"GeM":
A Maple module for symmetry
and conservation law computation for PDEs/ODEs

Alexei F. Cheviakov (A. Shevyakov)
University of Saskatchewan
 

     Since September 2004, I have been developing software for analysis of point symmetries and conservation laws of any given system of Ordinary or Partial Differential Equations, with minimal human intervention. Maple pas been chosen as a symbolic platform.

     The GeM software is designed to automate generation of determining equations and related operations, in order to compute symmetries and conservation laws for any ODE/PDE system, generally without limitations in DE order and number of variables.

     ODE/PDE systems containing arbitrary functions and/or constants can be analyzed, and classes of functions for which additional symmetries / conservation laws occur can be isolated.

     GeM output (determining equations) is usually fed to Maple "rifsimp" (a highly stable routine for differential reduction) which simplifies determining equations (and performs case splits when the given system contains arbitrary functions and/or constants.)

     GeM also contains special routines to output computed symmetries as well as fluxes/densities of computed conservation laws.

     Licensing: The GEM module is free for all research or study purposes. However I ask that a proper reference is made in any work that uses it.
 

Download: 

• GeM Module: Maple input file(mpl):  
  please request by email
  
• DESCRIPTION (PDF file): gem31desc_v03.pdf
  (nothing important has changed in v.32 compared to v.31.)

Maple version compatibility:

• Maple 12-16, tested

Current capabilities:

• Input: any ODE/PDE system, may involve arbitrary constants and functions.
• Automatic computation of differential consequences to necessary order.
• Computation of local (point, contact and higher-order) symmetries.
• Computation of multipliers of local conservation laws.
• Computation of fluxes/densities of local conservation laws.

Computation examples:

• Point symmetries of polytropic gas dynamics: 2D case
  symm_E_polytr_gam3.mws
• Point symmetries of a nonlinear heat equation
  symm_exp_heat.mws
• Point symmetry classification of nonlinear diffusion equations
  symmclass_diffus.mws
   
• Some higher-order symmetries of the KdV
  symm_KdV.mws
   
• Local conservation laws of polytropic gas dynamics: 2D case
  conslaw_E_polytr_gam3.mws
• Local conservation laws of static plasma equations: 3D case
  conslaw_static_MHD_3D.mws
• Local conservation laws of the KdV:
  conslaw_KdV.mws
   
• Integrating factors and first integrals of a nonlinear ODE
  intfact_ODE_eg.mws

Bugs/problems of GeM

• Please report to *shevyakov at math point usask point ca*.
  Will try to respond promptly...

1. I have tested GeM using only Maple Classic input (not the 2D input). For 32-bit systems, I would use only Maple Classic (Gold leaf) (MWS) worksheets/environment.
    
2. All examples using GeM routines essentially rely on the "rifsimp" routine of Maple. It is often handy to use Maple "pdsolve" to solve PDE systems, as shown in all online examples.
    

3. "rifsimp" seems to be stable and correct in all versions of Maple (8 and above).
    

4. "rifsimp casesplit". In classification problems, when "rifsimp(casesplit)" is used, Maple versions up to 11 yield different case trees for different runs of identically the same programs. This is due to the fact that sets are ordered depending on computer memory configuration. Starting from Maple 12, sets are ordered, and this issue seems to be resolved.
    
5. Maple "pdsolve" issues. Though GeM does not depend on Maple pdsolve routine, it is often handy to use it to solve determining equations. The following issues are known with pdsolve. The developers have been informed, and hopefully, the problems will be resolved in Maple 14.
    

   1. Note that "pdsolve" is known to return, in some cases, only particular solutions, and it will not tell whether the solution it returned is general or not.
       

   2. A number of bugs have been found in "pdsolve" over time. Maple versions 12 and 13 are known to have such bugs; the known ones have been corrected in ver.14.
       
   3. Starting from ver.14, Maple "pdsolve" has a special routine to compute special forms of lower-dimensional solution subspaces. It can often be handy in symmetry analysis. See DEtools[particularsol].

• Adjoint symmetries
• Approximate symmetries and conservation laws
• Computation of Frechet linearization and adjoint linearization of a given DE system
• Symmetry computations for linear DE systems
• Use of symmetries to generate invariant solutions
  

• GeM v. 021.b3 
   
• GeM v. 019.y