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\title{Manual for LIEPDE}
\author{Thomas Wolf \\
Department of Mathematics \\
Brock University \\
St.Catharines \\
Ontario, Canada L2S 3A1 \\
twolf@brocku.ca}
\begin{document}
\maketitle
\section{Purpose}
The procedure {\tt LIEPDE} computes infinitesimal symmetries
for a given single/system of differential equation(s) (ODEs or PDEs)
%\begin{equation}
% u^{\alpha}_J = w^{\alpha}(x,u^{\beta},...,u^{\beta}_K,...) \label{a1}
%\end{equation}
\begin{equation}
H_A = 0. \label{PDEs}
\end{equation}
To obey symmetries, differential equations (\ref{PDEs})
for unknown functions $u^\alpha$ of independent variables $x^i$
must be form-invariant under infinitesimal transformations
\begin{equation}
\tilde{x}^i = x^i + \varepsilon \xi^i, \;\; \;\;\;
\tilde{u}^\alpha = u^\alpha + \varepsilon \eta^\alpha \label{tran}
\end{equation}
of first order in $\varepsilon.$ To transform the equations (\ref{PDEs})
by (\ref{tran}), derivatives of $u^\alpha$ 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{u}^\alpha_{j_1\ldots j_k} & = &
u^\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} -
u^\alpha_{ij_1\ldots j_{k-1}}\frac{D \xi^i}{D x^k} \label{recur}
\end{eqnarray}
and the complete symmetry condition then takes the form
\begin{eqnarray}
X H_A & = & 0 \;\; \;\pmod{ H_A = 0} \label{sbed1} \\
X & = & \xi^i \frac{\partial}{\partial x^i} +
\eta^\alpha \frac{\partial}{\partial u^\alpha} +
\eta^\alpha_m \frac{\partial}{\partial u^\alpha_m} +
\eta^\alpha_{mn} \frac{\partial}{\partial u^\alpha_{mn}} + \ldots +
\eta^\alpha_{mn\ldots p} \frac{\partial}{\partial u^\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 $u^\alpha$ to reduce the number of independent
variables, because the symmetry condition (\ref{sbed1}) must be
fulfilled identically in $x^i, u^\alpha$ and all partial
derivatives of $u^\alpha.$
For point symmetries $\xi^i, \eta^\alpha$ are functions of $x^j,
u^\beta$ only. For more general higher order symmetries $\xi^i, \eta^\alpha$
may depend on derivatives of $u^\beta$. For those symmetries one can
without loss of generality set $\xi^i=0$ due to a symmetry of the symmetry
conditions on the manifold of solutions of $H_A=0$
themselves (e.g.\ $\S$5.1 in \cite{Olv}). The shifted generators
\[\tilde{\xi^i} = \xi^i + h^i, \; \; \; \;
\tilde{\eta^{\alpha}} = \eta^{\alpha} + h^i u^{\alpha},_i\]
with arbitrary $h^i=h^i(x^j, u^{\beta},\ldots, u^{\beta}_K)$
represent generators of the same symmetry.
\section{Syntax of {\tt LIEPDE}}
The procedure {\tt LIEPDE} is called through \\
{\tt LIEPDE({\it problem,symtype,flist,inequ}); } \\
All parameters are lists. \vspace{6pt} \\
The first parameter specifies the DEs to be investigated: \\
{\it problem} has the form \{{\it equations, ulist, xlist}\} where
\begin{tabbing}
\hspace{0.5cm}
{\it equations } \= is a list of equations,
each has the form {\tt df(ui,..)=...} where \\
\> the LHS (left hand side) {\tt df(ui,..)} is selected such that \\
\> - The RHS (right h.s.) of an equations must not include \\
\>$\;\,$ the derivative on the LHS nor a derivative of it. \\
\> - Neither the LHS nor any derivative of it of any equation \\
\>$\;\,$ may occur in any other equation.\\
\> - Each of the unknown functions occurs on the LHS of \\
\>$\;\,$ exactly one equation. \\
\hspace{0.5cm}
{\it ulist} \> is a list of function names, which can be chosen freely \\
\hspace{0.5cm}
{\it xlist} \> is a list of variable names, which can be chosen freely
\end{tabbing}
Equations can be given as a list of single differential expressions and then
the program will try to bring them into the `solved form' {\tt df(ui,..)=...}
automatically. If equations are given in the solved form then the above
conditions are checked and execution is stopped it they are not satisfied.
An easy way to get the equations in the desired form is to use \\
\verb+ FIRST SOLVE({+{\it eq1,eq2,}...\verb+},{+{\it one highest
derivative for each function u}\verb+})+ \\
(see the example of the Karpman equations in {\tt LIEPDE.TST}).
The example of the Burgers equation in {\tt LIEPDE.TST} demonstrates
that the number of symmetries for a given maximal order of the infinitesimal
generators depends on the derivative chosen for the LHS.
The second parameter {\it symtype} of {\tt LIEPDE} is a list $\{\;\}$ that
specifies the symmetry to be calculated. {\it symtype} can have the following
values and meanings:
\begin{tabbing}
\verb+{"point"} + \= Point symmetries with $\xi^i=\xi^i(x^j,u^{\beta}),\;
\eta^{\alpha}=\eta^{\alpha}(x^j,u^{\beta})$ are \\
\> determined.\\
\verb+{"contact"}+ \> Contact symmetries with $\xi^i=0, \;
\eta=\eta(x^j,u,u_k)$ are \\
\> determined $(u_k = \partial u/\partial x^k)$, which is only applicable if a \\
\> single equation (\ref{PDEs}) with an order $>1$ for a
single function \\
\> $u$ is to be investigated. (The {\it symtype}
\verb+{"contact"}+ \\
\> is equivalent to \verb+{"general",1}+ (see below) apart from \\
\> the additional checks done for \verb+{"contact"}+.)\\
\verb+{"general"+,{\it order}\verb+}+ \> where {\it order} is an integer $>0$.
Generalized symmetries $\xi^i=0,$ \\
\> $\eta^{\alpha}=\eta^{\alpha}(x^j,u^{\beta},\ldots,u^{\beta}_K)$
of a specified order are determined \\
\> (where $_K$ is a multiple index representing {\it order} many indices.) \\
\> NOTE: Characteristic functions of generalized symmetries \\
\> ($= \eta^{\alpha}$ if $\xi^i=0$) are equivalent if they are equal on\\
\> the solution manifold. Therefore, all dependences of\\
\> characteristic functions on the substituted derivatives \\
\> and their derivatives are dropped. For example, if the heat \\
\> equation is given as $u_t=u_{xx}$ (i.e.\ $u_t$ is substituted by $u_{xx}$) \\
\> then \verb+{"general",2}+ would not include characteristic \\
\> functions depending on $u_{tx}$ or $u_{xxx}$. \\
\> THEREFORE: \\
\> If you want to find {\it all} symmetries up to a given order then either \\
\> - avoid using $H_A=0$ to substitute lower order \\
\> $\;\,$derivatives by expressions involving higher derivatives, or \\
\> - increase the order specified in {\it symtype}. \\
\> For an illustration of this effect see the two symmetry \\
\> determinations of the Burgers equation in the file \\
\> {\tt LIEPDE.TST}. \\
\verb+{xi!_+{\it x1}\verb+ =...,..., + \> \\
\verb+ eta!_+{\it u1}\verb+=...,...}+ \> It is possible to specify an
ansatz for the symmetry. Such \\
\> an ansatz must specify all $\xi^i$ for all independent variables and \\
\> all $\eta^{\alpha}$ for all dependent variables in terms of differential \\
\> expressions which may involve unknown functions/constants. \\
\> The dependences of the unknown functions have to be declared \\
\> in advance by using the {\tt DEPEND} command. For example, \\
\> \verb+ DEPEND f, t, x, u$ + \\
\> specifies $f$ to be a function of $t,x,u$. If one wants to have $f$ as \\
\> a function of derivatives of $u(t,x)$, say $f$ depending on $u_{txx}$, \\
\> then one \underline{{\it cannot}} write \\
\> \verb+ DEPEND f, df(u,t,x,2)$ + \\
\> but instead must write \\
\> \verb+ DEPEND f, u!`1!`2!`2$ + \\
\> assuming {\it xlist} has been specified as \verb+ {t,x}+.
Because $t$ is the \\
\> first variable and $x$ is the second variable in {\it xlist} and $u$ is \\
\> differentiated oncs wrt.\ $t$ and twice wrt.\ $x$ we therefore \\
\> use \verb+ u!`1!`2!`2+. The character {\tt !} is the escape character \\
\> to allow special characters like ` to occur in an identifier. \\
\> \hspace{4mm} For generalized symmetries one usually sets all $\xi^i=0$.\\
\> Then the $\eta^{\alpha}$ are equal to the characteristic functions.
\end{tabbing}
\noindent The third parameter {\it flist} of {\tt LIEPDE} is a list $\{\;\}$
that includes
\begin{itemize}
\item all parameters and functions in the equations which are to
be determined such that symmetries exist (if any such
parameters/functions are
specified in {\it flist} then the symmetry conditions
formulated in {\tt LIEPDE}
become non-linear conditions which may be much harder for
{\tt CRACK} to solve with many cases and subcases to be considered.)
\item all unknown functions and constants in the ansatz
\verb+xi!_..+ and \verb+eta!_..+
if that has been specified in {\it symtype}.
\end{itemize}
\noindent The fourth parameter {\it inequ} of {\tt LIEPDE} is a list $\{\;\}$
that includes all non-vanishing expressions which represent
inequalities for the functions in flist.
The procedure {\tt LIEPDE} returns a list containing a list of
unsolved conditions if any, a list containing the general solution for
$\xi^i, \eta^{\alpha}$ and a list of constants and functions appearing
in the general solution or in the remaining unsolved conditions.
\section{Flags, parameters}
Two flags specify whether symmetry conditions are formulated
and solved in stages or in one go.
If the equation to be investigated is of higher than first order
and point symmetries are investigated then {\tt LIEPDE} allows
a set of preliminary conditions
to be formulated and solved before formulating and
solving the full set of conditions for this equation (more
details in \cite{Step},
\cite{Wo}). This successive execution is enabled by setting \\
\verb+ LISP(PRELIM_:=t)$+. \\
The default value is \\
\verb+ LISP(PRELIM_:=NIL)$+. \\
If the preliminary conditions are easy to solve completely then
it is advantageous
to formulate and solve them first, otherwise the formulation of
the complete more
overdetermined condition is better. Examples for both cases are
given together with comments in {\tt LIEPDE.TST}.
If symmetries of a system of equations are to be investigated
then with the setting \\
\verb+LISP(INDIVIDUAL_:=t)$+ conditions for the equations are formulated
and solved individually which provides a speed up if symmetry conditions are
very overdetermined. The default value is \\
\verb+ LISP(INDIVIDUAL_:=NIL)$+. \\
By default {\tt LIEPDE} computes $\xi$ and $\eta$ for each
symmetry. If a prolongation of the symmetry vector shall be calculated
then the order of this prolongation can be specified by the setting
\verb+LISP(PROLONG_ORDER:= ...)$+. \\
Flags that control the solution of the symmetry conditions by {\tt CRACK}
are displayed with \verb+CRACKHELP()$+. Among them are: \\
\verb+ LISP (PRINT_:= NIL/0/1/ ...)$+ \\
\verb+PRINT_=NIL+ suppresses all CRACK output, for \verb+PRINT_=n+
($n$ a positive integer)
{\tt CRACK} prints only equations with at most $n$ factors in their terms, and
%\verb+ LISP (LOGOPRINT_:=t/nil)$+ \\
%to print/not print a logo at the start of {\tt LIEPDE} \\
\verb+ OFF BATCH_MODE$+
enables the interactive solution of the system of conditions with {\tt CRACK}.
\section{Requirements}
{\tt REDUCE 3.6} and
the files {\tt CRACK.RED, LIEPDE.RED} and
all files {\tt CR*.RED} which are read in from {\tt CRACK.RED}. \\
\verb+ IN "crack.red","liepde.red"$+ \\
(and appropriate paths) or compilation with \\
\verb+ FASLOUT "crack"$+ \\
\verb+ IN "crack.red"$+ \\
\verb+ FASLEND$+ \\
\verb+ FASLOUT "liepde"$+ \\
\verb+ IN "liepde.red"$+ \\
\verb+ FASLEND$+ \\
\verb+ BYE$+ \\
and loading afterwards with \verb+ LOAD_PACKAGE crack,liepde$.+
\begin{thebibliography}{99}
\bibitem{Olv} P.J. Olver, Applications of Lie Groups to Differential
Equations, Springer-Verlag, New York (1986).
\bibitem{Step} H. Stephani, Differential Equations, Their solution using
symmetries, Ed. M.A.H. MacCallum, Cambridge Univ. Press (1989).
\bibitem{Wo} T. Wolf, An efficiency improved program LIEPDE for
determining Lie-symmetries
of PDEs, Proceedings of ``Modern Group Analysis: advanced analytical
and computational methods in mathematical physics'', Acireale, Italy,
October 1992, Kluwer Academic Publishers, pP 377-385, 1993.
\end{thebibliography}
\end{document}