@@ -1,598 +1,598 @@
-\documentstyle[11pt,reduce]{article}
-\title{The Package SPDE for Determining Symmetries of Partial
-Differential Equations}
-\date{}
-\author{Fritz Schwarz \\ GMD, Institut F1 \\
-Postfach 1240 \\ 5205 St. Augustin \\
-GERMANY \\[0.05in]
-Telephone: +49-2241-142782 \\
-Email: fritz.schwarz@gmd.de}
-\begin{document}
-\maketitle
-
-The package SPDE provides a set of functions which may be applied
-to determine the symmetry group of Lie- or point-symmetries of a
-given system of partial differential equations.  Preferably it is
-used interactively on a computer terminal. In many cases the
-determining system is solved completely automatically. In some
-other cases the user has to provide some additional input
-information for the solution algorithm to terminate. The package
-should only be used in compiled form.
-
-For all theoretical questions, a description of the algorithm and
-numerous examples the following articles should be consulted:
-``Automatically Determining Symmetries of Partial Differential
-Equations'', Computing vol. 34, page 91-106(1985) and vol. 36, page
-279-280(1986), ``Symmetries of Differential Equations: From Sophus
-Lie to Computer Algebra'', SIAM Review, to appear, and Chapter 2
-of the Lecture Notes ``Computer Algebra and Differential Equations
-of Mathematical Physics'', to appear.
-
-
-\section{Description of the System Functions and Variables}
-
-The symmetry analysis of partial differential equations logically
-falls into three parts. Accordingly the most important functions
-provided by the package are:
-
-\begin{table}
-\begin{center}
-\begin{tabular}{| c | c | }\hline
-Function name & Operation \\ \hline \hline
-\ttindex{CRESYS}
-CRESYS(\s{arguments}) & Constructs determining system \\ \hline
-\ttindex{SIMPSYS}
-SIMPSYS() & Solves determining system \\ \hline
-\ttindex{RESULT}
-RESULT() & Prints infinitesimal generators \\
-&  and commutator table \\ \hline
-\end{tabular}
-\end{center}
-\caption{SPDE Functions}
-\end{table}
-
-Some other useful functions for obtaining various kinds of output
-are:
-
-\begin{table}
-\begin{center}
-\begin{tabular}{| c | c |} \hline
-Function name & Operation \\ \hline \hline
-\ttindex{PRSYS}
-PRSYS() & Prints determining system \\ \hline
-\ttindex{PRGEN}
-PRGEN() & Prints infinitesimal generators \\ \hline
-\ttindex{COMM}
-COMM(U,V) & Prints commutator of generators U and V \\ \hline
-\end{tabular}
-\end{center}
-\caption{SPDE Useful Output Functions}\label{spde:useful}
-\end{table}
-
-There are several global variables defined by the system which should
-not be used for any other purpose than that given in
-Table~\ref{spde:intt} and~\ref{spde:op}. The three globals of the type
-integer are:
-
-\begin{table}
-\begin{center}
-\begin{tabular}{| c | c |}\hline
-Variable name & Meaning \\ \hline \hline
-\ttindex{NN}
-NN & Number of independent variables \\ \hline
-\ttindex{MM}
-MM & Number of dependent variables \\ \hline
-\ttindex{PCLASS}
-PCLASS=0, 1 or 2 & Controls amount of output \\ \hline
-\end{tabular}
-\end{center}
-\caption{SPDE Integer valued globals}\label{spde:intt}
-\end{table}
-
-In addition there are the following global variables of type
-operator:
-
-\begin{table}
-\begin{center}
-\begin{tabular}{| c | c |}\hline
-Variable name & Meaning \\ \hline \hline
-\ttindex{X(I)}
-X(I) & Independent variable $x_i$ \\ \hline
-\ttindex{U(ALFA)}
-U(ALFA) & Dependent variable $u^{alfa}$ \\ \hline
-\ttindex{U(ALFA,I)}
-U(ALFA,I) & Derivative of $u^{alfa}$ w.r.t. $x_i$ \\ \hline
-\ttindex{DEQ(I)}
-DEQ(I) & i-th differential equation \\ \hline
-\ttindex{SDER(I)}
-SDER(I) & Derivative w.r.t. which DEQ(I) is resolved \\ \hline
-\ttindex{GL(I)}
-GL(I) & i-th equation of determining system \\ \hline
-\ttindex{GEN(I)}
-GEN(I) & i-th infinitesimal generator \\ \hline
-\ttindex{XI(I)} \ttindex{ETA(ALFA)} \ttindex{ZETA(ALFA,I)}
-XI(I), ETA(ALFA)  & See definition given in the \\
-ZETA(ALFA,I) & references quoted in the introduction. \\ \hline
-\ttindex{C(I)}
-C(I) & i-th function used for substitution \\ \hline
-\end{tabular}
-\end{center}
-\caption{SPDE Operator type global variables}\label{spde:op}
-\end{table}
-
-
-The differential equations of the system at issue have to be assigned
-as values to the operator deq i applying the notation which is defined
-in Table~\ref{spde:op}. The entries in the third and the last line of
-that Table have obvious extensions to higher derivatives.
-
-The derivative w.r.t. which the i-th differential equation deq i is
-resolved has to be assigned to sder i. Exception: If there is a single
-differential equation and no assignment has been made by the user, the
-highest derivative is taken by default.
-
-When the appropriate assignments are made to the variable deq, the
-values of NN and MM (Table~\ref{spde:useful}) are determined
-automatically, i.e. they have not to be assigned by the user.
-
-\ttindex{CRESYS}
-The function CRESYS may be called with any number of arguments, i.e.
-
-\begin{verbatim}
-  CRESYS(); or CRESYS(deq 1,deq 2,... );
-\end{verbatim}
-
- are legal calls. If it is called without any argument, all current
-assignments to deq are taken into account. Example: If deq 1, deq 2
-and deq 3 have been assigned a differential equation and the symmetry
-group of the full system comprising all three equations is desired,
-equivalent calls are
-
-\begin{verbatim}
-  CRESYS();   or   CRESYS(deq 1,deq 2,deq 3);
-\end{verbatim}
-
-The first alternative saves some typing. If later in the session the
-symmetry group of deq 1 alone has to be determined, the correct call
-is
-
-\begin{verbatim}
-  CRESYS deq 1;
-\end{verbatim}
-
-\ttindex{SIMPSYS}
-After the determining system has bee created, SIMPSYS which has no
-arguments may be called for solving it. The amount of intermediate
-output produced by SIMPSYS is controlled by the global variable PCLASS
-with the default value 0. \ttindex{PCLASS} With PCLASS equal to 0, no
-intermediate steps are shown. With PCLASS equal to 1, all intermediate
-steps are displayed so that the solution algorithm may be followed
-\index{tracing ! SPDE package} through in detail. Each time the algorithm
-passes through the top of the main solution loop the message
-
-\begin{verbatim}
-  Entering main loop
-\end{verbatim}
-
-is written. PCLASS equal 2 produces a lot of LISP output and is of no
-interest for the normal user.
-
-If with PCLASS=0 the procedure SIMPSYS terminates without any
-response, the determining system is completely solved.  In some cases
-SIMPSYS does not solve the determining system completely in a single
-run. In general this is true if there are only genuine differential
-equations left which the algorithm cannot handle at present. If a case
-like this occurs, SIMPSYS returns the remaining equations of the
-determining system. To proceed with the solution algorithm,
-appropriate assignments have to be transmitted by the user, e.g. the
-explicit solution for one of the returned differential equations. Any
-new functions which are introduced thereby must be operators of the
-form c(k) with the correct dependencies generated by a depend
-statement (see the ``REDUCE User's Guide''). Its enumeration has to be
-chosen in agreement with the current number of functions which have
-alreday been introduced.  This value is returned by SIMPSYS too.
-
-After the determining system has been solved, the procedure RESULT,
-which has no arguments, may be called. It displays the infinitesimal
-generators and its non-vanishing commutators.
-
-
-\section{How to Use the Package}
-
-In this Section it is explained by way of several examples how the
-package SPDE is used interactively to determine the symmetry group of
-partial differential equations. Consider first the diffusion equation
-which in the notation given above may be written as
-
-\begin{verbatim}
-  deq 1:=u(1,1)+u(1,2,2);
-\end{verbatim}
-
-It has been assigned as the value of deq 1 by this statement.  There
-is no need to assign a value to sder 1 here because the system
-comprises only a single equation.
-
-The determining system is constructed by calling
-
-\begin{verbatim}
-  CRESYS(); or CRESYS deq 1;
-\end{verbatim}
-
-The latter call is compulsory if there are other assignments to the
-operator deq i than for i=1.
-
-The error message
-
-\begin{verbatim}
-  ***** Differential equations not defined
-\end{verbatim}
-
-appears if there are no differential equations assigned to any deq.
-
-If the user wants the determining system displayed for inspection
-before starting the solution algorithm he may call
-
-\ttindex{PRSYS}
-\begin{verbatim}
-  PRSYS();
-\end{verbatim}
-
-and gets the answer
-
-\begin{verbatim}
-  GL(1):=2*DF(ETA(1),U(1),X(2)) - DF(XI(2),X(2),2) -
-         DF(XI(2),X(1))
-
-  GL(2):=DF(ETA(1),U(1),2) - 2*DF(XI(2),U(1),X(2))
-
-  GL(3):=DF(ETA(1),X(2),2) + DF(ETA(1),X(1))
-
-  GL(4):=DF(XI(2),U(1),2)
-
-  GL(5):=DF(XI(2),U(1)) - DF(XI(1),U(1),X(2))
-
-  GL(6):=2*DF(XI(2),X(2)) - DF(XI(1),X(2),2) - DF(XI(1),X(1))
-
-  GL(7):=DF(XI(1),U(1),2)
-
-  GL(8):=DF(XI(1),U(1))
-
-  GL(9):=DF(XI(1),X(2))
-
-The remaining dependencies
-
-  XI(2) depends on U(1),X(2),X(1)
-
-  XI(1) depends on U(1),X(2),X(1)
-
-  ETA(1) depends on U(1),X(2),X(1)
-\end{verbatim}
-
-The last message means that all three functions XI(1), XI(2) and
-ETA(1) depend on X(1), X(2) and U(1). Without this information the
-nine equations GL(1) to GL(9) forming the determining system are
-meaningless. Now the solution algorithm may be activated by calling
-
-\ttindex{SIMPSYS}
-\begin{verbatim}
-   SIMPSYS();
-\end{verbatim}
-
-\ttindex{PCLASS}
-If the print flag PCLASS has its default value which is 0 no
-intermediate output is produced and the answer is
-
-\begin{verbatim}
-  Determining system is not completely solved
-
-  The remaining equations are
-
-  GL(1):=DF(C(1),X(2),2) + DF(C(1),X(1))
-
-  Number of functions is 16
-
-  The remaining dependencies
-
-  C(1) depends on X(2),X(1)
-\end{verbatim}
-
-With PCLASS equal to 1 about 6 pages of intermediate output are
-obtained. It allows the user to follow through each step of the
-solution algorithm.
-
-In this example the algorithm did not solve the determining system
-completely as it is shown by the last message. This was to be expected
-because the diffusion equation is linear and therefore the symmetry
-group contains a generator depending on a function which solves the
-original differential equation. In cases like this the user has to
-provide some additional information to the system so that the solution
-algorithm may continue. In the example under consideration the
-appropriate input is
-
-\begin{verbatim}
-   DF(C(1),X(1)) := - DF(C(1),X(2),2);
-\end{verbatim}
-
-If now the solution algorithm is activated again by
-
-\begin{verbatim}
-  SIMPSYS();
-\end{verbatim}
-
-the solution algorithm terminates without any further message, i.e.
-there are no equations of the determining system left unsolved. To
-obtain the symmetry generators one has to say finally
-
-\begin{verbatim}
-  RESULT();
-\end{verbatim}
-
-and obtains the answer
-
-\begin{verbatim}
-  The differential equation
-
-  DEQ(1):=U(1,2,2) + U(1,1)
-
-
-  The symmetry generators are
-
-  GEN(1):= DX(1)
-
-  GEN(2):= DX(2)
-
-  GEN(3):= 2*DX(2)*X(1) + DU(1)*U(1)*X(2)
-
-  GEN(4):= DU(1)*U(1)
-
-  GEN(5):= 2*DX(1)*X(1) + DX(2)*X(2)
-
-                       2
-  GEN(6):= 4*DX(1)*X(1)
-
-         + 4*DX(2)*X(2)*X(1)
-
-                           2
-           + DU(1)*U(1)*(X(2)  - 2*X(1))
-
-  GEN(7):= DU(1)*C(1)
-
-  The remaining dependencies
-
-  C(1) depends on X(2),X(1)
-
-
-  Constraints
-
-  DF(C(1),X(1)):= - DF(C(1),X(2),2)
-
-
-  The non-vanishing commutators of the finite subgroup
-
-
-  COMM(1,3):= 2*DX(2)
-
-  COMM(1,5):= 2*DX(1)
-
-  COMM(1,6):= 8*DX(1)*X(1) + 4*DX(2)*X(2) - 2*DU(1)*U(1)
-
-  COMM(2,3):= DU(1)*U(1)
-
-  COMM(2,5):= DX(2)
-
-  COMM(2,6):= 4*DX(2)*X(1) + 2*DU(1)*U(1)*X(2)
-
-  COMM(3,5):=  - (2*DX(2)*X(1) + DU(1)*U(1)*X(2))
-
-                          2
-  COMM(5,6):= 8*DX(1)*X(1)
-
-            + 8*DX(2)*X(2)*X(1)
-
-                                2
-            + 2*DU(1)*U(1)*(X(2)  - 2*X(1))
-\end{verbatim}
-
-The message ``Constraints'' which appears after the symmetry generators
-are displayed means that the function c(1) depends on x(1) and x(2)
-and satisfies the diffusion equation.
-
-More examples which may used for test runs are given in the final
-section.
-
-\index{ansatz of symmetry generator}
-If the user wants to test a certain ansatz of a symmetry generator for
-given differential equations, the correct proceeding is as follows.
-Create the determining system as described above. Make the appropriate
-assignments for the generator and call PRSYS() after that.  The
-determining system with this ansatz substituted is returned.  Example:
-Assume again that the determining system for the diffusion equation
-has been created. To check the correctness for example of generator GEN
-3 which has been obtained above, the assignments
-
-\begin{verbatim}
-  XI(1):=0;  XI(2):=2*X(1);  ETA(1):=X(2)*U(1);
-\end{verbatim}
-
-have to be made. If now PRSYS() is called all GL(K) are zero
-proving the correctness of this generator.
-
-Sometimes a user only wants to know some of the functions ZETA for for
-various values of its possible arguments and given values of MM and
-NN. In these cases the user has to assign the desired values of MM and
-NN and may call the ZETAs after that. Example:
-
-\begin{verbatim}
-  MM:=1;  NN:=2;
-
-  FACTOR U(1,2),U(1,1),U(1,1,2),U(1,1,1);
-
-  ON LIST;
-
-  ZETA(1,1);
-
-  -U(1,2)*U(1,1)*DF(XI(2),U(1))
-
-  -U(1,2)*DF(XI(2),X(1))
-
-         2
-  -U(1,1) *DF(XI(1),U(1))
-
-  +U(1,1)*(DF(ETA(1),U(1)) -DF(XI(1),X(1)))
-
-  +DF(ETA(1),X(1))
-
-
-  ZETA(1,1,1);
-
-  -2*U(1,1,2)*U(1,1)*DF(XI(2),U(1))
-
-  -2*U(1,1,2)*DF(XI(2),X(1))
-
-  -U(1,1,1)*U(1,2)*DF(XI(2),U(1))
-
-  -3*U(1,1,1)*U(1,1)*DF(XI(1),U(1))
-
-  +U(1,1,1)*(DF(ETA(1),U(1)) -2*DF(XI(1),X(1)))
-
-                2
-  -U(1,2)*U(1,1) *DF(XI(2),U(1),2)
-
-  -2*U(1,2)*U(1,1)*DF(XI(2),U(1),X(1))
-
-  -U(1,2)*DF(XI(2),X(1),2)
-
-         3
-  -U(1,1) *DF(XI(1),U(1),2)
-
-         2
-  +U(1,1) *(DF(ETA(1),U(1),2) -2*DF(XI(1),U(1),X(1)))
-
-  +U(1,1)*(2*DF(ETA(1),U(1),X(1)) -DF(XI(1),X(1),2))
-
-  +DF(ETA(1),X(1),2)
-\end{verbatim}
-
-If by error no values to MM or NN and have been assigned the message
-
-\begin{verbatim}
-  ***** Number of variables not defined
-\end{verbatim}
-
-is returned. Often the functions ZETA are desired for special values
-of its arguments ETA(ALFA) and XI(K). To this end they have to be
-assigned first to some other variable. After that they may be
-evaluated for the special arguments. In the previous example this may
-be achieved by
-
-\begin{verbatim}
-  Z11:=ZETA(1,1)$   Z111:=ZETA(1,1,1)$
-\end{verbatim}
-
-Now assign the following values to XI 1, XI 2 and ETA 1:
-
-\begin{verbatim}
-  XI 1:=4*X(1)**2; XI 2:=4*X(2)*X(1);
-
-  ETA 1:=U(1)*(X(2)**2  - 2*X(1));
-\end{verbatim}
-
-They correspond to the generator GEN 6 of the diffusion equation which
-has been obtained above. Now the desired expressions are obtained by
-calling
-
-\begin{verbatim}
-  Z11;
-
-                               2
- - (4*U(1,2)*X(2) - U(1,1)*X(2)  + 10*U(1,1)*X(1) + 2*U(1))
-
-  Z111;
-
-                                   2
- - (8*U(1,1,2)*X(2) - U(1,1,1)*X(2)  + 18*U(1,1,1)*X(1) +
-   12*U(1,1))
-\end{verbatim}
-
-
-\section{Test File}
-
-This appendix is a test file. The symmetry groups for various
-equations or systems of equations are determined. The variable PCLASS
-has the default value 0 and may be changed by the user before running
-it. The output may be compared with the results which are given in the
-references.
-
-\begin{verbatim}
-%The Burgers equations
-
-deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
-
-cresys deq 1$ simpsys()$ result()$
-
-%The Kadomtsev-Petviashvili equation
-
-deq 1:=3*u(1,3,3)+u(1,2,2,2,2)+6*u(1,2,2)*u 1
-
-       +6*u(1,2)**2+4*u(1,1,2)$
-
-cresys deq 1$ simpsys()$ result()$
-
-%The modified Kadomtsev-Petviashvili equation
-
-deq 1:=u(1,1,2)-u(1,2,2,2,2)-3*u(1,3,3)
-
-       +6*u(1,2)**2*u(1,2,2)+6*u(1,3)*u(1,2,2)$
-
-cresys deq 1$ simpsys()$ result()$
-
-%The real- and the imaginary part of the nonlinear
-%Schroedinger equation
-
-deq 1:= u(1,1)+u(2,2,2)+2*u 1**2*u 2+2*u 2**3$
-
-deq 2:=-u(2,1)+u(1,2,2)+2*u 1*u 2**2+2*u 1**3$
-
-%Because this is not a single equation the two assignments
-
-sder 1:=u(2,2,2)$  sder 2:=u(1,2,2)$
-
-%are necessary.
-
-cresys()$ simpsys()$ result()$
-
-%The symmetries of the system comprising the four equations
-
-deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
-
-deq 2:=u(2,1)+u(2,2,2)$
-
-deq 3:=u 1*u 2-2*u(2,2)$
-
-deq 4:=4*u(2,1)+u 2*(u 1**2+2*u(1,2))$
-
-sder 1:=u(1,2,2)$ sder 2:=u(2,2,2)$ sder 3:=u(2,2)$
-sder 4:=u(2,1)$
-
-%is obtained by calling
-
-cresys()$ simpsys()$
-
-df(c 5,x 1):=-df(c 5,x 2,2)$
-
-df(c 5,x 2,x 1):=-df(c 5,x 2,3)$
-
-simpsys()$  result()$
-
-% The symmetries of the subsystem comprising equation 1
-%  and 3 are obtained by
-
-cresys(deq 1,deq 3)$ simpsys()$ result()$
-
-% The result for all possible subsystems is discussed in
-% detail in ``Symmetries and Involution Systems: Some
-% Experiments in Computer Algebra'', contribution to the
-% Proceedings of the Oberwolfach Meeting on Nonlinear
-% Evolution Equations, Summer 1986, to appear.
-\end{verbatim}
-\end{document}
+\documentstyle[11pt,reduce]{article}
+\title{The Package SPDE for Determining Symmetries of Partial
+Differential Equations}
+\date{}
+\author{Fritz Schwarz \\ GMD, Institut F1 \\
+Postfach 1240 \\ 5205 St. Augustin \\
+GERMANY \\[0.05in]
+Telephone: +49-2241-142782 \\
+Email: fritz.schwarz@gmd.de}
+\begin{document}
+\maketitle
+
+The package SPDE provides a set of functions which may be applied
+to determine the symmetry group of Lie- or point-symmetries of a
+given system of partial differential equations.  Preferably it is
+used interactively on a computer terminal. In many cases the
+determining system is solved completely automatically. In some
+other cases the user has to provide some additional input
+information for the solution algorithm to terminate. The package
+should only be used in compiled form.
+
+For all theoretical questions, a description of the algorithm and
+numerous examples the following articles should be consulted:
+``Automatically Determining Symmetries of Partial Differential
+Equations'', Computing vol. 34, page 91-106(1985) and vol. 36, page
+279-280(1986), ``Symmetries of Differential Equations: From Sophus
+Lie to Computer Algebra'', SIAM Review, to appear, and Chapter 2
+of the Lecture Notes ``Computer Algebra and Differential Equations
+of Mathematical Physics'', to appear.
+
+
+\section{Description of the System Functions and Variables}
+
+The symmetry analysis of partial differential equations logically
+falls into three parts. Accordingly the most important functions
+provided by the package are:
+
+\begin{table}
+\begin{center}
+\begin{tabular}{| c | c | }\hline
+Function name & Operation \\ \hline \hline
+\ttindex{CRESYS}
+CRESYS(\s{arguments}) & Constructs determining system \\ \hline
+\ttindex{SIMPSYS}
+SIMPSYS() & Solves determining system \\ \hline
+\ttindex{RESULT}
+RESULT() & Prints infinitesimal generators \\
+&  and commutator table \\ \hline
+\end{tabular}
+\end{center}
+\caption{SPDE Functions}
+\end{table}
+
+Some other useful functions for obtaining various kinds of output
+are:
+
+\begin{table}
+\begin{center}
+\begin{tabular}{| c | c |} \hline
+Function name & Operation \\ \hline \hline
+\ttindex{PRSYS}
+PRSYS() & Prints determining system \\ \hline
+\ttindex{PRGEN}
+PRGEN() & Prints infinitesimal generators \\ \hline
+\ttindex{COMM}
+COMM(U,V) & Prints commutator of generators U and V \\ \hline
+\end{tabular}
+\end{center}
+\caption{SPDE Useful Output Functions}\label{spde:useful}
+\end{table}
+
+There are several global variables defined by the system which should
+not be used for any other purpose than that given in
+Table~\ref{spde:intt} and~\ref{spde:op}. The three globals of the type
+integer are:
+
+\begin{table}
+\begin{center}
+\begin{tabular}{| c | c |}\hline
+Variable name & Meaning \\ \hline \hline
+\ttindex{NN}
+NN & Number of independent variables \\ \hline
+\ttindex{MM}
+MM & Number of dependent variables \\ \hline
+\ttindex{PCLASS}
+PCLASS=0, 1 or 2 & Controls amount of output \\ \hline
+\end{tabular}
+\end{center}
+\caption{SPDE Integer valued globals}\label{spde:intt}
+\end{table}
+
+In addition there are the following global variables of type
+operator:
+
+\begin{table}
+\begin{center}
+\begin{tabular}{| c | c |}\hline
+Variable name & Meaning \\ \hline \hline
+\ttindex{X(I)}
+X(I) & Independent variable $x_i$ \\ \hline
+\ttindex{U(ALFA)}
+U(ALFA) & Dependent variable $u^{alfa}$ \\ \hline
+\ttindex{U(ALFA,I)}
+U(ALFA,I) & Derivative of $u^{alfa}$ w.r.t. $x_i$ \\ \hline
+\ttindex{DEQ(I)}
+DEQ(I) & i-th differential equation \\ \hline
+\ttindex{SDER(I)}
+SDER(I) & Derivative w.r.t. which DEQ(I) is resolved \\ \hline
+\ttindex{GL(I)}
+GL(I) & i-th equation of determining system \\ \hline
+\ttindex{GEN(I)}
+GEN(I) & i-th infinitesimal generator \\ \hline
+\ttindex{XI(I)} \ttindex{ETA(ALFA)} \ttindex{ZETA(ALFA,I)}
+XI(I), ETA(ALFA)  & See definition given in the \\
+ZETA(ALFA,I) & references quoted in the introduction. \\ \hline
+\ttindex{C(I)}
+C(I) & i-th function used for substitution \\ \hline
+\end{tabular}
+\end{center}
+\caption{SPDE Operator type global variables}\label{spde:op}
+\end{table}
+
+
+The differential equations of the system at issue have to be assigned
+as values to the operator deq i applying the notation which is defined
+in Table~\ref{spde:op}. The entries in the third and the last line of
+that Table have obvious extensions to higher derivatives.
+
+The derivative w.r.t. which the i-th differential equation deq i is
+resolved has to be assigned to sder i. Exception: If there is a single
+differential equation and no assignment has been made by the user, the
+highest derivative is taken by default.
+
+When the appropriate assignments are made to the variable deq, the
+values of NN and MM (Table~\ref{spde:useful}) are determined
+automatically, i.e. they have not to be assigned by the user.
+
+\ttindex{CRESYS}
+The function CRESYS may be called with any number of arguments, i.e.
+
+\begin{verbatim}
+  CRESYS(); or CRESYS(deq 1,deq 2,... );
+\end{verbatim}
+
+ are legal calls. If it is called without any argument, all current
+assignments to deq are taken into account. Example: If deq 1, deq 2
+and deq 3 have been assigned a differential equation and the symmetry
+group of the full system comprising all three equations is desired,
+equivalent calls are
+
+\begin{verbatim}
+  CRESYS();   or   CRESYS(deq 1,deq 2,deq 3);
+\end{verbatim}
+
+The first alternative saves some typing. If later in the session the
+symmetry group of deq 1 alone has to be determined, the correct call
+is
+
+\begin{verbatim}
+  CRESYS deq 1;
+\end{verbatim}
+
+\ttindex{SIMPSYS}
+After the determining system has bee created, SIMPSYS which has no
+arguments may be called for solving it. The amount of intermediate
+output produced by SIMPSYS is controlled by the global variable PCLASS
+with the default value 0. \ttindex{PCLASS} With PCLASS equal to 0, no
+intermediate steps are shown. With PCLASS equal to 1, all intermediate
+steps are displayed so that the solution algorithm may be followed
+\index{tracing ! SPDE package} through in detail. Each time the algorithm
+passes through the top of the main solution loop the message
+
+\begin{verbatim}
+  Entering main loop
+\end{verbatim}
+
+is written. PCLASS equal 2 produces a lot of LISP output and is of no
+interest for the normal user.
+
+If with PCLASS=0 the procedure SIMPSYS terminates without any
+response, the determining system is completely solved.  In some cases
+SIMPSYS does not solve the determining system completely in a single
+run. In general this is true if there are only genuine differential
+equations left which the algorithm cannot handle at present. If a case
+like this occurs, SIMPSYS returns the remaining equations of the
+determining system. To proceed with the solution algorithm,
+appropriate assignments have to be transmitted by the user, e.g. the
+explicit solution for one of the returned differential equations. Any
+new functions which are introduced thereby must be operators of the
+form c(k) with the correct dependencies generated by a depend
+statement (see the ``REDUCE User's Guide''). Its enumeration has to be
+chosen in agreement with the current number of functions which have
+alreday been introduced.  This value is returned by SIMPSYS too.
+
+After the determining system has been solved, the procedure RESULT,
+which has no arguments, may be called. It displays the infinitesimal
+generators and its non-vanishing commutators.
+
+
+\section{How to Use the Package}
+
+In this Section it is explained by way of several examples how the
+package SPDE is used interactively to determine the symmetry group of
+partial differential equations. Consider first the diffusion equation
+which in the notation given above may be written as
+
+\begin{verbatim}
+  deq 1:=u(1,1)+u(1,2,2);
+\end{verbatim}
+
+It has been assigned as the value of deq 1 by this statement.  There
+is no need to assign a value to sder 1 here because the system
+comprises only a single equation.
+
+The determining system is constructed by calling
+
+\begin{verbatim}
+  CRESYS(); or CRESYS deq 1;
+\end{verbatim}
+
+The latter call is compulsory if there are other assignments to the
+operator deq i than for i=1.
+
+The error message
+
+\begin{verbatim}
+  ***** Differential equations not defined
+\end{verbatim}
+
+appears if there are no differential equations assigned to any deq.
+
+If the user wants the determining system displayed for inspection
+before starting the solution algorithm he may call
+
+\ttindex{PRSYS}
+\begin{verbatim}
+  PRSYS();
+\end{verbatim}
+
+and gets the answer
+
+\begin{verbatim}
+  GL(1):=2*DF(ETA(1),U(1),X(2)) - DF(XI(2),X(2),2) -
+         DF(XI(2),X(1))
+
+  GL(2):=DF(ETA(1),U(1),2) - 2*DF(XI(2),U(1),X(2))
+
+  GL(3):=DF(ETA(1),X(2),2) + DF(ETA(1),X(1))
+
+  GL(4):=DF(XI(2),U(1),2)
+
+  GL(5):=DF(XI(2),U(1)) - DF(XI(1),U(1),X(2))
+
+  GL(6):=2*DF(XI(2),X(2)) - DF(XI(1),X(2),2) - DF(XI(1),X(1))
+
+  GL(7):=DF(XI(1),U(1),2)
+
+  GL(8):=DF(XI(1),U(1))
+
+  GL(9):=DF(XI(1),X(2))
+
+The remaining dependencies
+
+  XI(2) depends on U(1),X(2),X(1)
+
+  XI(1) depends on U(1),X(2),X(1)
+
+  ETA(1) depends on U(1),X(2),X(1)
+\end{verbatim}
+
+The last message means that all three functions XI(1), XI(2) and
+ETA(1) depend on X(1), X(2) and U(1). Without this information the
+nine equations GL(1) to GL(9) forming the determining system are
+meaningless. Now the solution algorithm may be activated by calling
+
+\ttindex{SIMPSYS}
+\begin{verbatim}
+   SIMPSYS();
+\end{verbatim}
+
+\ttindex{PCLASS}
+If the print flag PCLASS has its default value which is 0 no
+intermediate output is produced and the answer is
+
+\begin{verbatim}
+  Determining system is not completely solved
+
+  The remaining equations are
+
+  GL(1):=DF(C(1),X(2),2) + DF(C(1),X(1))
+
+  Number of functions is 16
+
+  The remaining dependencies
+
+  C(1) depends on X(2),X(1)
+\end{verbatim}
+
+With PCLASS equal to 1 about 6 pages of intermediate output are
+obtained. It allows the user to follow through each step of the
+solution algorithm.
+
+In this example the algorithm did not solve the determining system
+completely as it is shown by the last message. This was to be expected
+because the diffusion equation is linear and therefore the symmetry
+group contains a generator depending on a function which solves the
+original differential equation. In cases like this the user has to
+provide some additional information to the system so that the solution
+algorithm may continue. In the example under consideration the
+appropriate input is
+
+\begin{verbatim}
+   DF(C(1),X(1)) := - DF(C(1),X(2),2);
+\end{verbatim}
+
+If now the solution algorithm is activated again by
+
+\begin{verbatim}
+  SIMPSYS();
+\end{verbatim}
+
+the solution algorithm terminates without any further message, i.e.
+there are no equations of the determining system left unsolved. To
+obtain the symmetry generators one has to say finally
+
+\begin{verbatim}
+  RESULT();
+\end{verbatim}
+
+and obtains the answer
+
+\begin{verbatim}
+  The differential equation
+
+  DEQ(1):=U(1,2,2) + U(1,1)
+
+
+  The symmetry generators are
+
+  GEN(1):= DX(1)
+
+  GEN(2):= DX(2)
+
+  GEN(3):= 2*DX(2)*X(1) + DU(1)*U(1)*X(2)
+
+  GEN(4):= DU(1)*U(1)
+
+  GEN(5):= 2*DX(1)*X(1) + DX(2)*X(2)
+
+                       2
+  GEN(6):= 4*DX(1)*X(1)
+
+         + 4*DX(2)*X(2)*X(1)
+
+                           2
+           + DU(1)*U(1)*(X(2)  - 2*X(1))
+
+  GEN(7):= DU(1)*C(1)
+
+  The remaining dependencies
+
+  C(1) depends on X(2),X(1)
+
+
+  Constraints
+
+  DF(C(1),X(1)):= - DF(C(1),X(2),2)
+
+
+  The non-vanishing commutators of the finite subgroup
+
+
+  COMM(1,3):= 2*DX(2)
+
+  COMM(1,5):= 2*DX(1)
+
+  COMM(1,6):= 8*DX(1)*X(1) + 4*DX(2)*X(2) - 2*DU(1)*U(1)
+
+  COMM(2,3):= DU(1)*U(1)
+
+  COMM(2,5):= DX(2)
+
+  COMM(2,6):= 4*DX(2)*X(1) + 2*DU(1)*U(1)*X(2)
+
+  COMM(3,5):=  - (2*DX(2)*X(1) + DU(1)*U(1)*X(2))
+
+                          2
+  COMM(5,6):= 8*DX(1)*X(1)
+
+            + 8*DX(2)*X(2)*X(1)
+
+                                2
+            + 2*DU(1)*U(1)*(X(2)  - 2*X(1))
+\end{verbatim}
+
+The message ``Constraints'' which appears after the symmetry generators
+are displayed means that the function c(1) depends on x(1) and x(2)
+and satisfies the diffusion equation.
+
+More examples which may used for test runs are given in the final
+section.
+
+\index{ansatz of symmetry generator}
+If the user wants to test a certain ansatz of a symmetry generator for
+given differential equations, the correct proceeding is as follows.
+Create the determining system as described above. Make the appropriate
+assignments for the generator and call PRSYS() after that.  The
+determining system with this ansatz substituted is returned.  Example:
+Assume again that the determining system for the diffusion equation
+has been created. To check the correctness for example of generator GEN
+3 which has been obtained above, the assignments
+
+\begin{verbatim}
+  XI(1):=0;  XI(2):=2*X(1);  ETA(1):=X(2)*U(1);
+\end{verbatim}
+
+have to be made. If now PRSYS() is called all GL(K) are zero
+proving the correctness of this generator.
+
+Sometimes a user only wants to know some of the functions ZETA for for
+various values of its possible arguments and given values of MM and
+NN. In these cases the user has to assign the desired values of MM and
+NN and may call the ZETAs after that. Example:
+
+\begin{verbatim}
+  MM:=1;  NN:=2;
+
+  FACTOR U(1,2),U(1,1),U(1,1,2),U(1,1,1);
+
+  ON LIST;
+
+  ZETA(1,1);
+
+  -U(1,2)*U(1,1)*DF(XI(2),U(1))
+
+  -U(1,2)*DF(XI(2),X(1))
+
+         2
+  -U(1,1) *DF(XI(1),U(1))
+
+  +U(1,1)*(DF(ETA(1),U(1)) -DF(XI(1),X(1)))
+
+  +DF(ETA(1),X(1))
+
+
+  ZETA(1,1,1);
+
+  -2*U(1,1,2)*U(1,1)*DF(XI(2),U(1))
+
+  -2*U(1,1,2)*DF(XI(2),X(1))
+
+  -U(1,1,1)*U(1,2)*DF(XI(2),U(1))
+
+  -3*U(1,1,1)*U(1,1)*DF(XI(1),U(1))
+
+  +U(1,1,1)*(DF(ETA(1),U(1)) -2*DF(XI(1),X(1)))
+
+                2
+  -U(1,2)*U(1,1) *DF(XI(2),U(1),2)
+
+  -2*U(1,2)*U(1,1)*DF(XI(2),U(1),X(1))
+
+  -U(1,2)*DF(XI(2),X(1),2)
+
+         3
+  -U(1,1) *DF(XI(1),U(1),2)
+
+         2
+  +U(1,1) *(DF(ETA(1),U(1),2) -2*DF(XI(1),U(1),X(1)))
+
+  +U(1,1)*(2*DF(ETA(1),U(1),X(1)) -DF(XI(1),X(1),2))
+
+  +DF(ETA(1),X(1),2)
+\end{verbatim}
+
+If by error no values to MM or NN and have been assigned the message
+
+\begin{verbatim}
+  ***** Number of variables not defined
+\end{verbatim}
+
+is returned. Often the functions ZETA are desired for special values
+of its arguments ETA(ALFA) and XI(K). To this end they have to be
+assigned first to some other variable. After that they may be
+evaluated for the special arguments. In the previous example this may
+be achieved by
+
+\begin{verbatim}
+  Z11:=ZETA(1,1)$   Z111:=ZETA(1,1,1)$
+\end{verbatim}
+
+Now assign the following values to XI 1, XI 2 and ETA 1:
+
+\begin{verbatim}
+  XI 1:=4*X(1)**2; XI 2:=4*X(2)*X(1);
+
+  ETA 1:=U(1)*(X(2)**2  - 2*X(1));
+\end{verbatim}
+
+They correspond to the generator GEN 6 of the diffusion equation which
+has been obtained above. Now the desired expressions are obtained by
+calling
+
+\begin{verbatim}
+  Z11;
+
+                               2
+ - (4*U(1,2)*X(2) - U(1,1)*X(2)  + 10*U(1,1)*X(1) + 2*U(1))
+
+  Z111;
+
+                                   2
+ - (8*U(1,1,2)*X(2) - U(1,1,1)*X(2)  + 18*U(1,1,1)*X(1) +
+   12*U(1,1))
+\end{verbatim}
+
+
+\section{Test File}
+
+This appendix is a test file. The symmetry groups for various
+equations or systems of equations are determined. The variable PCLASS
+has the default value 0 and may be changed by the user before running
+it. The output may be compared with the results which are given in the
+references.
+
+\begin{verbatim}
+%The Burgers equations
+
+deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
+
+cresys deq 1$ simpsys()$ result()$
+
+%The Kadomtsev-Petviashvili equation
+
+deq 1:=3*u(1,3,3)+u(1,2,2,2,2)+6*u(1,2,2)*u 1
+
+       +6*u(1,2)**2+4*u(1,1,2)$
+
+cresys deq 1$ simpsys()$ result()$
+
+%The modified Kadomtsev-Petviashvili equation
+
+deq 1:=u(1,1,2)-u(1,2,2,2,2)-3*u(1,3,3)
+
+       +6*u(1,2)**2*u(1,2,2)+6*u(1,3)*u(1,2,2)$
+
+cresys deq 1$ simpsys()$ result()$
+
+%The real- and the imaginary part of the nonlinear
+%Schroedinger equation
+
+deq 1:= u(1,1)+u(2,2,2)+2*u 1**2*u 2+2*u 2**3$
+
+deq 2:=-u(2,1)+u(1,2,2)+2*u 1*u 2**2+2*u 1**3$
+
+%Because this is not a single equation the two assignments
+
+sder 1:=u(2,2,2)$  sder 2:=u(1,2,2)$
+
+%are necessary.
+
+cresys()$ simpsys()$ result()$
+
+%The symmetries of the system comprising the four equations
+
+deq 1:=u(1,1)+u 1*u(1,2)+u(1,2,2)$
+
+deq 2:=u(2,1)+u(2,2,2)$
+
+deq 3:=u 1*u 2-2*u(2,2)$
+
+deq 4:=4*u(2,1)+u 2*(u 1**2+2*u(1,2))$
+
+sder 1:=u(1,2,2)$ sder 2:=u(2,2,2)$ sder 3:=u(2,2)$
+sder 4:=u(2,1)$
+
+%is obtained by calling
+
+cresys()$ simpsys()$
+
+df(c 5,x 1):=-df(c 5,x 2,2)$
+
+df(c 5,x 2,x 1):=-df(c 5,x 2,3)$
+
+simpsys()$  result()$
+
+% The symmetries of the subsystem comprising equation 1
+%  and 3 are obtained by
+
+cresys(deq 1,deq 3)$ simpsys()$ result()$
+
+% The result for all possible subsystems is discussed in
+% detail in ``Symmetries and Involution Systems: Some
+% Experiments in Computer Algebra'', contribution to the
+% Proceedings of the Oberwolfach Meeting on Nonlinear
+% Evolution Equations, Summer 1986, to appear.
+\end{verbatim}
+\end{document}