equation to an ordinary differential equation (ODE) or to a PDE in less independent coordinates. The first case occurs for an equation with two independent variables. The second case occurs if a PDE contains more than two independent variables. Then a PDE is reduced to a PDE in fewer independent variables. However, in both cases the reduction procedure generates a similarity representation of the original equation. This reduction procedure works independently of the type of the PDE, linear or nonlinear. The procedure is also independent of the order of the PDE and does not need information on any boundary conditions. The benefit of the reduction is a simpler equation allowing an analytic or numeric solution. The reduced equation is known as a similarity solution. Examples of similarity solutions are often found in mechanics, hydrodynamics, in the general theory of relativity, etc. We will show that the calculation of a similarity reduction is an algorithmic procedure delivering large number of solutions frequently discussed in the literature as ansaÈtze. The main idea behind a similarity reduction is the term invariance. Invariance is discussed in this section. To simplify the mathematical representation, we restrict the examination to cases with two independent variables. A generalization to more independent variables is straightforward. Let us consider a general PDE with two independent variables and one dependent variable in the form x; t; u; ux ; ut ; ux;x ; . . . 0
(17)
which is invariant under the one-parameter Lie group of transformations x X x; t; u;
(18)
t T x; t; u;
(19)
u U x; t; u;
(20)
Let us further assume that u x; t is a solution of Eq. (17). Inserting this solution into the transformations (18)±(20), we get x X x; t; x; t;
(21)
t T x; t; x; t;
(22)
u U x; t; x; t;
(23)
stating that u* is also a solution of the transformed PDE. The use of these facts allows to define the invariance in the following way. Definition (Invariance). A PDE is invariant under a one-parameter Lie group transformation if u U x; t; x; t; satisfies the transformed PDE whenever u x; t is a solution of the original PDE. If we additionally want to solve this problem uniquely, we end up by the functional equation X x; t; x; t; ; T x; t; x; t; U x; t; x; t;
(24)
The solution of this functional equation can be derived by introducing the infinitesimal
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transformations of (21)±(23) X x 1 x; t; u O 2 ;
T t 2 x; t; u O 2 ;
U u 1 x; t; u O 2 (25)
Inserting the infinitesimal representation of the transformations into the functional Eq. (24), we get the simplification x 1 x; t; ; t 2 x; t; 1 x; t;
(26)
Where represents the group parameter. Expression (26) contains on the left-hand side the solution we are looking for. In fact the above expression also contains in a functional form. However, the functional representation depends on the infinitesimal parameter allowing a Taylor expansion around the identity 0. If we additionally subtract the right-hand side from the left-hand side and extract the terms of lowest order in , we get the result x 1 t 2 ÿ 1 0
(27)
This first-order PDE is called the invariant surface condition. The problem we face now is to solve this first-order partial differential equation. The essential point is that the solution of a first-order partial differential equation is represented by a family of surfaces u ÿ x; t const. If F x; t; 0 defines a surface satisfying the first-order partial differential equation, then this surface is an invariant of the one-parameter Lie group transformation. This is obvious from the condition vF x; t; 0
(28)
where v is the vector field of the transformation given by v 1 @x 2 @t 1 @
(29)
The result is that the vector field v applied to the surface F x; t; delivers the determining equation for the surface. These facts are stated in the following theorem. Theorem (Invariance condition). The function F x; t; is an invariant of a one-parameter Lie group transformation if the condition vF 0
(30)
is satisfied. This condition always results into a first-order partial differential equation independent of the number of dependent and independent variables. The equation is solvable by applying the method of characteristics. We demonstrate the application of the theorem by the following example. Let us apply the theorem to three arbitrary infinitesimals
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where c is a real parameter. The invariance condition (27) in Mathematica reads
The solution of this PDE follows by
representing the general solution of the first-order partial differential equation. The arbitrary function C[1] depends on two invariants given by I1 t=x2 and I2 u=xc . These two invariants are the basis for the reduction. The two invariants allow us to reduce the original equation to an ordinary differential equation. The reduction procedure itself is based on the theorem. Theorem (Invariant representation). Let the equation 0 be invariant under a one-parameter group G and let the infinitesimals i ; j ; i 1; 2; . . . ; p, and j 1; 2; . . . ; q be non-vanishing functions on the solution surface F of the equation. In this case the surface F can be represented by equations of the form k I1 x; u; . . . ; Ipÿ1 x; u 0;
k 1; 2; . . . ; q
(31)
where I1 ; . . . ; Ipÿ1 define a basis of invariants of the group G The steps discussed in the above theorems are incorporated in the MathLie function LieReduction[]. From the theoretical discussion above it is obvious that the reduction procedure works as long as we can solve the invariance surface condition and the expressions of the invariants. MathLie fails if the invariants contain transcendental functions because Mathematica is not able to solve transcendental equations. The following examples demonstrate applications of the function LieReduction[]. The first example discusses the reduction of the HD equation in connection with a simple solution. Example 2 demonstrates the application of LieReduction[] to a system of equations. In the third example we examine a PDE with three independent variables. Example 1. The reduction process for the HD equation starts with selecting a subgroup from the total symmetry group. Let us select the infinitesimals in a way that only translations are considered in the reduction; i.e. k3 and k1 1 while the remaining group parameters vanish. This subgroup is represented by
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The selected subgroup represents a transformation allowing only translations in the independent variables. The real parameter is an arbitrary constant describing a reciprocal velocity. Reduction of the HD equation follows by applying function LieReduction[] to the HD equation in connection with the information on the dependent and independent variables and the selected subgroup for these variables.
The result is a similarity representation of the solution in terms of the similarity variable and the similarity function F1 F1 . This function has to satisfy a third-order ODE. A solution of the HD equation is known if we can solve the third-order ODE. However, this third-order ODE is neither solved by Mathematica nor by Lie's method. The reason why Lie's method fails to provide a solution is due to the lack of a sufficient number of symmetries for the third-order ODE. We can check the leak of symmetries by carrying out a symmetry analysis of the equation by
The infinitesimals contain two group constants which means that only a reduction of the order of the equation is possible. However, another subgroup of the symmetries for the HD equation follows by selecting k3 1; k4 , and the other constants equal to zero
The reduction for this subgroup follows by
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The similarity representation of the solution shows that a solution of the HD equation follows from the solution of a first-order ODE.
The solution is given by DSolve[] in a pure function representation denoted by &. The similarity solution is a constant C[1]. The solution in original variables reads
which is a solution of the HD equation. Example 2. The second example discusses the application of MathLie to a system of equations. It shows that the application of the functions Infinitesimals[] and LieReduction[] is as simple as that of a single equation. The system we examine is a 11-dimensional model describing a turbulent burst in a fluid [13]. The equations of motion for the turbulent energy and the turbulent energy dissipation b are
The symmetries of the coupled system of PDEs are determined by
The result from 21 determining equations for the infinitesimals is a four-dimensional symmetry group. Let us select the scaling group characterized by k4 and k2 to reduce the original equations.
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Reduction of the turbulent vortex field equations follows by
We observe that the similarity representation consists of the similarity variable tzÿ2= 3 and two similarity functions F1 and F2 satisfying a coupled nonlinear system of second-order ODEs. These ODEs determine the solution of the original equation. We will not dwell on the solution of this system. The example demonstrates that MathLie is capable to handle systems of equations. Function LieReduction[] works even if simple transcendental functions in the similarity solution are present. However, a reduction fails if the similarity representation contains complicated transcendental functions. In this case we have to carry out the reduction by hand. As to the number of equations in the
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system, MathLie has no limitation. However, it may happen that the memory of the computer is too small to carry out calculations for large systems. Example 3. The third example applies the theory of reduction to a 21-dimensional equation to a single PDE. From the theorems and the discussion above, we expect that the 21-dimensional PDE first is reduced to a 11-dimensional PDE and in a second step simplifies to a single ODE. The equation under discussion is the 21-dimensional Zabolotskaya±Khoklov equation (ZK) [14] 6u2x ux;t 6uux;x ÿ uy;y ÿ ux;x;x 0:
(32)
Eq. (32) is used to describe shallow water waves. The equation is written in Mathematica by
The infinitesimals for ZK equation are determined by
We realize that the ZK equation allows an infinite symmetry group depending on two arbitrary functions F1 and F2. A subgroup embedded in the infinite group is the symmetry group of translations. The following calculations are based on this kind of primary symmetry. The first reduction of the ZK model allowing translation F1 1, F2 1, and k1 c thus follows by
The resulting similarity representation consists of four parts: the two similarity variables 1 t ÿ cx, 2 y ÿ x, the similarity representation u F1 1 ; 2 , and a third-order nonlinear PDE for the similarity function F1. To solve the once reduced PDE for F1, we again apply Lie's symmetry method a
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second time to the once reduced PDE. The symmetries of the first reduction follow with
The result is a finite three-dimensional symmetry group. This group can be used again to reduce the once reduced PDE a second time and thus reduces the ZK equation to an ODE. Before carrying out this second reduction, we rename variables in the once reduced PDE to clarify the notation
The second reduction of the equation eqv is carried out by selecting the subgroup of translations from the secondary symmetry group. The simplified infinitesimals for this case are given by
The corresponding reduction follows from
The result is a third-order nonlinear ODE. The examination of the symmetries for the ODE results in a two-dimensional discrete symmetry group depending on group parameters c and v of the preceding
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reductions
The derived result indicates that the third-order ODE is reducible to a second-order ODE which in fact is possible by an integration with respect to zeta1.
In principle the resulting second-order ODE can be treated again by Lie's procedure. However, here we prefer using DSolve[] an internal function of Mathematica in order to solve the problem. Function DSolve[] can solve the ODE resulting from the first integral by
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The result is a complicated expression containing special functions of the Airy type. However, the solution is simplified if we set the integration constant c1 equal to zero.
The solution for this special case reads
containing the Tan[] as a special function. At this stage of the calculation we derived a solution for a
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special subgroup embedded into the infinite symmetries of the original equation. This special solution helps us to represent the solution in original variables. To derive the representation of the solution in x, y, t, and u, we have to invert the similarity transformation carried out above. Due to limitations of space, we restrict the discussion to the second solution sol2 which in original variables read
The complicated symbolic solution simplifies in x; y; t space if we choose a special set of parameters Thus the simplified solution reads
The Tanh[] is one of many other solutions of the ZK-equation representing a propagating step in (x,y)-space. The ZK equation is an example that Lie's method of symmetry analysis is capable to derive analytic solutions for a model in 21 dimensions. The steps of the symmetry analysis are supported by MathLie shortening the calculation time and increasing the reliability. 5. Conclusion In this paper, we have carried out a comprehensive discussion of Lie's classical theory in connection with MathLie. We briefly discussed the theoretical background and demonstrated the application of
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MathLie functions to partial differential equations. The functions of MathLie are designed in a way to allow a symmetry analysis to be carried out step by step or completely automatically. Each theoretical step has a counterpart in MathLie. We demonstrated that two functions Infinitesimals[] and LieReductin[] are sufficient to solve an equation. These two functions are used to determine the infinitesimal transformations and the similarity reduction of the original equations. The symmetries and the reductions are automatically calculated by MathLie. The user only had to decide which subgroup of the symmetry transformation should be used in the reduction step. We restricted the presentation to point symmetries because these symmetries are widely used in practical applications. However, MathLie is capable to support another five different types of symmetry analysis based on non-classical, potential, approximate, and generalized symmetries. A discussion of these additional symmetry methods are given in [3]. References [1] W. Hereman, Review of symbolic software for the computation of Lie symmetries of differential equations, Euromath. Bull. 2 (1994) 45±82. [2] W. Hereman, Symbolic software for Lie symmetry analysis, in: N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3: New Trends in Theoretical Developments and Computational Methods, chapter 13, CRC Press, Boca Raton, 1996, pp. 367±413. [3] G. Baumann, Symmetry Analysis of Differential Equations Using Mathematica, TELOS/Springer, New York, 1998. [4] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin, 1986. [5] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Dordrecht, 1985. [6] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. [7] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. [8] G. Baumann, Symmetry Analysis of Differential Equations with Mathematica, Math. Comput. Modelling 25 (1997) 25± 37. [9] M. Janet, Sur les systeÁmes d'eÂriveeÂs partielles, J. Math. Pure Appl. 3 (1920) 65±151. [10] Ch. Riquier, Les systeÁmes d'eÂquations aux deÂriveÂes partielles, Gauthier-Villars, Paris, 1910. [11] L.P. Kadanoff, Exact solutions for the Saffman±Taylor problem with surface tension, Phys. Rev. Lett. 65 (1990) 2986± 2988. [12] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics, vol. 149, Cambridge University Press, Cambridge, 1991. [13] G.I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996. [14] V.I. Karpman, V.Yu. Belashov, Dynamics of two-dimensional solutions in weakly dispersive media, Phys. Lett. 154 (1991) 131±139.