The expressions (15a), (16a), and (17a) were substituted into the evolution equation (15). Using the DIMSYM (Sherring, 1993) routines under Reduce a search was made for point symmetries. Only one generator was found for each case and these are listed below in the same order as (15), (16), and (17) respectively.
The existence of only one generator for each set of expressions indicated that the evolution equation (15) could be reduced to at most a third order ODE by use of point symmetries.
It was decided to select one pair expressions (17) for further
investigation as the invariants were the simplest to calculate from
the generators. More importantly, the constitutive relations (20)
display degeneracy and
Equation (15) may be simplified to give
where
From the generator (23) the invariants may be found be solving (Olver, 1993)
Substituting this into equation (25) we get
where Let
The evolution equation (29) is now a third order ODE. |