REDUCE

20.14 CONLAW: Find Conservation Laws for Differential Equations

This package computes first integrals of ordinary differential equations (ODEs) or conservation laws (CLs) of partial differential equations (PDEs) or systems of both. Four different approaches to compute CLs have been implemented in four different procedures CONLAW1CONLAW4. All use the package CRACK to solve the overdetermined system of conditions they generate.

Author: Thomas Wolf

20.14.1 Purpose

The procedures CONLAW1, CONLAW2, CONLAW3, CONLAW4try to find conservation laws for a given single/system of differential equation(s) (ODEs or PDEs)

\begin{equation} u^{\alpha }_J = w^{\alpha }(x,u^{\beta },\ldots ,u^{\beta }_K,\ldots ) \label {conlaw-a1} \end{equation}
CONLAW1tries to find the conserved current \(P^i\) by solving
\begin{equation} \text {Div}\,P = 0 \quad \text {modulo} \quad (\ref {conlaw-a1}) \label {conlaw-a2} \end{equation}
directly. CONLAW3tries to find \(P^i\) and the characteristic functions (integrating factors) \(Q^{\nu }\) by solving
\begin{equation} \mbox {Div}\,P = \sum _{\nu } Q^{\nu }\cdot (u^{\nu }_J - w^{\nu }) \label {conlaw-a3} \end{equation}
identically in all \(u\)-derivatives. Applying the Euler operator (variational derivative) for each \(u^{\nu }\) on (20.57) gives a zero left hand side and therefore conditions involving only \(Q^{\nu }\). CONLAW4tries to solve these conditions identically in all \(u\)-derivatives and to compute \(P^i\) afterwards. CONLAW2does substitutions based on (20.55) before solving these conditions on \(Q^{\nu }\) and therefore computes adjoined symmetries. These are completed, if possible, to conservation laws by computing \(P^i\) from the \(Q^{\nu }\).

All four procedures have the same syntax. They have two parameters, both lists. The first parameter specifies the equations (20.55), the second specifies the computation to be done. One can either specify an ansatz for \(P^i, Q^{\nu }\) or investigate a general situation, only specifying the order of the characteristic functions or the conserved current.

20.14.2 Syntax

The procedures CONLAWi, \(i=1,2,3,4\), are called through

CONLAWi(problem, runmode);

where both parameters are lists.

The first parameter problem specifies the DEs to be investigated. It has the form {equations, ulist, xlist}where

equations is a list of equations, each of the form df(ui,…)=where the left-hand side df(ui,…)is selected such that

If CONLAW3or CONLAW4are run where no substitutions are made then the left-hand sides of equations can be df(ui,…)**n=where nis a number. No distinction is made between equations and constraints.

ulist is a list of function names, which can be chosen freely; the number of functions and equations need not be equal.

xlist is a list of variable names, which can be chosen freely.

The second parameter runmode specifies the calculation to be done. It has the form {minord, maxord, expl, flist, inequ}where

minord and maxord are respectively the minimum and maximum of the highest order of derivatives of \(u\)

expl is tor nilto indicate whether or not the characteristic functions q_ior conserved current may depend explicitly on the variables of xlist.

flist is a list of unknown functions in any ansatz for p_i, q_j, also all parameters and parametric functions in the equation that are to be calculated such that conservation laws exist; if there are no such unknown functions then flist is the empty list {}.

inequ is a list of expressions none of which may be identically zero for the conservation law to be found; if there is no such expression then inequ is an empty list {}.

The procedures CONLAWireturn a list of conservation laws {\(C_1,C_2,\ldots \)}; if no non-trivial conservation law is found they return the empty list {}. Each \(C_i\) representing a conservation law has the form {{\(P^1,P^2,\ldots \)},{\(Q^1,Q^2,\ldots \)}}.

An ansatz for a conservation law can be formulated by specifying one or more of the components \(P^i\) for CONLAW1, one or more of the functions \(Q^{\mu }\) for CONLAW2and CONLAW4, or one or more of \(P^i, Q^{\mu }\) for CONLAW3. The \(P^i\) are input as p_iwhere istands for a variable name, and the \(Q^{\mu }\) are input as q_i where istands for an index, which is the number of the equation in the input list equations with which q_iis multiplied.

There is a restriction in the structure of all the expressions for p_iand q_jthat are specified: they must be homogeneous linear in the unknown functions or constants which appear in these expressions. The reason for this restriction is not for CRACK to be able to solve the resulting overdetermined system but for CONLAWito be able afterwards to extract the individual conservation laws from the general solution of the determining conditions.

All such unknown functions and constants must be listed in flist (see above). The dependencies of such functions must be defined before calling CONLAWi. This is done with the command DEPEND, e.g.

     DEPEND f,t,x,u$

to specify \(f\) as a function of \(t,x,u\). If one wants to have \(f\) as a function of derivatives of \(u(t,x)\), say \(f\) depending on \(u_{txx}\), then one cannot write

     DEPEND f, df(u,t,x,2)$

but instead must write

     DEPEND f, u!`1!`2!`2$

if xlist has been specified as {t,x}, because tis the first variable and xis the second variable in xlist and uis differentiated once wrt. tand twice wrt. x; we therefore get u!`1!`2!`2. The character !is the escape character to allow special characters like ` to occur in an identifier name.

It is possible to add extra conditions like PDEs for \(P^i, Q^{\mu }\) as a list cl_condiof expressions that shall vanish.

Remarks

20.14.3 Flags and Parameters

     lisp(print_ := nil/0/1/ ...)$

print_:=nilsuppresses all CRACK output; for print_:=\(n\)(\(n\) an integer) CRACK prints only equations with at most \(n\) factors in their terms.

     crackhelp()$

shows other flags controlling the solution of the overdetermined PDE-system.

     off batch_mode$

solves the system of conditions with CRACK interactively.

     lisp(quasilin_rhs := t)$

reduces in the ansatz for \(P^i\) the order to \(m-1\) if the order of the right-hand side is \(m\). This can be used to speed up computations if the right-hand side is known to be linear in the highest derivatives (see the note above).

20.14.4 Requirements

     load_package crack, conlaw0, conlaw1$

where conlaw1can be replaced by conlaw2, conlaw3or conlaw4as appropriate.

20.14.5 Examples

Below a CRACK procedure nodepndis used to clean up after each run and delete all dependencies of each function in the list of functions in the argument of nodepnd. More details concerning these examples are given when running the file conlaw.tst(in the REDUCE packages/crackdirectory).

     lisp(print_:=nil);  % to suppress output from CRACK


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