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This package provides programs APPLYSYM, QUASILINPDE and DETRAFO for applying infinitesimal symmetries of differential equations, the generalization of special solutions and the calculation of symmetry and similarity variables.
In this paper the programs APPLYSYM, QUASILINPDE and DETRAFO are described which aim at the utilization of infinitesimal symmetries of differential equations. The purpose of QUASILINPDE is the general solution of quasilinear PDEs. This procedure is used by APPLYSYM for the application of point symmetries for either
calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables in a PDE(-system) or for
generalizing given special solutions of ODEs / PDEs with new constant parameters.
The program DETRAFO performs arbitrary point- and contact transformations of ODEs / PDEs and is applied if similarity and symmetry variables have been found. The program APPLYSYM is used in connection with the program LIEPDE for formulating and solving the conditions for point- and contact symmetries which is described in [Wol93]. The actual problem solving is done in all these programs through a call to the package CRACK for solving overdetermined PDE-systems.
The investigation of infinitesimal symmetries of differential equations (DEs) with computer algebra programs attrackted considerable attention over the last years. Corresponding programs are available in all major computer algebra systems. In a review article by W. Hereman [Her95] about 200 references are given, many of them describing related software.
One reason for the popularity of the symmetry method is the fact that Sophus Lie’s method [Lie75, Lie67] is the most widely used method for computing exact solutions of non-linear DEs. Another reason is that the first step in this method, the formulation of the determining equation for the generators of the symmetries, can already be very cumbersome, especially in the case of PDEs of higher order and/or in case of many dependent and independent variables. Also, the formulation of the conditions is a straight forward task involving only differentiations and basic algebra - an ideal task for computer algebra systems. Less straight forward is the automatic solution of the symmetry conditions which is the strength of the program LIEPDE (for a comparison with another program see [Wol93]).
The novelty described in this paper are programs aiming at the final third step: Applying symmetries for
calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables of a PDE(-system) or for
generalizing given special solutions of ODEs/PDEs with new constant parameters.
Programs which run on their own but also allow interactive user control are indispensible for these calculations. On one hand the calculations can become quite lengthy, like variable transformations of PDEs (of higher order, with many variables). On the other hand the freedom of choosing the right linear combination of symmetries and choosing the optimal new symmetry- and similarity variables makes it necessary to ‘play’ with the problem interactively.
The focus in this paper is directed on questions of implementation and efficiency, no principally new mathematics is presented.
In the following subsections a review of the first two steps of the symmetry method is given as well as the third, i.e. the application step is outlined. Each of the remaining sections is devoted to one procedure.
To obey classical Lie-symmetries, differential equations \begin {equation} H_A = 0 \label {PDEs} \end {equation} for unknown functions \(y^\alpha ,\;\;1\leq \alpha \leq p\) of independent variables \(x^i,\;\;1\leq i \leq q\) must be forminvariant against infinitesimal transformations \begin {equation} \tilde {x}^i = x^i + \varepsilon \xi ^i, \;\; \;\;\; \tilde {y}^\alpha = y^\alpha + \varepsilon \eta ^\alpha \label {tran} \end {equation} in first order of \(\varepsilon .\) To transform the equations (\ref {PDEs}) by (\ref {tran}), derivatives of \(y^\alpha \) must be transformed, i.e. the part linear in \(\varepsilon \) must be determined. The corresponding formulas are (see e.g. [Olv86, Ste89]) \begin {align} \tilde {y}^\alpha _{j_1\ldots j_k} &= y^\alpha _{j_1\ldots j_k} + \varepsilon \eta ^\alpha _{j_1\ldots j_k} + O(\varepsilon ^2) \nonumber \\[3mm] \eta ^\alpha _{j_1\ldots j_{k-1}j_k} &= \frac {D \eta ^\alpha _{j_1\ldots j_{k-1}}}{D x^k} - y^\alpha _{ij_1\ldots j_{k-1}}\frac {D \xi ^i}{D x^k} \label {recur} \end {align}
where \(D/Dx^k\) means total differentiation w.r.t. \(x^k\) and from now on lower latin indices of functions \(y^\alpha ,\) (and later \(u^\alpha \)) denote partial differentiation w.r.t. the independent variables \(x^i,\) (and later \(v^i\)). The complete symmetry condition then takes the form \begin {align} X H_A &= 0 \bmod H_A = 0\ \label {sbed1} \\ X &= \xi ^i \frac {\partial }{\partial x^i} + \eta ^\alpha \frac {\partial }{\partial y^\alpha } + \eta ^\alpha _m \frac {\partial }{\partial y^\alpha _m} + \eta ^\alpha _{mn} \frac {\partial }{\partial y^\alpha _{mn}} + \ldots + \eta ^\alpha _{mn\ldots p} \frac {\partial }{\partial y^\alpha _{mn\ldots p}}. \label {sbed2} \end {align}
where mod \(H_A = 0\) means that the original PDE-system is used to replace some partial derivatives of \(y^\alpha \) to reduce the number of independent variables, because the symmetry condition (\ref {sbed1}) must be fulfilled identically in \(x^i, y^\alpha \) and all partial derivatives of \(y^\alpha .\)
For point symmetries, \(\xi ^i, \eta ^\alpha \) are functions of \(x^j, y^\beta \) and for contact symmetries they depend on \(x^j, y^\beta \) and \(y^\beta _k.\) We restrict ourself to point symmetries as those are the only ones that can be applied by the current version of the program APPLYSYM (see below). For literature about generalized symmetries see [Her95].
Though the formulation of the symmetry conditions (\ref {sbed1}), (\ref {sbed2}), (\ref {recur}) is straightforward and handled in principle by all related programs [Her95], the computational effort to formulate the conditions (\ref {sbed1}) may cause problems if the number of \(x^i\) and \(y^\alpha \) is high. This can partially be avoided if at first only a few conditions are formulated and solved such that the remaining ones are much shorter and quicker to formulate.
A first step in this direction is to investigate one PDE \(H_A = 0\) after another, as done in [CHW91]. Two methods to partition the conditions for a single PDE are described by Bocharov/Bronstein [BB89] and Stephani [Ste89].
In the first method only those terms of the symmetry condition \(X H_A = 0\) are calculated which contain at least a derivative of \(y^\alpha \) of a minimal order \(m.\) Setting coefficients of these \(u\)-derivatives to zero provides symmetry conditions. Lowering the minimal order \(m\) successively then gradually provides all symmetry conditions.
The second method is even more selective. If \(H_A\) is of order \(n\) then only terms of the symmetry condition \(X H_A = 0\) are generated which contain \(n'\)th order derivatives of \(y^\alpha .\) Furthermore these derivatives must not occur in \(H_A\) itself. They can therefore occur in the symmetry condition (\ref {sbed1}) only in \(\eta ^\alpha _{j_1\ldots j_n},\) i.e. in the terms \[\eta ^\alpha _{j_1\ldots j_n} \frac {\partial H_A}{\partial y^\alpha _{j_1\ldots j_n}}. \] If only coefficients of \(n'\)th order derivatives of \(y^\alpha \) need to be accurate to formulate preliminary conditions then from the total derivatives to be taken in (\ref {recur}) only that part is performed which differentiates w.r.t. the highest \(y^\alpha \)-derivatives. This means, for example, to form only \(y^\alpha _{mnk} \partial /\partial y^\alpha _{mn} \) if the expression, which is to be differentiated totally w.r.t. \(x^k\), contains at most second order derivatives of \(y^\alpha .\)
The second method is applied in LIEPDE. Already the formulation of the remaining conditions is speeded up considerably through this iteration process. These methods can be applied if systems of DEs or single PDEs of at least second order are investigated concerning symmetries.
The second step in applying the whole method consists in solving the determining conditions (\ref {sbed1}), (\ref {sbed2}), (\ref {recur}) which are linear homogeneous PDEs for \(\xi ^i, \eta ^\alpha \). The complete solution of this system is not algorithmic any more because the solution of a general linear PDE-system is as difficult as the solution of its non-linear characteristic ODE-system which is not covered by algorithms so far.
Still algorithms are used successfully to simplify the PDE-system by calculating its standard normal form and by integrating exact PDEs if they turn up in this simplification process [Wol93]. One problem in this respect, for example, concerns the optimization of the symbiosis of both algorithms. By that we mean the ranking of priorities between integrating, adding integrability conditions and doing simplifications by substitutions - all depending on the length of expressions and the overall structure of the PDE-system. Also the extension of the class of PDEs which can be integrated exactly is a problem to be pursuit further.
The program LIEPDE which formulates the symmetry conditions calls the program CRACK to solve them. This is done in a number of successive calls in order to formulate and solve some first order PDEs of the overdetermined system first and use their solution to formulate and solve the next subset of conditions as described in the previous subsection. Also, LIEPDE can work on DEs that contain parametric constants and parametric functions. An ansatz for the symmetry generators can be formulated. For more details see [Wol93] or [BW92].
The procedure LIEPDE is called through
LIEPDE(problem,symtype,flist,inequ);
All parameters are lists.
The first parameter specifies the DEs to be investigated:
problem has the form {equations, ulist, xlist} where
equations
is a list of equations, each has the form df(ui,..)=... where the LHS (left hand side) df(ui,..) is selected such that
The RHS (right h.s.) of an equations must not include the derivative on the LHS nor a derivative of it.
Neither the LHS nor any derivative of it of any equation may occur in any other equation.
Each of the unknown functions occurs on the LHS of exactly one equation.
ulist
is a list of function names, which can be chosen freely.
xlist
is a list of variable names, which can be chosen freely.
Equations can be given as a list of single differential expressions and then the program
will try to bring them into the ‘solved form’ df(ui,..)=... automatically. If
equations are given in the solved form then the above conditions are checked and
execution is stopped it they are not satisfied. An easy way to get the equations in the
desired form is to use
FIRST SOLVE({eq1,eq2,...},{one highest derivative for each function u})
(see the example of the Karpman equations in LIEPDE.TST). The example of the
Burgers equation in LIEPDE.TST demonstrates that the number of symmetries for a
given maximal order of the infinitesimal generators depends on the derivative chosen for
the LHS.
The second parameter symtype of LIEPDE is a list \(\{\;\}\) that specifies the symmetry to be calculated. symtype can have the following values and meanings:
{"point"}
Point symmetries with \(\xi ^i=\xi ^i(x^j,u^{\beta }),\; \eta ^{\alpha }=\eta ^{\alpha }(x^j,u^{\beta })\) are determined.
{"contact"}
Contact symmetries with \(\xi ^i=0,\;\eta =\eta (x^j,u,u_k)\) are determined \((u_k = \partial u/\partial x^k)\), which is only applicable if a single equation (\ref {PDEs}) with an order \(>1\) for a single function \(u\) is to be investigated. (The symtype {"contact"} is equivalent to {"general", 1} (see below) apart from the additional checks done for {"contact"}.)
{"general", order}
where order is an integer \(>0\). Generalized symmetries \(\xi ^i=0,\) \(\eta ^{\alpha }=\eta ^{\alpha }(x^j,u^{\beta },\ldots ,u^{\beta }_K)\) of a specified order are
determined (where \(_K\) is a multiple index representing order many indices.)
NOTE: Characteristic functions of generalized symmetries (\(= \eta ^{\alpha }\) if \(\xi ^i=0\)) are equivalent if
they are equal on the solution manifold. Therefore, all dependences of
characteristic functions on the substituted derivatives and their derivatives are
dropped. For example, if the heat equation is given as \(u_t=u_{xx}\) (i.e. \(u_t\) is substituted by \(u_{xx}\)) then
{"general", 2} would not include characteristic functions depending on \(u_{tx}\) or \(u_{xxx}\).
THEREFORE:
If you want to find all symmetries up to a given order then either
avoid using \(H_A=0\) to substitute lower order derivatives by expressions involving higher derivatives, or
increase the order specified in symtype.
For an illustration of this effect see the two symmetry determinations of the Burgers equation in the file LIEPDE.TST.
{xi!_x1 =...,..., eta!_u1 =...,...}
It is possible to specify an ansatz for the symmetry. Such an ansatz must specify all
\(\xi ^i\) for all independent variables and all \(\eta ^{\alpha }\) for all dependent variables in terms of
differential expressions which may involve unknown functions/constants. The
dependences of the unknown functions have to be declared in advance by using the
DEPEND command. For example,
DEPEND f, t, x, u$
specifies \(f\) to be a function of \(t,x,u\). If one wants to have \(f\) as a function of derivatives of \(u(t,x)\),
say \(f\) depending on \(u_{txx}\), then one cannot write
DEPEND f, df(u,t,x,2)$
but instead must write
DEPEND f, u!‘1!‘2!‘2$
assuming xlist has been specified as {t,x}. Because \(t\) is the first variable and \(x\) is
the second variable in xlist and \(u\) is differentiated oncs wrt. \(t\) and twice wrt. \(x\) we
therefore use u!‘1!‘2!‘2. The character ! is the escape character to allow
special characters like ‘ to occur in an identifier.
For generalized symmetries one usually sets all \(\xi ^i=0\). Then the \(\eta ^{\alpha }\) are equal to the characteristic functions.
The third parameter flist of LIEPDE is a list \(\{\;\}\) that includes
all parameters and functions in the equations which are to be determined such that symmetries exist (if any such parameters/functions are specified in flist then the symmetry conditions formulated in LIEPDE become non-linear conditions which may be much harder for CRACK to solve with many cases and subcases to be considered.)
all unknown functions and constants in the ansatz xi!_.. and eta!_.. if that has been specified in symtype.
The fourth parameter inequ of LIEPDE is a list \(\{\;\}\) that includes all non-vanishing expressions which represent inequalities for the functions in flist.
The result of LIEPDE is a list with 3 elements, each of which is a list: \[ \{\{\textit {con}_1,\textit {con}_2,\ldots \}, \{\texttt {xi}\__{\ldots }=\ldots , \ldots , \texttt {eta}\__{\ldots }=\ldots , \ldots \}, \{\textit {flist}\}\}. \] The first list contains remaining unsolved symmetry conditions con\(_i\). It is the empty list {} if all conditions have been solved. The second list gives the symmetry generators, i.e. expressions for \(\xi _i\) and \(\eta _j\). The last list contains all free constants and functions occuring in the first and second list.
If infinitesimal symmetries have been found then the program APPLYSYM can use them for the following purposes:
Both methods are described in the following section.
In the following we assume that a symmetry generator \(X\), given in (\ref {sbed2}), is known such that ODE(s)/PDE(s) \(H_A=0\) satisfy the symmetry condition (\ref {sbed1}). The aim is to find new dependent functions \(u^\alpha = u^\alpha (x^j,y^\beta )\) and new independent variables \(v^i = v^i(x^j,y^\beta ),\;\; 1\leq \alpha ,\beta \leq p,\;1\leq i,j \leq q\) such that the symmetry generator \(X = \xi ^i(x^j,y^\beta )\partial _{x^i} + \eta ^\alpha (x^j,y^\beta )\partial _{y^\alpha }\) transforms to \begin {equation} X = \partial _{v^1}. \label {sbed3} \end {equation}
Inverting the above transformation to \(x^i=x^i(v^j,u^\beta ), y^\alpha =y^\alpha (v^j,u^\beta )\) and setting
\(H_A(x^i(v^j,u^\beta ), y^\alpha (v^j,u^\beta ),\ldots ) = h_A(v^j, u^\beta ,\ldots )\) this means that \begin {align*} 0 &= X H_A(x^i,y^\alpha ,y^\beta _j,\ldots )\;\;\; \bmod \;\;\; H_A=0 \\ &= X h_A(v^i,u^\alpha ,u^\beta _j,\ldots )\;\;\; \bmod \;\;\; h_A=0 \\ &= \partial _{v^1}h_A(v^i,u^\alpha ,u^\beta _j,\ldots )\;\;\; \bmod \;\;\; h_A=0. \end {align*}
Consequently, the variable \(v^1\) does not occur explicitly in \(h_A\). In the case of an ODE(-system) \((v^1=v)\) the new equations \(0=h_A(v,u^\alpha ,du^\beta /dv,\ldots )\) are then of lower total order after the transformation \(z = z(u^1) = du^1/dv\) with now \(z, u^2,\ldots u^p\) as unknown functions and \(u^1\) as independent variable.
The new form (\ref {sbed3}) of \(X\) leads directly to conditions for the symmetry variable \(v^1\) and the similarity variables \(v^i|_{i\neq 1}, u^\alpha \) (all functions of \(x^k,y^\gamma \)): \begin {align} X v^1 = 1 &= \xi ^i(x^k,y^\gamma )\partial _{x^i}v^1 + \eta ^\alpha (x^k,y^\gamma )\partial _{y^\alpha }v^1 \label {ql1} \\ X v^j|_{j\neq 1} = X u^\beta = 0 &= \xi ^i(x^k,y^\gamma )\partial _{x^i}u^\beta + \eta ^\alpha (x^k,y^\gamma )\partial _{y^\alpha }u^\beta \label {ql2} \end {align}
The general solutions of (\ref {ql1}), (\ref {ql2}) involve free functions of \(p+q-1\) arguments. From the general solution of equation (\ref {ql2}), \(p+q-1\) functionally independent special solutions have to be selected (\(v^2,\ldots ,v^p\) and \(u^1,\ldots ,u^q\)), whereas from (\ref {ql1}) only one solution \(v^1\) is needed. Together, the expressions for the symmetry and similarity variables must define a non-singular transformation \(x,y \rightarrow u,v\).
Different special solutions selected at this stage will result in different resulting DEs which are equivalent under point transformations but may look quite differently. A transformation that is more difficult than another one will in general only complicate the new DE(s) compared with the simpler transformation. We therefore seek the simplest possible special solutions of (\ref {ql1}), (\ref {ql2}). They also have to be simple because the transformation has to be inverted to solve for the old variables in order to do the transformations.
The following steps are performed in the corresponding mode of the program APPLYSYM:
The user is asked to specify a symmetry by selecting one symmetry from all the known symmetries or by specifying a linear combination of them.
Through a call of the procedure QUASILINPDE (described in a later section) the two linear first order PDEs (\ref {ql1}), (\ref {ql2}) are investigated and, if possible, solved.
From the general solution of (\ref {ql1}) 1 special solution is selected and from (\ref {ql2}) \(p+q-1\) special solutions are selected which should be as simple as possible.
The user is asked whether the symmetry variable should be one of the independent variables (as it has been assumed so far) or one of the new functions (then only derivatives of this function and not the function itself turn up in the new DE(s)).
Through a call of the procedure DETRAFO the transformation \(x^i,y^\alpha \rightarrow v^j,u^\beta \) of the DE(s) \(H_A=0\) is finally done.
The program returns to the starting menu.
A second application of infinitesimal symmetries is the generalization of a known special solution given in implicit form through \(0 = F(x^i,y^\alpha )\). If one knows a symmetry variable \(v^1\) and similarity variables \(v^r, u^\alpha ,\;\;2\leq r\leq p\) then \(v^1\) can be shifted by a constant \(c\) because of \(\partial _{v^1}H_A = 0\) and therefore the DEs \(0 = H_A(v^r,u^\alpha ,u^\beta _j,\ldots )\) are unaffected by the shift. Hence from \[0 = F(x^i, y^\alpha ) = F(x^i(v^j,u^\beta ), y^\alpha (v^j,u^\beta )) = \bar {F}(v^j,u^\beta )\] follows that \[ 0 = \bar {F}(v^1+c,v^r,u^\beta ) = \bar {F}(v^1(x^i,y^\alpha )+c, v^r(x^i,y^\alpha ), u^\beta (x^i,y^\alpha ))\] defines implicitly a generalized solution \(y^\alpha =y^\alpha (x^i,c)\).
This generalization works only if \(\partial _{v^1}\bar {F} \neq 0\) and if \(\bar {F}\) does not already have a constant additive to \(v^1\).
The method above needs to know \(x^i=x^i(u^\beta ,v^j),\; y^\alpha =y^\alpha (u^\beta ,v^j)\) and \(u^\alpha = u^\alpha (x^j,y^\beta ), v^\alpha = v^\alpha (x^j,y^\beta )\) which may be practically impossible. Better is, to integrate \(x^i,y^\alpha \) along \(X\): \begin {equation} \frac {d\bar {x}^i}{d\varepsilon } = \xi ^i(\bar {x}^j(\varepsilon ), \bar {y}^\beta (\varepsilon )), \;\;\;\;\; \frac {d\bar {y}^\alpha }{d\varepsilon } = \eta ^\alpha (\bar {x}^j(\varepsilon ), \bar {y}^\beta (\varepsilon )) \label {ODEsys} \end {equation} with initial values \(\bar {x}^i = x^i, \bar {y}^\alpha = y^\alpha \) for \(\varepsilon = 0.\) (This ODE-system is the characteristic system of (\ref {ql2}).)
Knowing only the finite transformations \begin {equation} \bar {x}^i = \bar {x}^i(x^j,y^\beta ,\varepsilon ),\;\; \bar {y}^\alpha = \bar {y}^\alpha (x^j,y^\beta ,\varepsilon ) \label {ODEsol} \end {equation} gives immediately the inverse transformation \(\bar {x}^i = \bar {x}^i(x^j,y^\beta ,\varepsilon ),\;\; \bar {y}^\alpha = \bar {y}^\alpha (x^j,y^\beta ,\varepsilon )\) just by \(\varepsilon \rightarrow -\varepsilon \) and renaming \(x^i,y^\alpha \leftrightarrow \bar {x}^i,\bar {y}^\alpha .\)
The special solution \(0 = F(x^i,y^\alpha )\) is generalized by the new constant \(\varepsilon \) through \[ 0 = F(x^i,y^\alpha ) = F(x^i(\bar {x}^j,\bar {y}^\beta ,\varepsilon ), y^\alpha (\bar {x}^j,\bar {y}^\beta ,\varepsilon )) \] after dropping the \(\bar {~}\).
The steps performed in the corresponding mode of the program APPLYSYM show features of both techniques:
The user is asked to specify a symmetry by selecting one symmetry from all the known symmetries or by specifying a linear combination of them.
The special solution to be generalized and the name of the new constant have to be put in.
Through a call of the procedure QUASILINPDE, the PDE (\ref {ql1}) is solved which amounts to a solution of its characteristic ODE system (\ref {ODEsys}) where \(v^1=\varepsilon \).
QUASILINPDE returns a list of constant expressions \begin {equation} c_i = c_i(x^k, y^\beta , \varepsilon ),\;\;1\leq i\leq p+q \end {equation} which are solved for \(x^j=x^j(c_i,\varepsilon ),\;\; y^\alpha =y^\alpha (c_i,\varepsilon )\) to obtain the generalized solution through \[ 0 = F(x^j, y^\alpha ) = F( x^j(c_i(x^k, y^\beta , 0), \varepsilon ), y^\alpha (c_i(x^k, y^\beta , 0), \varepsilon )). \]
The new solution is availabe for further generalizations w.r.t. other symmetries.
If one would like to generalize a given special solution with \(m\) new constants because \(m\) symmetries are known, then one could run the whole program \(m\) times, each time with a different symmetry or one could run the program once with a linear combination of \(m\) symmetry generators which again is a symmetry generator. Running the program once adds one constant but we have in addition \(m-1\) arbitrary constants in the linear combination of the symmetries, so \(m\) new constants are added. Usually one will generalize the solution gradually to make solving (\ref {ODEsys}) gradually more difficult.
The call of APPLYSYM is APPLYSYM({de, fun, var}, {sym, cons});
de is a single DE or a list of DEs in the form of a vanishing expression or in the form \(\ldots =\ldots \;\;\).
fun is the single function or the list of functions occuring in de.
var is the single variable or the list of variables in de.
sym is a linear combination of all symmetries, each with a different constant coefficient, in form of a list of the \(\xi ^i\) and \(\eta ^\alpha \): {xi_…=…,…,eta_…=…,…}, where the indices after ‘xi_’ are the variable names and after ‘eta_’ the function names.
cons is the list of constants in sym, one constant for each symmetry.
The list that is the first argument of APPLYSYM is the same as the first argument of LIEPDE and the second argument is the list that LIEPDE returns without its first element (the unsolved conditions). An example is given below.
What APPLYSYM returns depends on the last performed modus. After modus 1 the
return is
{{newde, newfun, newvar}, trafo}
where
newde lists the transformed equation(s)
newfun lists the new function name(s)
newvar lists the new variable name(s)
trafo lists the transformations \(x^i=x^i(v^j,u^\beta ), y^\alpha =y^\alpha (v^j,u^\beta )\)
After modus 2, APPLYSYM returns the generalized special solution.
Weyl’s class of solutions of Einsteins field equations consists of axialsymmetric time independent metrics of the form \begin {equation} {\mathrm {d}} s^2 = e^{-2 U} \left [ e^{2 k} \left ( \mathrm {d} \rho ^2 + \mathrm {d} z^2 \right )+\rho ^2 \mathrm {d} \varphi ^2 \right ] - e^{2 U} \mathrm {d} t^2, \end {equation} where \(U\) and \(k\) are functions of \(\rho \) and \(z\). If one is interested in generalizing these solutions to have a time dependence then the resulting DEs can be transformed such that one longer third order ODE for \(U\) results which contains only \(\rho \) derivatives [Kub]. Because \(U\) appears not alone but only as derivative, a substitution \begin {equation} g = dU/d\rho \label {g1dgl} \end {equation} lowers the order and the introduction of a function \begin {equation} h = \rho g - 1 \label {g2dgl} \end {equation} simplifies the ODE to \begin {equation} 0 = 3\rho ^2h\,h'' -5\rho ^2\,h'^2+5\rho \,h\,h'-20\rho \,h^3h'-20\,h^4+16\,h^6+4\,h^2. \label {hdgl} \end {equation} where \('= d/d\rho \). Calling LIEPDE through
depend h,r;
prob:={{-20*h**4+16*h**6+3*r**2*h*df(h,r,2)+5*r*h*df(h,r)
-20*h**3*r*df(h,r)+4*h**2-5*r**2*df(h,r)**2},
{h}, {r}};
sym:=liepde(prob, {"point"},{},{});
end;
gives
sym := {{},
3 2
{xi_r= - c10*r - c11*r, eta_h=c10*h*r },
{c10,c11}}.
All conditions have been solved because the first element of sym is \(\{\}\). The two existing symmetries are therefore \begin {equation} - \rho ^3 \partial _{\rho } + h \rho ^2 \,\partial _{h} \;\;\;\;\;\;\mbox {and} \;\;\;\;\;\;\rho \partial _{\rho }. \end {equation} Corresponding finite transformations can be calculated with APPLYSYM through
newde:=applysym(prob,rest sym);
The interactive session is given below with the user input following the prompt ‘:’ or following ‘?’. (Empty lines have been deleted.)
Do you want to find similarity and symmetry variables (1) or generalize a special solution with new parameters (2) or exit the program (3) Input:3: 1;
We enter ‘1’ because we want to reduce dependencies by finding similarity variables and one symmetry variable and then doing the transformation such that the symmetry variable does not explicitly occur in the DE.
---------------------- The 1. symmetry is: 3 xi_r= - r 2 eta_h=h*r ---------------------- The 2. symmetry is: xi_r= - r ---------------------- Which single symmetry or linear combination of symmetries do you want to apply? Enter an expression with ‘sy_(i)’ for the i’th symmetry. Terminate input with ‘$’ or ‘;’. sy_(1);
We could have entered ‘sy_(2);’ or a combination of both as well with the calculation running then differently.
The symmetry to be applied in the following is
3 2
{xi_r = - r ,eta_h = h*r }
Terminate the following input with $ or ; .
Enter the name of the new dependent variable
(which will get an index attached): u;
Enter the name of the new independent variables:
(which will get an index attached): v;
This was the input part, now the real calculation starts.
The ODE/PDE (-system) under investigation is : 2 2 2 3 0 = 3*df(h,r,2)*h*r - 5*df(h,r) *r - 20*df(h,r)*h *r 6 4 2 + 5*df(h,r)*h*r + 16*h - 20*h + 4*h for the function(s) : h. It will be looked for a new dependent variable u and an independent variable v such that the transformed de(-system) does not depend on u or v. 1. Determination of the similarity variable 2 The quasilinear PDE: 0 = r *(df(u_,h)*h - df(u_,r)*r). The equivalent characteristic system: 3 0= - df(u_,r)*r 2 0= - r *(df(h,r)*r + h) for the functions: h(r) u_(r).
The PDE is equation (\ref {ql2}).
The general solution of the PDE is given through 0 = ff(u_,h*r) with arbitrary function ff(..). A suggestion for this function ff provides: 0 = - h*r + u_ Do you like this choice? (Y or N) y
For the following calculation only a single special solution of the PDE is necessary and this has to be specified from the general solution by choosing a special function ff. (This function is called ff to prevent a clash with names of user variables/functions.) In principle any choice of ff would work, if it defines a non-singular coordinate transformation, i.e. here \(r\) must be a function of \(u\_\). If we have \(q\) independent variables and \(p\) functions of them then ff has \(p+q\) arguments. Because of the condition \(0 = \)ff one has essentially the freedom of choosing a function of \(p+q-1\) arguments freely. This freedom is also necessary to select \(p+q-1\) different functions ff and to find as many functionally independent solutions \(u\_\) which all become the new similarity variables. \(q\) of them become the new functions \(u^\alpha \) and \(p-1\) of them the new variables \(v^2,\ldots ,v^p\). Here we have \(p=q=1\) (one single ODE).
Though the program could have done that alone, once the general solution ff(..) is known, the user can interfere here to enter a simpler solution, if possible.
2. Determination of the symmetry variable 2 3 The quasilinear PDE: 0 = df(u_,h)*h*r - df(u_,r)*r - 1. The equivalent characteristic system: 3 0=df(r,u_) + r 2 0=df(h,u_) - h*r for the functions: r(u_) h(u_) . New attempt with a different independent variable The equivalent characteristic system: 2 0=df(u_,h)*h*r - 1 2 0=r *(df(r,h)*h + r) for the functions: r(h) u_(h) . The general solution of the PDE is given through 2 2 2 - 2*h *r *u_ + h 0 = ff(h*r,--------------------) 2 with arbitrary function ff(..). A suggestion for this function ff(..) yields: 2 2 h *( - 2*r *u_ + 1) 0 = --------------------- 2 Do you like this choice? (Y or N) y
Similar to above.
The suggested solution of the algebraic system which will
do the transformation is:
sqrt(v)*sqrt(2)
{h=sqrt(v)*sqrt(2)*u,r=-----------------}
2*v
Is the solution ok? (Y or N) y
In the intended transformation shown above the dependent
variable is u and the independent variable is v.
The symmetry variable is v, i.e. the transformed expression
will be free of v.
Is this selection of dependent and independent variables ok? (Y or N) n
We so far assumed that the symmetry variable is one of the new variables, but, of course we also could choose it to be one of the new functions. If it is one of the functions then only derivatives of this function occur in the new DE, not the function itself. If it is one of the variables then this variable will not occur explicitly.
In our case we prefer (without strong reason) to have the function as symmetry variable. We therefore answered with ‘no’. As a consequence, \(u\) and \(v\) will exchange names such that still all new functions have the name \(u\) and the new variables have name \(v\):
Please enter a list of substitutions. For example, to
make the variable, which is so far call u1, to an
independent variable v2 and the variable, which is
so far called v2, to an dependent variable u1,
enter: ‘{u1=v2, v2=u1};’{u=v,v=u};
The transformed equation which should be free of u:
3 6 2 3
0=3*u *v - 16*u *v - 20*u *v + 5*u
2v v v v
Do you want to find similarity and symmetry variables (1)
or generalize a special solution with new parameters (2)
or exit the program (3) :
We stop here. The following is returned from our APPLYSYM call:
3 6
{{{3*df(u,v,2)*v - 16*df(u,v) *v
2 3
- 20*df(u,v) *v + 5*df(u,v)},
{u},
{v}},
1 2*u*v
{r=-----------------,h=-----------------
sqrt(u)*sqrt(2) sqrt(u)*sqrt(2)
}}
The use of APPLYSYM effectively provided us the finite transformation \begin {equation} \rho =(2\,u)^{-1/2},\;\;\;\;\;h=(2\,u)^{1/2}\,v \label {trafo1}. \end {equation} and the new ODE \begin {equation} 0 = 3u''v - 16u'^3v^6 - 20u'^2v^3 + 5u' \label {udgl} \end {equation} where \(u=u(v)\) and \('=d/dv.\) Using one symmetry we reduced the 2. order ODE (\ref {hdgl}) to a first order ODE (\ref {udgl}) for \(u'\) plus one integration. The second symmetry can be used to reduce the remaining ODE to an integration too by introducing a variable \(w\) through \(v^3d/dv = d/dw\), i.e. \(w = -1/(2v^2)\). With \begin {equation} p=du/dw \label {udot} \end {equation} the remaining ODE is \[0 = 3\,w\,\frac {dp}{dw} + 2\,p\,(p+1)(4\,p+1) \] with solution \[ \tilde {c}w^{-2}/4 = \tilde {c}v^4 = \frac {p^3(p+1)}{(4\,p+1)^4},\;\;\; \tilde {c}=const. \] Writing (\ref {udot}) as \(p = v^3(du/dp)/(dv/dp)\) we get \(u\) by integration and with (\ref {trafo1}) further a parametric solution for \(\rho ,h\): \begin {align} \rho & = \left (\frac {3c_1^2(2p-1)}{p^{1/2}(p+1)^{1/2}}+c_2\right )^{-1/2} \\ h & = \frac {(c_2p^{1/2}(p+1)^{1/2}+6c_1^2p-3c_1^2)^{1/2}p^{1/2}}{c_1(4p+1)} \end {align}
where \(c_1, c_2 = const.\) and \(c_1=\tilde {c}^{1/4}.\) Finally, the metric function \(U(p)\) is obtained as an integral from (\ref {g1dgl}),(\ref {g2dgl}).
Restrictions of the applicability of the program APPLYSYM result from limitations of the program QUASILINPDE described in a section below. Essentially this means that symmetry generators may only be polynomially non-linear in \(x^i, y^\alpha \). Though even then the solvability can not be guaranteed, the generators of Lie-symmetries are mostly very simple such that the resulting PDE (\ref {PDE}) and the corresponding characteristic ODE-system have good chances to be solvable.
Apart from these limitations implied through the solution of differential equations with CRACK and algebraic equations with SOLVE the program APPLYSYM itself is free of restrictions, i.e. if once new versions of CRACK, SOLVE would be available then APPLYSYM would not have to be changed.
Currently, whenever a computational step could not be performed the user is informed and has the possibility of entering interactively the solution of the unsolved algebraic system or the unsolved linear PDE.
The generalization of special solutions of DEs as well as the computation of similarity and symmetry variables involve the general solution of single first order linear PDEs. The procedure QUASILINPDE is a general procedure aiming at the general solution of PDEs \begin {equation} a_1(w_i,\phi )\phi _{w_1} + a_2(w_i,\phi )\phi _{w_2} + \ldots + a_n(w_i,\phi )\phi _{w_n} = b(w_i,\phi ) \label {PDE} \end {equation} in \(n\) independent variables \(w_i, i=1\ldots n\) for one unknown function \(\phi =\phi (w_i)\).
for \(\phi , w_i\) regarded now as functions of one variable \(\varepsilon \).
Because the \(a_i\) and \(b\) do not depend explicitly on \(\varepsilon \), one of the equations (\ref {char1}),(\ref {char2}) with non-vanishing right hand side can be used to divide all others through it and by that having a system with one less ODE to solve. If the equation to divide through is one of (\ref {char1}) then the remaining system would be \begin {align} \frac {dw_i}{dw_k} & = \frac {a_i}{a_k} , \;\;\;i=1,2,\ldots k-1,k+1,\ldots n \label {char3} \\ \frac {d\phi }{dw_k} & = \frac {b}{a_k} \label {char4} \end {align}
with the independent variable \(w_k\) instead of \(\varepsilon \). If instead we divide through equation (\ref {char2}) then the remaining system would be \[ \frac {dw_i}{d\phi } = \frac {a_i}{b} , \;\;\;i=1,2,\ldots n \label {char3a} \] with the independent variable \(\phi \) instead of \(\varepsilon \).
The equation to divide through is chosen by a subroutine with a heuristic to find the “simplest” non-zero right hand side (\(a_k\) or \(b\)), i.e. one which
is constant or
depends only on one variable or
is a product of factors, each of which depends only on one variable.
One purpose of this division is to reduce the number of ODEs by one. Secondly, the general solution of (\ref {char1}), (\ref {char2}) involves an additive constant to \(\varepsilon \) which is not relevant and would have to be set to zero. By dividing through one ODE we eliminate \(\varepsilon \) and lose the problem of identifying this constant in the general solution before we would have to set it to zero.
If the characteristic ODE-system can not be solved in the form (\ref {char3}), (\ref {char4}) or (\ref {char3a}) then successively all other ODEs of (\ref {char1}), (\ref {char2}) with non-vanishing right hand side are used for division until one is found such that the resulting ODE-system can be solved completely. Otherwise the PDE can not be solved by QUASILINPDE.
On either way one ends up with \(n\) equations \begin {equation} 0=g_i(\phi ,w_j,c_k),\;\;i,j,k=1\ldots n \label {charsol2} \end {equation} involving \(n\) constants \(c_k\).
The final step is to solve (\ref {charsol2}) for the \(c_i\) to obtain \begin {equation} c_i = c_i(\phi , w_1,\ldots ,w_n) \;\;\;\;\;i=1,\ldots n . \label {cons} \end {equation} The final solution \(\phi = \phi (w_i)\) of the PDE (\ref {PDE}) is then given implicitly through \[ 0 = F(c_1(\phi ,w_i),c_2(\phi ,w_i),\ldots ,c_n(\phi ,w_i)) \] where \(F\) is an arbitrary function with \(n\) arguments.
The call of QUASILINPDE is
QUASILINPDE(de, fun, varlist);
de is the differential expression which vanishes due to the PDE de\(\; = 0\) or, de may be the differential equation itself in the form \(\;\;\ldots = \ldots \;\;\).
fun is the unknown function.
varlist is the list of variables of fun.
The result of QUASILINPDE is a list of general solutions \[ \{\textit {sol}_1, \textit {sol}_2, \ldots \}. \] If QUASILINPDE can not solve the PDE then it returns \(\{\}\). Each solution \(\textit {sol}_i\) is a list of expressions \[ \{\textit {ex}_1, \textit {ex}_2, \ldots \} \] such that the dependent function (\(\phi \) in (\ref {PDE})) is determined implicitly through an arbitrary function \(F\) and the algebraic equation \[ 0 = F(\textit {ex}_1, \textit {ex}_2, \ldots ). \]
Example 1:
To solve the quasilinear first order PDE \[1 = xu,_x + uu,_y - zu,_z\] for the function \(u = u(x,y,z),\) the input would be
depend u,x,y,z;
de:=x*df(u,x)+u*df(u,y)-z*df(u,z) - 1;
varlist:={x,y,z};
QUASILINPDE(de,u,varlist);
In this example the procedure returns \[\{ \{ x/e^u, ze^u, u^2 - 2y \} \},\] i.e. there is one general solution (because the
outer list has only one element which itself is a list) and \(u\) is given implicitly through the
algebraic equation \[ 0 = F(x/e^u, ze^u, u^2 - 2y)\] with arbitrary function \(F.\)
Example 2:
For the linear inhomogeneous PDE \[ 0 = y z,_x + x z,_y - 1, \;\;\;\;\mbox {for}\;\;\;\;z=z(x,y)\] QUASILINPDE returns the result that for an
arbitrary function \(F,\) the equation \[ 0 = F\left (\frac {x+y}{e^z},e^z(x-y)\right ) \] defines the general solution for \(z\).
Example 3:
For the linear inhomogeneous PDE (3.8) from [Kam59] \[ 0 = x w,_x + (y+z)(w,_y - w,_z), \;\;\;\;\mbox {for}\;\;\;\;w=w(x,y,z)\] QUASILINPDE returns the
result that for an arbitrary function \(F,\) the equation \[ 0 = F\left (w, \;y+z, \;\ln (x)(y+z)-y\right ) \] defines the general solution for \(w\), i.e. for
any function \(f\) \[ w = f\left (y+z, \;\ln (x)(y+z)-y\right ) \] solves the PDE.
One restriction on the applicability of QUASILINPDE results from the program CRACK which tries to solve the characteristic ODE-system of the PDE. So far CRACK can be applied only to polynomially non-linear DE’s, i.e. the characteristic ODE-system (\ref {char3}),(\ref {char4}) or (\ref {char3a}) may only be polynomially non-linear, i.e. in the PDE (\ref {PDE}) the expressions \(a_i\) and \(b\) may only be rational in \(w_j,\phi \).
The task of CRACK is simplified as (\ref {charsol1}) does not have to be solved for \(w_j, \phi \). On the other hand (\ref {charsol1}) has to be solved for the \(c_i\). This gives a second restriction coming from the REDUCE function SOLVE. Though SOLVE can be applied to polynomial and transzendential equations, again no guarantee for solvability can be given.
Finally, after having found the finite transformations, the program APPLYSYM calls the procedure DETRAFO to perform the transformations. DETRAFO can also be used alone to do point- or higher order transformations which involve a considerable computational effort if the differential order of the expression to be transformed is high and if many dependent and independent variables are involved. This might be especially useful if one wants to experiment and try out different coordinate transformations interactively, using DETRAFO as standalone procedure.
To run DETRAFO, the old functions \(y^{\alpha }\) and old variables \(x^i\) must be known explicitly in terms of algebraic or differential expressions of the new functions \(u^{\beta }\) and new variables \(v^j\). Then for point transformations the identity \begin {align} dy^{\alpha } & = \left (y^{\alpha },_{v^i} + y^{\alpha },_{u^{\beta }}u^{\beta },_{v^i}\right ) dv^i \\ & = y^{\alpha },_{x^j}dx^j \\ & = y^{\alpha },_{x^j}\left (x^j,_{v^i} + x^j,_{u^{\beta }}u^{\beta },_{v^i}\right ) dv^i \end {align}
provides the transformation \begin {equation} y^{\alpha },_{x^j} = \frac {dy^\alpha }{dv^i}\cdot \left (\frac {dx^j}{dv^i}\right )^{-1} \label {trafo} \end {equation} with det\(\left (dx^j/dv^i\right ) \neq 0\) because of the regularity of the transformation which is checked by DETRAFO. Non-regular transformations are not performed.
DETRAFO is not restricted to point transformations. In the case of contact- or higher order transformations, the total derivatives \(dy^{\alpha }/dv^i\) and \(dx^j/dv^i\) then only include all \(v^i-\) derivatives of \(u^{\beta }\) which occur in \begin {align*} y^{\alpha } &= y^{\alpha }(v^i,u^{\beta },u^{\beta },_{v^j},\ldots ) \\ x^k &= x^k(v^i,u^{\beta },u^{\beta },_{v^j},\ldots ). \end {align*}
The call of DETRAFO is
DETRAFO({ex\(_1\), ex\(_2\), …, ex\(_m\)},
{ofun\(_1=\)fex\(_1\), ofun\(_2=\)fex\(_2\), …,ofun\(_p=\)fex\(_p\)},
{ovar\(_1=\)vex\(_1\), ovar\(_2=\)vex\(_2\), …, ovar\(_q=\)vex\(_q\)},
{nfun\(_1\), nfun\(_2\), …, nfun\(_p\)},
{nvar\(_1\), nvar\(_2\), …, nvar\(_q\)});
where \(m,p,q\) are arbitrary.
The ex\(_i\) are differential expressions to be transformed.
The second list is the list of old functions ofun expressed as expressions fex in terms of new functions nfun and new independent variables nvar.
Similarly the third list expresses the old independent variables ovar as expressions vex in terms of new functions nfun and new independent variables nvar.
The last two lists include the new functions nfun and new independent variables nvar.
Names for ofun, ovar, nfun and nvar can be arbitrarily chosen.
As the result DETRAFO returns the first argument of its input, i.e. the list \[\{\textit {ex}_1, \textit {ex}_2, \ldots , \textit {ex}_m\}\] where all \(\textit {ex}_i\) are transformed.
The only requirement is that the old independent variables \(x^i\) and old functions \(y^\alpha \) must be given explicitly in terms of new variables \(v^j\) and new functions \(u^\beta \) as indicated in the syntax. Then all calculations involve only differentiations and basic algebra.
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