@@ -1,1194 +1,1194 @@ -% This is a LaTeX file -\documentstyle[12pt]{article} - -%Sets size of page and margins -\oddsidemargin 10mm \evensidemargin 10mm -\topmargin 0pt \headheight 0pt \headsep 0pt -\footheight 14pt \footskip 40pt -\textheight 23cm \textwidth 15cm -%\textheight 15cm \textwidth 10cm - -%spaces lines at one and a half spacing -%\def\baselinestretch{1.5} - -%\parskip = \baselineskip - -%Defines capital R for the reals, ... -%\font\Edth=msym10 -%\def\Integer{\hbox{\Edth Z}} -%\def\Rational{\hbox{\Edth Q}} -%\def\Real{\hbox{\Edth R}} -%\def\Complex{\hbox{\Edth C}} - -\title{Programs for Applying Symmetries of PDEs} -\author{Thomas Wolf \\ - School of Mathematical Sciences \\ - Queen Mary and Westfield College \\ - University of London \\ - London E1 4NS \\ - T.Wolf@maths.qmw.ac.uk -} - -\begin{document} -\maketitle -\begin{abstract} -In this paper the programs {\tt APPLYSYM}, {\tt QUASILINPDE} and -{\tt DETRAFO} are described which aim at the utilization -of infinitesimal symmetries of differential equations. The purpose -of {\tt QUASILINPDE} is the general solution of -quasilinear PDEs. This procedure is used by {\tt APPLYSYM} -for the application of point symmetries for either -\begin{itemize} -\item 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 -\item generalizing given special solutions of ODEs/PDEs with new constant -parameters. -\end{itemize} - -The program {\tt DETRAFO} performs arbitrary point- and contact -transformations of ODEs/PDEs and is applied if similarity -and symmetry variables have been found. -The program {\tt APPLYSYM} is used in connection with the program -{\tt LIEPDE} for formulating and solving the conditions for point- and -contact symmetries which is described in \cite{LIEPDE}. -The actual problem solving is done in all these programs through a call -to the package {\tt CRACK} for solving overdetermined PDE-systems. -\end{abstract} - -\tableofcontents -%------------------------------------------------------------------------- -\section{Introduction and overview of the symmetry method} -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 -\cite{WHer} 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 -\cite{lie1},\cite{lie2} 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 {\tt LIEPDE} -(for a comparison with another program see \cite{LIEPDE}). - -The novelty described in this paper are programs aiming at -the final third step: Applying symmetries for -\begin{itemize} -\item 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 -\item generalizing given special solutions of ODEs/PDEs with new constant -parameters. -\end{itemize} -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. -%--------------------------------------- -\subsection{The first step: Formulating the symmetry conditions} - -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. \cite{Olv}, \cite{Step}) -\begin{eqnarray} -\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 \\ \vspace{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{eqnarray} -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{eqnarray} -X H_A & = & 0 \;\; \; \; \mbox{mod} \; \; \; 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{eqnarray} -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 {\tt APPLYSYM} -(see below). For literature about generalized symmetries see \cite{WHer}. - -Though the formulation of the symmetry conditions (\ref{sbed1}), -(\ref{sbed2}), (\ref{recur}) -is straightforward and handled in principle by all related -programs \cite{WHer}, 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 \cite{Cham}. Two methods to partition the -conditions for a single PDE are described by Bocharov/Bronstein -\cite{Alex} and Stephani \cite{Step}. - -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 {\tt 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. -%--------------------------------------- -\subsection{The second step: Solving the symmetry conditions} -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 \cite{LIEPDE}. -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 {\tt LIEPDE} which formulates the symmetry conditions calls the -program {\tt 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, {\tt 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 \cite{LIEPDE} or \cite{WoBra}. - - -The call of {\tt LIEPDE} is \\ -{\tt LIEPDE}(\{{\it de}, {\it fun}, {\it var}\}, -\{{\it od}, {\it lp}, {\it fl}\}); \\ -where -\begin{itemize} -\item {\it de} is a single DE or a list of DEs in the form of a vanishing - expression or in the form $\ldots=\ldots\;\;$. -\item {\it fun} is the single function or the list of functions occuring - in {\it de}. -\item {\it var} is the single variable or the list of variables in {\it de}. -\item {\it od} is the order of the ansatz for $\xi, \eta.$ It is = 0 for -point symmetries and = 1 for contact symmetries (accepted by -{\tt LIEPDE} only in case of one ODE/PDE for one unknown function). -% and $>1$ for dynamical symmetries -%(only in case of one ODE for one unknown function) -\item If {\it lp} is $nil$ then the standard ansatz for $\xi^i, \eta^\alpha$ -is taken which is - \begin{itemize} - \item for point symmetries ({\it od} =0) is $\xi^i = \xi^i(x^j,y^\beta), - \eta^\alpha = \eta^\alpha(x^j,y^\beta)$ - \item for contact symmetries ({\it od} =1) is - $ \xi^i := \Omega_{u_i}, \;\;\; - \eta := u_i\Omega_{u_i} \; - \; \Omega, $ \\ - $\Omega:=\Omega(x^i, u, u_j)$ -%\item for dynamical symmetries ({\it od}$>1$) \\ -% $ \xi := \Omega,_{u'}, \;\;\; -% \eta := u'\Omega,_{u'} \; - \; \Omega, \;\;\; -% \Omega:=\Omega(x, u, u',\ldots, y^{({\it od}-1)})$ -% where {\it od} must be less than the order of the ODE. - \end{itemize} - - If {\it lp} is not $nil$ then {\it lp} is the ansatz for - $\xi^i, \eta^\alpha$ and must have the form - \begin{itemize} - \item for point symmetries - {\tt \{xi\_\mbox{$x1$} = ..., ..., eta\_\mbox{$u1$} = ..., ...\}} - where {\tt xi\_, eta\_ } - are fixed and $x1, \ldots, u1$ are to be replaced by the actual names - of the variables and functions. - \item otherwise {\tt spot\_ = ...} where the expression on the right hand - side is the ansatz for the Symmetry-POTential $\Omega.$ - \end{itemize} - -\item {\it fl} is the list of free functions in the ansatz -in case {\it lp} is not $nil.$ -\end{itemize} - - -The result of {\tt LIEPDE} is a list with 3 elements, each of which -is a list: -\[ \{\{{\it con}_1,{\it con}_2,\ldots\}, - \{{\tt xi}\__{\ldots}=\ldots, \ldots, - {\tt eta}\__{\ldots}=\ldots, \ldots\}, - \{{\it flist}\}\}. \] -The first list contains remaining unsolved symmetry conditions {\it 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. - -%That the automatic calculation of symmetries run in most practical cases -%is shown with the following example. It is insofar difficult, as many -%symmetries exist and the solution consequently more difficult is to deriv. -% -%--------------------------------------- -%\subsection{Example} -%For the following PDE-system, which takes its simplest form in the -%formalism of exterior forms: -% -%\begin{eqnarray*} -%0 & = & 3k_t,_{tt}-2k_t,_{xx}-2k_t,_{yy}-2k_t,_{zz}-k_x,_{tx}-2k_zk_x,_y \\ -% & & +2k_yk_x,_z-k_y,_{ty}+2k_zk_y,_x-2k_xk_y,_z-k_z,_{tz}-2k_yk_z,_x+2k_xk_z,_y \\ -%0 & = & k_t,_{tx}-2k_zk_t,_y+2k_yk_t,_z+2k_x,_{tt}-3k_x,_{xx}-2k_x,_{yy} \\ -% & & -2k_x,_{zz}+2k_zk_y,_t-k_y,_{xy}-2k_tk_y,_z-2k_yk_z,_t-k_z,_{xz}+2k_tk_z,_y \\ -%0 & = & k_t,_{ty}+2k_zk_t,_x-2k_xk_t,_z-2k_zk_x,_t-k_x,_{xy}+2k_tk_x,_z \\ -% & & +2k_y,_{tt}-2k_y,_{xx}-3k_y,_{yy}-2k_y,_{zz}+2k_xk_z,_t-2k_tk_z,_x-k_z,_{yz} \\ -%0 & = & k_t,_{tz}-2k_yk_t,_x+2k_xk_t,_y+2k_yk_x,_t-k_x,_{xz}-2k_tk_x,_y \\ -% & & -2k_xk_y,_t+2k_tk_y,_x-k_y,_{yz}+2k_z,_{tt}-2k_z,_{xx}-2k_z,_{yy}-3k_z,_{zz} -%\end{eqnarray*} -%--------------------------------------- -\subsection{The third step: Application of infinitesimal symmetries} -If infinitesimal symmetries have been found then -the program {\tt APPLYSYM} can use them for the following purposes: -\begin{enumerate} -\item Calculation of one symmetry variable and further similarity variables. -After transforming -the DE(-system) to these variables, the symmetry variable will not occur -explicitly any more. For ODEs this has the consequence that their order has -effectively been reduced. -\item Generalization of a special solution by one or more constants of -integration. -\end{enumerate} -Both methods are described in the following section. -%------------------------------------------------------------------------- -\section{Applying symmetries with {\tt APPLYSYM}} -%--------------------------------------- -\subsection{The first mode: Calculation of similarity and symmetry variables} -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{eqnarray*} - 0 & = & X H_A(x^i,y^\alpha,y^\beta_j,\ldots)\;\;\; \mbox{mod} \;\;\; H_A=0 \\ - & = & X h_A(v^i,u^\alpha,u^\beta_j,\ldots)\;\;\; \mbox{mod} \;\;\; h_A=0 \\ - & = & \partial_{v^1}h_A(v^i,u^\alpha,u^\beta_j,\ldots)\;\;\; \mbox{mod} - \;\;\; h_A=0. -\end{eqnarray*} -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{eqnarray} - 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{eqnarray} -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 {\tt APPLYSYM}: -\begin{itemize} -\item 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. -\item Through a call of the procedure {\tt QUASILINPDE} (described in a later -section) the two linear first order PDEs (\ref{ql1}), (\ref{ql2}) are -investigated and, if possible, solved. -\item 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. -\item 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)). -\item Through a call of the procedure {\tt DETRAFO} the transformation -$x^i,y^\alpha \rightarrow v^j,u^\beta$ of the DE(s) $H_A=0$ is finally done. -\item The program returns to the starting menu. -\end{itemize} -%--------------------------------------- -\subsection{The second mode: Generalization of special solutions} -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 $x^i=x^i(u^\beta,v^j),\; -y^\alpha=y^\alpha(u^\beta,v^j)$ \underline{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 {\tt APPLYSYM} show features of both techniques: -\begin{itemize} -\item 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. -\item The special solution to be generalized and the name of the new -constant have to be put in. -\item Through a call of the procedure {\tt QUASILINPDE}, the PDE (\ref{ql1}) -is solved which amounts to a solution of its characteristic ODE system -(\ref{ODEsys}) where $v^1=\varepsilon$. -\item {\tt 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 -\[ 0 = F(x^j, y^\alpha) - = F( x^j(c_i(x^k, y^\beta, 0), \varepsilon)), - y^\alpha(c_i(x^k, y^\beta, 0), \varepsilon))). \] -\item The new solution is availabe for further generalizations w.r.t.\ other -symmetries. -\end{itemize} -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. -%--------------------------------------- -\subsection{Syntax} -The call of {\tt APPLYSYM} is -{\tt APPLYSYM}(\{{\it de}, {\it fun}, {\it var}\}, \{{\it sym}, {\it cons}\}); -\begin{itemize} -\item {\it de} is a single DE or a list of DEs in the form of a vanishing - expression or in the form $\ldots=\ldots\;\;$. -\item {\it fun} is the single function or the list of functions occuring - in {\it de}. -\item {\it var} is the single variable or the list of variables in {\it de}. -\item {\it 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\_\ldots=\ldots,\ldots,eta\_\ldots=\ldots,\ldots\}, - where the indices after `xi\_' are the variable names and after `eta\_' - the function names. -\item {\it cons} is the list of constants in {\it sym}, one constant for each - symmetry. -\end{itemize} -The list that is the first argument of {\tt APPLYSYM} is the same as the -first argument of {\tt LIEPDE} and the -second argument is the list that {\tt LIEPDE} returns without its first -element (the unsolved conditions). An example is given below. - -What {\tt APPLYSYM} returns depends on the last performed modus. -After modus 1 the return is \\ -\{\{{\it newde}, {\it newfun}, {\it newvar}\}, {\it trafo}\} \\ -where -\begin{itemize} -\item {\it newde} lists the transformed equation(s) -\item {\it newfun} lists the new function name(s) -\item {\it newvar} lists the new variable name(s) -\item {\it trafo} lists the transformations $x^i=x^i(v^j,u^\beta), - y^\alpha=y^\alpha(v^j,u^\beta)$ -\end{itemize} -After modus 2, {\tt APPLYSYM} returns the generalized special solution. -%--------------------------------------- -\subsection{Example: A second order ODE} -Weyl's class of solutions of Einsteins field equations consists of -axialsymmetric time independent metrics of the form -\begin{equation} -{\rm{d}} s^2 = e^{-2 U} \left[ e^{2 k} \left( \rm{d} \rho^2 + \rm{d} -z^2 \right)+\rho^2 \rm{d} \varphi^2 \right] - e^{2 U} \rm{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 \cite{Markus}. 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 {\tt LIEPDE} through -\small \begin{verbatim} -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,{0,nil,nil}); -end; \end{verbatim} \normalsize -gives \small \begin{verbatim} - 3 2 -sym := {{}, {xi_r= - c10*r - c11*r, eta_h=c10*h*r }, {c10,c11}}. -\end{verbatim} \normalsize -All conditions have been solved because the first element of {\tt 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 {\tt APPLYSYM} through -\small \begin{verbatim} -newde:=applysym(de,rest sym); -\end{verbatim} \normalsize -The interactive session is given below with the user input following -the prompt `{\tt Input:3:}' or following `?'. (Empty lines have been deleted.) -\small \begin{verbatim} -Do you want to find similarity and symmetry variables (enter `1;') -or generalize a special solution with new parameters (enter `2;') -or exit the program (enter `;') -Input:3: 1; -\end{verbatim} \normalsize -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. -\small \begin{verbatim} ----------------------- 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. -sy_(1); -\end{verbatim} \normalsize -We could have entered `sy\_(2);' or a combination of both -as well with the calculation running then -differently. -\small \begin{verbatim} -The symmetry to be applied in the following is - 3 2 -{xi_r= - r ,eta_h=h*r } -Enter the name of the new dependent variables: -Input:3: u; -Enter the name of the new independent variables: -Input:3: v; -\end{verbatim} \normalsize -This was the input part, now the real calculation starts. -\small \begin{verbatim} -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). -\end{verbatim} \normalsize -The PDE is equation (\ref{ql2}). -\small \begin{verbatim} -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 -\end{verbatim} \normalsize -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 {\tt ff}. (This function is called {\tt ff} to prevent a clash with -names of user variables/functions.) In principle any choice of {\tt 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 {\tt ff} has $p+q$ arguments. Because of the -condition $0 = ${\tt 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 {\tt 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 -{\tt ff(..)} is known, the user can interfere here to enter a simpler solution, -if possible. -\small \begin{verbatim} -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 -\end{verbatim} \normalsize -Similar to above. -\small \begin{verbatim} -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 -\end{verbatim} \normalsize -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 $v$: -\small \begin{verbatim} -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};' -Input:3: {u=v,v=u}; - -The transformed equation which should be free of u: - 3 6 2 3 -0=3*df(u,v,2)*v - 16*df(u,v) *v - 20*df(u,v) *v + 5*df(u,v) -Do you want to find similarity and symmetry variables (enter `1;') -or generalize a special solution with new parameters (enter `2;') -or exit the program (enter `;') -Input:3: ; -\end{verbatim} -We stop here. The following is returned from our {\tt APPLYSYM} call: -\small \begin{verbatim} - 3 6 2 3 -{{{3*df(u,v,2)*v - 16*df(u,v) *v - 20*df(u,v) *v + 5*df(u,v)}, - {u}, - {v}}, - sqrt(u)*sqrt(2) - {r=-----------------, h=sqrt(u)*sqrt(2)*v }} - 2*u -\end{verbatim} \normalsize -The use of {\tt 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{eqnarray} -\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{eqnarray} -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}). -%--------------------------------------- -\subsection{Limitations of {\tt APPLYSYM}} -Restrictions of the applicability of the program {\tt APPLYSYM} result -from limitations of the program {\tt 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 {\tt CRACK} and algebraic equations with {\tt SOLVE} -the program {\tt APPLYSYM} itself is free of restrictions, -i.e.\ if once new versions of {\tt CRACK, SOLVE} -would be available then {\tt 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. -%------------------------------------------------------------------------- -\section{Solving quasilinear PDEs} -%--------------------------------------- -\subsection{The content of {\tt QUASILINPDE}} -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 {\tt 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 -$\phi=\phi(w_i)$. -\begin{enumerate} -\item -The first step in solving a quasilinear PDE (\ref{PDE}) -is the formulation of the corresponding characteristic ODE-system -\begin{eqnarray} -\frac{dw_i}{d\varepsilon} & = & a_i(w_j,\phi) \label{char1} \\ -\frac{d\phi}{d\varepsilon} & = & b(w_j,\phi) \label{char2} -\end{eqnarray} -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{eqnarray} -\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{eqnarray} -with the independent variable $w_k$ instead of $\varepsilon$. -If instead we divide through equation -(\ref{char2}) then the remaining system would be -\begin{eqnarray} -\frac{dw_i}{d\phi} & = & \frac{a_i}{b} , \;\;\;i=1,2,\ldots n - \label{char3a} -\end{eqnarray} -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 -\begin{itemize} -\item is constant or -\item depends only on one variable or -\item is a product of factors, each of which depends only on -one variable. -\end{itemize} - -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. - -\item % from enumerate -To solve the system (\ref{char3}), (\ref{char4}) or (\ref{char3a}), -the procedure {\tt CRACK} is called. -Although being designed primarily for the solution of overdetermined -PDE-systems, {\tt CRACK} can also be used to solve simple not -overdetermined ODE-systems. This solution -process is not completely algorithmic. Improved versions of {\tt CRACK} -could be used, without making any changes of {\tt QUASILINPDE} -necessary. - -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 {\tt QUASILINPDE}. - -\item % from enumerate -If the characteristic ODE-system (\ref{char1}), (\ref{char2}) has been -integrated completely and in full generality to the implicit solution -\begin{equation} -0 = G_i(\phi, w_j, c_k, \varepsilon),\;\; -i,k=1,\ldots,n+1,\;\;j=1,\ldots,n \label{charsol1} -\end{equation} -then according to the general theory for solving first order PDEs, -$\varepsilon$ has -to be eliminated from one of the equations and to be substituted in the -others to have left $n$ equations. -Also the constant that turns up additively to $\varepsilon$ -is to be set to zero. Both tasks are automatically -fulfilled, if, as described above, $\varepsilon$ is already eliminated -from the beginning by dividing all equations of (\ref{char1}), -(\ref{char2}) -through one of them. - -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. -\end{enumerate} -%--------------------------------------- -\subsection{Syntax} -The call of {\tt QUASILINPDE} is \\ -{\tt QUASILINPDE}({\it de}, {\it fun}, {\it varlist}); -\begin{itemize} -\item -{\it de} is the differential expression which vanishes due to the PDE -{\it de}$\; = 0$ or, {\it de} may be the differential equation itself in the -form $\;\;\ldots = \ldots\;\;$. -\item -{\it fun} is the unknown function. -\item -{\it varlist} is the list of variables of {\it fun}. -\end{itemize} -The result of {\tt QUASILINPDE} is a list of general solutions -\[ \{{\it sol}_1, {\it sol}_2, \ldots \}. \] -If {\tt QUASILINPDE} can not solve the PDE then it returns $\{\}$. -Each solution ${\it sol}_i$ is a list of expressions -\[ \{{\it ex}_1, {\it ex}_2, \ldots \} \] -such that the dependent function ($\phi$ in (\ref{PDE})) is determined -implicitly through an arbitrary function $F$ and the algebraic -equation \[ 0 = F({\it ex}_1, {\it ex}_2, \ldots). \] -%--------------------------------------- -\subsection{Examples} -{\em 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 -\small \begin{verbatim} -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); -\end{verbatim} \normalsize -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 $F.$ \\ -{\em Example 2:}\\ -For the linear inhomogeneous PDE -\[ 0 = y z,_x + x z,_y - 1, \;\;\;\;\mbox{for}\;\;\;\;z=z(x,y)\] -{\tt 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 $z$. \\ -{\em Example 3:}\\ -For the linear inhomogeneous PDE (3.8) from \cite{KamkePDE} -\[ 0 = x w,_x + (y+z)(w,_y - w,_z), \;\;\;\;\mbox{for}\;\;\;\;w=w(x,y,z)\] -{\tt 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. -%--------------------------------------- -\subsection{Limitations of {\tt QUASILINPDE}} -One restriction on the applicability of {\tt QUASILINPDE} results from -the program {\tt CRACK} which tries to solve the -characteristic ODE-system of the PDE. So far {\tt 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 {\tt 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 {\tt SOLVE}. -Though {\tt SOLVE} can be applied -to polynomial and transzendential equations, again no guarantee for -solvability can be given. -%------------------------------------------------------------------------- -\section{Transformation of DEs} -%--------------------------------------- -\subsection{The content of {\tt DETRAFO}} -Finally, after having found the finite transformations, -the program {\tt APPLYSYM} calls the procedure -{\tt DETRAFO} to perform the transformations. {\tt 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 {\tt DETRAFO} as standalone procedure. - -To run {\tt 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{eqnarray} -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{eqnarray} -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 {\it det}$\left(dx^j/dv^i\right) \neq 0$ because of the regularity -of the transformation which is checked by {\tt DETRAFO}. Non-regular -transformations are not performed. - -{\tt 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{eqnarray*} -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{eqnarray*} -%--------------------------------------- -\subsection{Syntax} -The call of {\tt DETRAFO} is -\begin{tabbing} -{\tt DETRAFO}(\=\{{\it ex}$_1$, {\it ex}$_2$, \ldots , {\it ex}$_m$\}, \\ - \>\{{\it ofun}$_1=${\it fex}$_1$, {\it ofun}$_2=${\it fex}$_2$, - \ldots ,{\it ofun}$_p=${\it fex}$_p$\}, \\ - \>\{{\it ovar}$_1=${\it vex}$_1$, {\it ovar}$_2=${\it vex}$_2$, \ldots , - {\it ovar}$_q=${\it vex}$_q$\}, \\ - \>\{{\it nfun}$_1$, {\it nfun}$_2$, \ldots , {\it nfun}$_p$\},\\ - \>\{{\it nvar}$_1$, {\it nvar}$_2$, \ldots , {\it nvar}$_q$\}); -\end{tabbing} -where $m,p,q$ are arbitrary. -\begin{itemize} -\item -The {\it ex}$_i$ are differential expressions to be transformed. -\item -The second list is the list of old functions {\it ofun} expressed -as expressions {\it fex} in terms -of new functions {\it nfun} and new independent variables {\it nvar}. -\item -Similarly the third list expresses the old independent variables {\it ovar} -as expressions {\it vex} in terms of new functions -{\it nfun} and new independent variables {\it nvar}. -\item -The last two lists include the new functions {\it nfun} -and new independent variables {\it nvar}. -\end{itemize} -Names for {\it ofun, ovar, nfun} and {\it nvar} can be arbitrarily -chosen. - -As the result {\tt DETRAFO} returns the first argument of its input, -i.e.\ the list -\[\{{\it ex}_1, {\it ex}_2, \ldots , {\it ex}_m\}\] -where all ${\it ex}_i$ are transformed. -%--------------------------------------- -\subsection{Limitations of {\tt DETRAFO}} -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. -%------------------------------------------------------------------------- -\section{Availability} -The programs run under {\tt REDUCE 3.4.1} or later versions and are available -by anonymous ftp from 138.37.80.15, directory {\tt ~ftp/pub/crack}. -%The manual file {\tt APPLYSYM.TEX} gives more details on the syntax. - -\begin{thebibliography}{99} -\bibitem{WHer} W.\,Hereman, Chapter 13 in vol 3 of the CRC Handbook of -Lie Group Analysis of Differential Equations, Ed.: N.H.\,Ibragimov, -CRC Press, Boca Raton, Florida (1995). -Systems described in this paper are among others: \\ -DELiA (Alexei Bocharov et.al.) Pascal \\ -DIFFGROB2 (Liz Mansfield) Maple \\ -DIMSYM (James Sherring and Geoff Prince) REDUCE \\ -HSYM (Vladimir Gerdt) Reduce \\ -LIE (V. Eliseev, R.N. Fedorova and V.V. Kornyak) Reduce \\ -LIE (Alan Head) muMath \\ -Lie (Gerd Baumann) Mathematica \\ -LIEDF/INFSYM (Peter Gragert and Paul Kersten) Reduce \\ -Liesymm (John Carminati, John Devitt and Greg Fee) Maple \\ -MathSym (Scott Herod) Mathematica \\ -NUSY (Clara Nucci) Reduce \\ -PDELIE (Peter Vafeades) Macsyma \\ -SPDE (Fritz Schwarz) Reduce and Axiom \\ -SYM\_DE (Stanly Steinberg) Macsyma \\ -Symmgroup.c (Dominique Berube and Marc de Montigny) Mathematica \\ -STANDARD FORM (Gregory Reid and Alan Wittkopf) Maple \\ -SYMCAL (Gregory Reid) Macsyma and Maple \\ -SYMMGRP.MAX (Benoit Champagne, Willy Hereman and Pavel Winternitz) Macsyma \\ -LIE package (Khai Vu) Maple \\ -Toolbox for symmetries (Mark Hickman) Maple \\ -Lie symmetries (Jeffrey Ondich and Nick Coult) Mathematica. - -\bibitem{lie1} S.\,Lie, Sophus Lie's 1880 Transformation Group Paper, -Translated by M.\,Ackerman, comments by R.\,Hermann, Mathematical Sciences -Press, Brookline, (1975). - -\bibitem{lie2} S.\,Lie, Differentialgleichungen, Chelsea Publishing Company, -New York, (1967). - -\bibitem{LIEPDE} T.\,Wolf, An efficiency improved program {\tt LIEPDE} -for determining Lie - symmetries of PDEs, Proceedings of the workshop on -Modern group theory methods in Acireale (Sicily) Nov.\,(1992) - -\bibitem{Riq} C.\,Riquier, Les syst\`{e}mes d'\'{e}quations -aux d\'{e}riv\'{e}es partielles, Gauthier--Villars, Paris (1910). - -\bibitem{Th} J.\,Thomas, Differential Systems, AMS, Colloquium -publications, v.\,21, N.Y.\,(1937). - -\bibitem{Ja} M.\,Janet, Le\c{c}ons sur les syst\`{e}mes d'\'{e}quations aux -d\'{e}riv\'{e}es, Gauthier--Villars, Paris (1929). - -\bibitem{Topu} V.L.\,Topunov, Reducing Systems of Linear Differential -Equations to a Passive Form, Acta Appl.\,Math.\,16 (1989) 191--206. - -\bibitem{Alex} A.V.\,Bocharov and M.L.\,Bronstein, Efficiently -Implementing Two Methods of the Geometrical Theory of Differential -Equations: An Experience in Algorithm and Software Design, Acta.\,Appl. -Math.\,16 (1989) 143--166. - -\bibitem{Olv} P.J. Olver, Applications of Lie Groups to Differential -Equations, Springer-Verlag New York (1986). - -\bibitem{Reid1} G.J.\,Reid, A triangularization algorithm which -determines the Lie symmetry algebra of any system of PDEs, J.Phys.\,A: -Math.\,Gen.\,23 (1990) L853-L859. - -\bibitem{FS} F.\,Schwarz, Automatically Determining Symmetries of Partial -Differential Equations, Computing 34, (1985) 91-106. - -\bibitem{Fush} W.I.\,Fushchich and V.V.\,Kornyak, Computer Algebra -Application for Determining Lie and Lie--B\"{a}cklund Symmetries of -Differential Equations, J.\,Symb.\,Comp.\,7 (1989) 611--619. - -\bibitem{Ka} E.\,Kamke, Differentialgleichungen, L\"{o}sungsmethoden -und L\"{o}sungen, Band 1, Gew\"{o}hnliche Differentialgleichungen, -Chelsea Publishing Company, New York, 1959. - -\bibitem{KamkePDE} E.\,Kamke, Differentialgleichungen, L\"{o}sungsmethoden -und L\"{o}sungen, Band 2, Partielle Differentialgleichungen, 6.Aufl., -Teubner, Stuttgart:Teubner, 1979. - -\bibitem{Wo} T.\,Wolf, An Analytic Algorithm for Decoupling and Integrating -systems of Nonlinear Partial Differential Equations, J.\,Comp.\,Phys., -no.\,3, 60 (1985) 437-446 and, Zur analytischen Untersuchung und exakten -L\"{o}sung von Differentialgleichungen mit Computeralgebrasystemen, -Dissertation B, Jena (1989). - -\bibitem{WoBra} T.\,Wolf, A. Brand, The Computer Algebra Package {\tt CRACK} - for Investigating PDEs, Manual for the package {\tt CRACK} in the REDUCE - network library and in Proceedings of ERCIM School on Partial - Differential Equations and Group Theory, April 1992 in Bonn, GMD Bonn. - -\bibitem{WM} M.A.H.\,MacCallum, F.J.\,Wright, Algebraic Computing with REDUCE, -Clarendon Press, Oxford (1991). - -\bibitem{Mal} M.A.H.\,MacCallum, An Ordinary Differential Equation -Solver for REDUCE, Proc.\,ISAAC'88, Springer Lect.\,Notes in Comp Sci. -358, 196--205. - -\bibitem{Step} H.\,Stephani, Differential equations, Their solution using -symmetries, Cambridge University Press (1989). - -\bibitem{Karp} V.I.\,Karpman, Phys.\,Lett.\,A 136, 216 (1989) - -\bibitem{Cham} B.\,Champagne, W.\,Hereman and P.\,Winternitz, The computer - calculation of Lie point symmetries of large systems of differential - equations, Comp.\,Phys.\,Comm.\,66, 319-340 (1991) - -\bibitem{Markus} M.\,Kubitza, private communication - -\end{thebibliography} - -\end{document} - - +% This is a LaTeX file +\documentstyle[12pt]{article} + +%Sets size of page and margins +\oddsidemargin 10mm \evensidemargin 10mm +\topmargin 0pt \headheight 0pt \headsep 0pt +\footheight 14pt \footskip 40pt +\textheight 23cm \textwidth 15cm +%\textheight 15cm \textwidth 10cm + +%spaces lines at one and a half spacing +%\def\baselinestretch{1.5} + +%\parskip = \baselineskip + +%Defines capital R for the reals, ... +%\font\Edth=msym10 +%\def\Integer{\hbox{\Edth Z}} +%\def\Rational{\hbox{\Edth Q}} +%\def\Real{\hbox{\Edth R}} +%\def\Complex{\hbox{\Edth C}} + +\title{Programs for Applying Symmetries of PDEs} +\author{Thomas Wolf \\ + School of Mathematical Sciences \\ + Queen Mary and Westfield College \\ + University of London \\ + London E1 4NS \\ + T.Wolf@maths.qmw.ac.uk +} + +\begin{document} +\maketitle +\begin{abstract} +In this paper the programs {\tt APPLYSYM}, {\tt QUASILINPDE} and +{\tt DETRAFO} are described which aim at the utilization +of infinitesimal symmetries of differential equations. The purpose +of {\tt QUASILINPDE} is the general solution of +quasilinear PDEs. This procedure is used by {\tt APPLYSYM} +for the application of point symmetries for either +\begin{itemize} +\item 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 +\item generalizing given special solutions of ODEs/PDEs with new constant +parameters. +\end{itemize} + +The program {\tt DETRAFO} performs arbitrary point- and contact +transformations of ODEs/PDEs and is applied if similarity +and symmetry variables have been found. +The program {\tt APPLYSYM} is used in connection with the program +{\tt LIEPDE} for formulating and solving the conditions for point- and +contact symmetries which is described in \cite{LIEPDE}. +The actual problem solving is done in all these programs through a call +to the package {\tt CRACK} for solving overdetermined PDE-systems. +\end{abstract} + +\tableofcontents +%------------------------------------------------------------------------- +\section{Introduction and overview of the symmetry method} +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 +\cite{WHer} 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 +\cite{lie1},\cite{lie2} 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 {\tt LIEPDE} +(for a comparison with another program see \cite{LIEPDE}). + +The novelty described in this paper are programs aiming at +the final third step: Applying symmetries for +\begin{itemize} +\item 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 +\item generalizing given special solutions of ODEs/PDEs with new constant +parameters. +\end{itemize} +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. +%--------------------------------------- +\subsection{The first step: Formulating the symmetry conditions} + +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. \cite{Olv}, \cite{Step}) +\begin{eqnarray} +\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 \\ \vspace{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{eqnarray} +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{eqnarray} +X H_A & = & 0 \;\; \; \; \mbox{mod} \; \; \; 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{eqnarray} +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 {\tt APPLYSYM} +(see below). For literature about generalized symmetries see \cite{WHer}. + +Though the formulation of the symmetry conditions (\ref{sbed1}), +(\ref{sbed2}), (\ref{recur}) +is straightforward and handled in principle by all related +programs \cite{WHer}, 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 \cite{Cham}. Two methods to partition the +conditions for a single PDE are described by Bocharov/Bronstein +\cite{Alex} and Stephani \cite{Step}. + +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 {\tt 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. +%--------------------------------------- +\subsection{The second step: Solving the symmetry conditions} +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 \cite{LIEPDE}. +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 {\tt LIEPDE} which formulates the symmetry conditions calls the +program {\tt 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, {\tt 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 \cite{LIEPDE} or \cite{WoBra}. + + +The call of {\tt LIEPDE} is \\ +{\tt LIEPDE}(\{{\it de}, {\it fun}, {\it var}\}, +\{{\it od}, {\it lp}, {\it fl}\}); \\ +where +\begin{itemize} +\item {\it de} is a single DE or a list of DEs in the form of a vanishing + expression or in the form $\ldots=\ldots\;\;$. +\item {\it fun} is the single function or the list of functions occuring + in {\it de}. +\item {\it var} is the single variable or the list of variables in {\it de}. +\item {\it od} is the order of the ansatz for $\xi, \eta.$ It is = 0 for +point symmetries and = 1 for contact symmetries (accepted by +{\tt LIEPDE} only in case of one ODE/PDE for one unknown function). +% and $>1$ for dynamical symmetries +%(only in case of one ODE for one unknown function) +\item If {\it lp} is $nil$ then the standard ansatz for $\xi^i, \eta^\alpha$ +is taken which is + \begin{itemize} + \item for point symmetries ({\it od} =0) is $\xi^i = \xi^i(x^j,y^\beta), + \eta^\alpha = \eta^\alpha(x^j,y^\beta)$ + \item for contact symmetries ({\it od} =1) is + $ \xi^i := \Omega_{u_i}, \;\;\; + \eta := u_i\Omega_{u_i} \; - \; \Omega, $ \\ + $\Omega:=\Omega(x^i, u, u_j)$ +%\item for dynamical symmetries ({\it od}$>1$) \\ +% $ \xi := \Omega,_{u'}, \;\;\; +% \eta := u'\Omega,_{u'} \; - \; \Omega, \;\;\; +% \Omega:=\Omega(x, u, u',\ldots, y^{({\it od}-1)})$ +% where {\it od} must be less than the order of the ODE. + \end{itemize} + + If {\it lp} is not $nil$ then {\it lp} is the ansatz for + $\xi^i, \eta^\alpha$ and must have the form + \begin{itemize} + \item for point symmetries + {\tt \{xi\_\mbox{$x1$} = ..., ..., eta\_\mbox{$u1$} = ..., ...\}} + where {\tt xi\_, eta\_ } + are fixed and $x1, \ldots, u1$ are to be replaced by the actual names + of the variables and functions. + \item otherwise {\tt spot\_ = ...} where the expression on the right hand + side is the ansatz for the Symmetry-POTential $\Omega.$ + \end{itemize} + +\item {\it fl} is the list of free functions in the ansatz +in case {\it lp} is not $nil.$ +\end{itemize} + + +The result of {\tt LIEPDE} is a list with 3 elements, each of which +is a list: +\[ \{\{{\it con}_1,{\it con}_2,\ldots\}, + \{{\tt xi}\__{\ldots}=\ldots, \ldots, + {\tt eta}\__{\ldots}=\ldots, \ldots\}, + \{{\it flist}\}\}. \] +The first list contains remaining unsolved symmetry conditions {\it 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. + +%That the automatic calculation of symmetries run in most practical cases +%is shown with the following example. It is insofar difficult, as many +%symmetries exist and the solution consequently more difficult is to deriv. +% +%--------------------------------------- +%\subsection{Example} +%For the following PDE-system, which takes its simplest form in the +%formalism of exterior forms: +% +%\begin{eqnarray*} +%0 & = & 3k_t,_{tt}-2k_t,_{xx}-2k_t,_{yy}-2k_t,_{zz}-k_x,_{tx}-2k_zk_x,_y \\ +% & & +2k_yk_x,_z-k_y,_{ty}+2k_zk_y,_x-2k_xk_y,_z-k_z,_{tz}-2k_yk_z,_x+2k_xk_z,_y \\ +%0 & = & k_t,_{tx}-2k_zk_t,_y+2k_yk_t,_z+2k_x,_{tt}-3k_x,_{xx}-2k_x,_{yy} \\ +% & & -2k_x,_{zz}+2k_zk_y,_t-k_y,_{xy}-2k_tk_y,_z-2k_yk_z,_t-k_z,_{xz}+2k_tk_z,_y \\ +%0 & = & k_t,_{ty}+2k_zk_t,_x-2k_xk_t,_z-2k_zk_x,_t-k_x,_{xy}+2k_tk_x,_z \\ +% & & +2k_y,_{tt}-2k_y,_{xx}-3k_y,_{yy}-2k_y,_{zz}+2k_xk_z,_t-2k_tk_z,_x-k_z,_{yz} \\ +%0 & = & k_t,_{tz}-2k_yk_t,_x+2k_xk_t,_y+2k_yk_x,_t-k_x,_{xz}-2k_tk_x,_y \\ +% & & -2k_xk_y,_t+2k_tk_y,_x-k_y,_{yz}+2k_z,_{tt}-2k_z,_{xx}-2k_z,_{yy}-3k_z,_{zz} +%\end{eqnarray*} +%--------------------------------------- +\subsection{The third step: Application of infinitesimal symmetries} +If infinitesimal symmetries have been found then +the program {\tt APPLYSYM} can use them for the following purposes: +\begin{enumerate} +\item Calculation of one symmetry variable and further similarity variables. +After transforming +the DE(-system) to these variables, the symmetry variable will not occur +explicitly any more. For ODEs this has the consequence that their order has +effectively been reduced. +\item Generalization of a special solution by one or more constants of +integration. +\end{enumerate} +Both methods are described in the following section. +%------------------------------------------------------------------------- +\section{Applying symmetries with {\tt APPLYSYM}} +%--------------------------------------- +\subsection{The first mode: Calculation of similarity and symmetry variables} +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{eqnarray*} + 0 & = & X H_A(x^i,y^\alpha,y^\beta_j,\ldots)\;\;\; \mbox{mod} \;\;\; H_A=0 \\ + & = & X h_A(v^i,u^\alpha,u^\beta_j,\ldots)\;\;\; \mbox{mod} \;\;\; h_A=0 \\ + & = & \partial_{v^1}h_A(v^i,u^\alpha,u^\beta_j,\ldots)\;\;\; \mbox{mod} + \;\;\; h_A=0. +\end{eqnarray*} +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{eqnarray} + 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{eqnarray} +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 {\tt APPLYSYM}: +\begin{itemize} +\item 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. +\item Through a call of the procedure {\tt QUASILINPDE} (described in a later +section) the two linear first order PDEs (\ref{ql1}), (\ref{ql2}) are +investigated and, if possible, solved. +\item 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. +\item 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)). +\item Through a call of the procedure {\tt DETRAFO} the transformation +$x^i,y^\alpha \rightarrow v^j,u^\beta$ of the DE(s) $H_A=0$ is finally done. +\item The program returns to the starting menu. +\end{itemize} +%--------------------------------------- +\subsection{The second mode: Generalization of special solutions} +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 $x^i=x^i(u^\beta,v^j),\; +y^\alpha=y^\alpha(u^\beta,v^j)$ \underline{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 {\tt APPLYSYM} show features of both techniques: +\begin{itemize} +\item 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. +\item The special solution to be generalized and the name of the new +constant have to be put in. +\item Through a call of the procedure {\tt QUASILINPDE}, the PDE (\ref{ql1}) +is solved which amounts to a solution of its characteristic ODE system +(\ref{ODEsys}) where $v^1=\varepsilon$. +\item {\tt 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 +\[ 0 = F(x^j, y^\alpha) + = F( x^j(c_i(x^k, y^\beta, 0), \varepsilon)), + y^\alpha(c_i(x^k, y^\beta, 0), \varepsilon))). \] +\item The new solution is availabe for further generalizations w.r.t.\ other +symmetries. +\end{itemize} +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. +%--------------------------------------- +\subsection{Syntax} +The call of {\tt APPLYSYM} is +{\tt APPLYSYM}(\{{\it de}, {\it fun}, {\it var}\}, \{{\it sym}, {\it cons}\}); +\begin{itemize} +\item {\it de} is a single DE or a list of DEs in the form of a vanishing + expression or in the form $\ldots=\ldots\;\;$. +\item {\it fun} is the single function or the list of functions occuring + in {\it de}. +\item {\it var} is the single variable or the list of variables in {\it de}. +\item {\it 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\_\ldots=\ldots,\ldots,eta\_\ldots=\ldots,\ldots\}, + where the indices after `xi\_' are the variable names and after `eta\_' + the function names. +\item {\it cons} is the list of constants in {\it sym}, one constant for each + symmetry. +\end{itemize} +The list that is the first argument of {\tt APPLYSYM} is the same as the +first argument of {\tt LIEPDE} and the +second argument is the list that {\tt LIEPDE} returns without its first +element (the unsolved conditions). An example is given below. + +What {\tt APPLYSYM} returns depends on the last performed modus. +After modus 1 the return is \\ +\{\{{\it newde}, {\it newfun}, {\it newvar}\}, {\it trafo}\} \\ +where +\begin{itemize} +\item {\it newde} lists the transformed equation(s) +\item {\it newfun} lists the new function name(s) +\item {\it newvar} lists the new variable name(s) +\item {\it trafo} lists the transformations $x^i=x^i(v^j,u^\beta), + y^\alpha=y^\alpha(v^j,u^\beta)$ +\end{itemize} +After modus 2, {\tt APPLYSYM} returns the generalized special solution. +%--------------------------------------- +\subsection{Example: A second order ODE} +Weyl's class of solutions of Einsteins field equations consists of +axialsymmetric time independent metrics of the form +\begin{equation} +{\rm{d}} s^2 = e^{-2 U} \left[ e^{2 k} \left( \rm{d} \rho^2 + \rm{d} +z^2 \right)+\rho^2 \rm{d} \varphi^2 \right] - e^{2 U} \rm{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 \cite{Markus}. 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 {\tt LIEPDE} through +\small \begin{verbatim} +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,{0,nil,nil}); +end; \end{verbatim} \normalsize +gives \small \begin{verbatim} + 3 2 +sym := {{}, {xi_r= - c10*r - c11*r, eta_h=c10*h*r }, {c10,c11}}. +\end{verbatim} \normalsize +All conditions have been solved because the first element of {\tt 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 {\tt APPLYSYM} through +\small \begin{verbatim} +newde:=applysym(de,rest sym); +\end{verbatim} \normalsize +The interactive session is given below with the user input following +the prompt `{\tt Input:3:}' or following `?'. (Empty lines have been deleted.) +\small \begin{verbatim} +Do you want to find similarity and symmetry variables (enter `1;') +or generalize a special solution with new parameters (enter `2;') +or exit the program (enter `;') +Input:3: 1; +\end{verbatim} \normalsize +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. +\small \begin{verbatim} +---------------------- 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. +sy_(1); +\end{verbatim} \normalsize +We could have entered `sy\_(2);' or a combination of both +as well with the calculation running then +differently. +\small \begin{verbatim} +The symmetry to be applied in the following is + 3 2 +{xi_r= - r ,eta_h=h*r } +Enter the name of the new dependent variables: +Input:3: u; +Enter the name of the new independent variables: +Input:3: v; +\end{verbatim} \normalsize +This was the input part, now the real calculation starts. +\small \begin{verbatim} +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). +\end{verbatim} \normalsize +The PDE is equation (\ref{ql2}). +\small \begin{verbatim} +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 +\end{verbatim} \normalsize +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 {\tt ff}. (This function is called {\tt ff} to prevent a clash with +names of user variables/functions.) In principle any choice of {\tt 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 {\tt ff} has $p+q$ arguments. Because of the +condition $0 = ${\tt 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 {\tt 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 +{\tt ff(..)} is known, the user can interfere here to enter a simpler solution, +if possible. +\small \begin{verbatim} +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 +\end{verbatim} \normalsize +Similar to above. +\small \begin{verbatim} +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 +\end{verbatim} \normalsize +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 $v$: +\small \begin{verbatim} +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};' +Input:3: {u=v,v=u}; + +The transformed equation which should be free of u: + 3 6 2 3 +0=3*df(u,v,2)*v - 16*df(u,v) *v - 20*df(u,v) *v + 5*df(u,v) +Do you want to find similarity and symmetry variables (enter `1;') +or generalize a special solution with new parameters (enter `2;') +or exit the program (enter `;') +Input:3: ; +\end{verbatim} +We stop here. The following is returned from our {\tt APPLYSYM} call: +\small \begin{verbatim} + 3 6 2 3 +{{{3*df(u,v,2)*v - 16*df(u,v) *v - 20*df(u,v) *v + 5*df(u,v)}, + {u}, + {v}}, + sqrt(u)*sqrt(2) + {r=-----------------, h=sqrt(u)*sqrt(2)*v }} + 2*u +\end{verbatim} \normalsize +The use of {\tt 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{eqnarray} +\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{eqnarray} +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}). +%--------------------------------------- +\subsection{Limitations of {\tt APPLYSYM}} +Restrictions of the applicability of the program {\tt APPLYSYM} result +from limitations of the program {\tt 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 {\tt CRACK} and algebraic equations with {\tt SOLVE} +the program {\tt APPLYSYM} itself is free of restrictions, +i.e.\ if once new versions of {\tt CRACK, SOLVE} +would be available then {\tt 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. +%------------------------------------------------------------------------- +\section{Solving quasilinear PDEs} +%--------------------------------------- +\subsection{The content of {\tt QUASILINPDE}} +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 {\tt 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 +$\phi=\phi(w_i)$. +\begin{enumerate} +\item +The first step in solving a quasilinear PDE (\ref{PDE}) +is the formulation of the corresponding characteristic ODE-system +\begin{eqnarray} +\frac{dw_i}{d\varepsilon} & = & a_i(w_j,\phi) \label{char1} \\ +\frac{d\phi}{d\varepsilon} & = & b(w_j,\phi) \label{char2} +\end{eqnarray} +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{eqnarray} +\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{eqnarray} +with the independent variable $w_k$ instead of $\varepsilon$. +If instead we divide through equation +(\ref{char2}) then the remaining system would be +\begin{eqnarray} +\frac{dw_i}{d\phi} & = & \frac{a_i}{b} , \;\;\;i=1,2,\ldots n + \label{char3a} +\end{eqnarray} +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 +\begin{itemize} +\item is constant or +\item depends only on one variable or +\item is a product of factors, each of which depends only on +one variable. +\end{itemize} + +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. + +\item % from enumerate +To solve the system (\ref{char3}), (\ref{char4}) or (\ref{char3a}), +the procedure {\tt CRACK} is called. +Although being designed primarily for the solution of overdetermined +PDE-systems, {\tt CRACK} can also be used to solve simple not +overdetermined ODE-systems. This solution +process is not completely algorithmic. Improved versions of {\tt CRACK} +could be used, without making any changes of {\tt QUASILINPDE} +necessary. + +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 {\tt QUASILINPDE}. + +\item % from enumerate +If the characteristic ODE-system (\ref{char1}), (\ref{char2}) has been +integrated completely and in full generality to the implicit solution +\begin{equation} +0 = G_i(\phi, w_j, c_k, \varepsilon),\;\; +i,k=1,\ldots,n+1,\;\;j=1,\ldots,n \label{charsol1} +\end{equation} +then according to the general theory for solving first order PDEs, +$\varepsilon$ has +to be eliminated from one of the equations and to be substituted in the +others to have left $n$ equations. +Also the constant that turns up additively to $\varepsilon$ +is to be set to zero. Both tasks are automatically +fulfilled, if, as described above, $\varepsilon$ is already eliminated +from the beginning by dividing all equations of (\ref{char1}), +(\ref{char2}) +through one of them. + +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. +\end{enumerate} +%--------------------------------------- +\subsection{Syntax} +The call of {\tt QUASILINPDE} is \\ +{\tt QUASILINPDE}({\it de}, {\it fun}, {\it varlist}); +\begin{itemize} +\item +{\it de} is the differential expression which vanishes due to the PDE +{\it de}$\; = 0$ or, {\it de} may be the differential equation itself in the +form $\;\;\ldots = \ldots\;\;$. +\item +{\it fun} is the unknown function. +\item +{\it varlist} is the list of variables of {\it fun}. +\end{itemize} +The result of {\tt QUASILINPDE} is a list of general solutions +\[ \{{\it sol}_1, {\it sol}_2, \ldots \}. \] +If {\tt QUASILINPDE} can not solve the PDE then it returns $\{\}$. +Each solution ${\it sol}_i$ is a list of expressions +\[ \{{\it ex}_1, {\it ex}_2, \ldots \} \] +such that the dependent function ($\phi$ in (\ref{PDE})) is determined +implicitly through an arbitrary function $F$ and the algebraic +equation \[ 0 = F({\it ex}_1, {\it ex}_2, \ldots). \] +%--------------------------------------- +\subsection{Examples} +{\em 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 +\small \begin{verbatim} +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); +\end{verbatim} \normalsize +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 $F.$ \\ +{\em Example 2:}\\ +For the linear inhomogeneous PDE +\[ 0 = y z,_x + x z,_y - 1, \;\;\;\;\mbox{for}\;\;\;\;z=z(x,y)\] +{\tt 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 $z$. \\ +{\em Example 3:}\\ +For the linear inhomogeneous PDE (3.8) from \cite{KamkePDE} +\[ 0 = x w,_x + (y+z)(w,_y - w,_z), \;\;\;\;\mbox{for}\;\;\;\;w=w(x,y,z)\] +{\tt 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. +%--------------------------------------- +\subsection{Limitations of {\tt QUASILINPDE}} +One restriction on the applicability of {\tt QUASILINPDE} results from +the program {\tt CRACK} which tries to solve the +characteristic ODE-system of the PDE. So far {\tt 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 {\tt 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 {\tt SOLVE}. +Though {\tt SOLVE} can be applied +to polynomial and transzendential equations, again no guarantee for +solvability can be given. +%------------------------------------------------------------------------- +\section{Transformation of DEs} +%--------------------------------------- +\subsection{The content of {\tt DETRAFO}} +Finally, after having found the finite transformations, +the program {\tt APPLYSYM} calls the procedure +{\tt DETRAFO} to perform the transformations. {\tt 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 {\tt DETRAFO} as standalone procedure. + +To run {\tt 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{eqnarray} +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{eqnarray} +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 {\it det}$\left(dx^j/dv^i\right) \neq 0$ because of the regularity +of the transformation which is checked by {\tt DETRAFO}. Non-regular +transformations are not performed. + +{\tt 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{eqnarray*} +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{eqnarray*} +%--------------------------------------- +\subsection{Syntax} +The call of {\tt DETRAFO} is +\begin{tabbing} +{\tt DETRAFO}(\=\{{\it ex}$_1$, {\it ex}$_2$, \ldots , {\it ex}$_m$\}, \\ + \>\{{\it ofun}$_1=${\it fex}$_1$, {\it ofun}$_2=${\it fex}$_2$, + \ldots ,{\it ofun}$_p=${\it fex}$_p$\}, \\ + \>\{{\it ovar}$_1=${\it vex}$_1$, {\it ovar}$_2=${\it vex}$_2$, \ldots , + {\it ovar}$_q=${\it vex}$_q$\}, \\ + \>\{{\it nfun}$_1$, {\it nfun}$_2$, \ldots , {\it nfun}$_p$\},\\ + \>\{{\it nvar}$_1$, {\it nvar}$_2$, \ldots , {\it nvar}$_q$\}); +\end{tabbing} +where $m,p,q$ are arbitrary. +\begin{itemize} +\item +The {\it ex}$_i$ are differential expressions to be transformed. +\item +The second list is the list of old functions {\it ofun} expressed +as expressions {\it fex} in terms +of new functions {\it nfun} and new independent variables {\it nvar}. +\item +Similarly the third list expresses the old independent variables {\it ovar} +as expressions {\it vex} in terms of new functions +{\it nfun} and new independent variables {\it nvar}. +\item +The last two lists include the new functions {\it nfun} +and new independent variables {\it nvar}. +\end{itemize} +Names for {\it ofun, ovar, nfun} and {\it nvar} can be arbitrarily +chosen. + +As the result {\tt DETRAFO} returns the first argument of its input, +i.e.\ the list +\[\{{\it ex}_1, {\it ex}_2, \ldots , {\it ex}_m\}\] +where all ${\it ex}_i$ are transformed. +%--------------------------------------- +\subsection{Limitations of {\tt DETRAFO}} +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. +%------------------------------------------------------------------------- +\section{Availability} +The programs run under {\tt REDUCE 3.4.1} or later versions and are available +by anonymous ftp from 138.37.80.15, directory {\tt ~ftp/pub/crack}. +%The manual file {\tt APPLYSYM.TEX} gives more details on the syntax. + +\begin{thebibliography}{99} +\bibitem{WHer} W.\,Hereman, Chapter 13 in vol 3 of the CRC Handbook of +Lie Group Analysis of Differential Equations, Ed.: N.H.\,Ibragimov, +CRC Press, Boca Raton, Florida (1995). +Systems described in this paper are among others: \\ +DELiA (Alexei Bocharov et.al.) Pascal \\ +DIFFGROB2 (Liz Mansfield) Maple \\ +DIMSYM (James Sherring and Geoff Prince) REDUCE \\ +HSYM (Vladimir Gerdt) Reduce \\ +LIE (V. Eliseev, R.N. Fedorova and V.V. Kornyak) Reduce \\ +LIE (Alan Head) muMath \\ +Lie (Gerd Baumann) Mathematica \\ +LIEDF/INFSYM (Peter Gragert and Paul Kersten) Reduce \\ +Liesymm (John Carminati, John Devitt and Greg Fee) Maple \\ +MathSym (Scott Herod) Mathematica \\ +NUSY (Clara Nucci) Reduce \\ +PDELIE (Peter Vafeades) Macsyma \\ +SPDE (Fritz Schwarz) Reduce and Axiom \\ +SYM\_DE (Stanly Steinberg) Macsyma \\ +Symmgroup.c (Dominique Berube and Marc de Montigny) Mathematica \\ +STANDARD FORM (Gregory Reid and Alan Wittkopf) Maple \\ +SYMCAL (Gregory Reid) Macsyma and Maple \\ +SYMMGRP.MAX (Benoit Champagne, Willy Hereman and Pavel Winternitz) Macsyma \\ +LIE package (Khai Vu) Maple \\ +Toolbox for symmetries (Mark Hickman) Maple \\ +Lie symmetries (Jeffrey Ondich and Nick Coult) Mathematica. + +\bibitem{lie1} S.\,Lie, Sophus Lie's 1880 Transformation Group Paper, +Translated by M.\,Ackerman, comments by R.\,Hermann, Mathematical Sciences +Press, Brookline, (1975). + +\bibitem{lie2} S.\,Lie, Differentialgleichungen, Chelsea Publishing Company, +New York, (1967). + +\bibitem{LIEPDE} T.\,Wolf, An efficiency improved program {\tt LIEPDE} +for determining Lie - symmetries of PDEs, Proceedings of the workshop on +Modern group theory methods in Acireale (Sicily) Nov.\,(1992) + +\bibitem{Riq} C.\,Riquier, Les syst\`{e}mes d'\'{e}quations +aux d\'{e}riv\'{e}es partielles, Gauthier--Villars, Paris (1910). + +\bibitem{Th} J.\,Thomas, Differential Systems, AMS, Colloquium +publications, v.\,21, N.Y.\,(1937). + +\bibitem{Ja} M.\,Janet, Le\c{c}ons sur les syst\`{e}mes d'\'{e}quations aux +d\'{e}riv\'{e}es, Gauthier--Villars, Paris (1929). + +\bibitem{Topu} V.L.\,Topunov, Reducing Systems of Linear Differential +Equations to a Passive Form, Acta Appl.\,Math.\,16 (1989) 191--206. + +\bibitem{Alex} A.V.\,Bocharov and M.L.\,Bronstein, Efficiently +Implementing Two Methods of the Geometrical Theory of Differential +Equations: An Experience in Algorithm and Software Design, Acta.\,Appl. +Math.\,16 (1989) 143--166. + +\bibitem{Olv} P.J. 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