Systems of vector-scalar evolutionary PDE systems with higher order symmetries

The algebraic systems reported below result as conditions for a system of one vector and one scalar equation to have higher order symmetries also in the form of one vector and one scalar equation.

Computations reported in the following tables were performed with the package CRACK running in the computer algebra system REDUCE with 120 MB under Linux on a 1.7 GHz Pentium 4 PC. In CRACK one can customize solving strategies to the type of problem at hand by selecting a set of modules to be used and by specifying their relative priority, i.e. in which order they should be tried. The only directly problem specific hints that were given to CRACK were to do case distinctions whether some unknowns vanish or not (which are the coefficients of the leading derivatives in the corresponding integrable systems and their derivatives). More details on which modules were used and in which order can be obtained from Thomas Wolf (twolf@brocku.ca).

The quoted numbers of unknowns, equations and terms are the numbers after already first simplifications have been performed.

Conditions for 2nd order systems with a 3rd order symmetry

lambda of system, symmetry2, 2 1, 1 1/2, 1/2 1/3, 2/3 2/3, 1/3
# of unknowns in the system5 10 15 10 13
# of unknowns in the symmetry6 21 36 24 22
total # of unknowns11 31 51 34 35
# of conditions13 66 149 102 114
total # of terms in all conditions34 341 1093 529 694
average # of terms in a condition2.6 5.1 7.3 5.2 6.1
time to formulate alg. conditions0.5 s 1.8 s 8 s 3.2 s 6.3 s
time to solve conditions0.5 s 29 s 29 s 45 s 22 s
# of solutions0 3 0 0 1


Conditions for 2nd order systems with a 4th order symmetry

lambda of system, symmetry 2, 2 1, 1 1/2, 1/2 1/3, 2/3 2/3, 1/3
# of unknowns in the system5 10 15 10 13
# of unknowns in the symmetry12 39 79 54 66
total # of unknowns17 49 94 64 79
# of conditions26 123 313 215 276
total # of terms in all conditions77 770 3096 1462 2435
average # of terms in a condition2.9 6.3 9.9 6.8 8.8
time to formulate alg. conditions1 s 5 s 48 s 13 s 48 s
time to solve conditions0.4 s 1m 58s 3m 44s 1m 23s 3m 40s
# of solutions0 4 0 0 2


Conditions for 3rd order systems with a 5th order symmetry

lambda of system, symmetry 2, 2 1, 1 1/2, 1/2 1/3, 2/3 2/3, 1/3
# of unknowns in the system 6 21 36 24 22
# of unknowns in the symmetry 17 74 164 115 126
total # of unknowns 23 95 200 139 148
# of conditions 50 386 1154 798 955
total # of terms in all conditions 218 5000 27695 12694 17385
average # of terms in a condition 4.3 13 24 16 18
time to formulate alg. conditions 5s 2m 52s 2h 7m 23m 45s 41m 18s
time to solve conditions 6.5s 5h 47m 1 day 1h 20m 1h 7m
# of solutions 4 25 2 0 2


Conditions for 1st order systems with a 2nd order symmetry

For [lambda(u),lambda(v)]=[2,2], [1,1], [2/3,1/3], [1/3,2/3], [1/2,1/2] solutions do either not have a system of 1st order, or the symmetry is not of 2nd order or system and symmetry are triangular, i.e. u_t, u_s do not involve v or v_t, v_s do not involve u.

Conditions for 1st order systems with a 3rd order symmetry

For [lambda(u),lambda(v)]=[2,2], [2/3,1/3], [1/3,2/3] solutions do either not have a system of 1st order, or the symmetry is not of 3rd order or system and symmetry are triangular, i.e. u_t, u_s do not involve v or v_t, v_s do not involve u.

The following table lists only solution having a 1st order system, a 3rd order symmetry and for u_t or u_s to involve v and for v_t or v_s to involve u.

lambda of system, symmetry 1, 1 1/2, 1/2
# of unknowns in the system 5 6
# of unknowns in the symmetry 21 36
total # of unknowns 26 42
# of conditions 35 66
total # of terms in all conditions 98 221
average # of terms in a condition 2.8 3.3
time to formulate alg. conditions 1s 2.3s
time to solve conditions 2.7s 9.6s
# of solutions 1 1


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This page is maintained by

Thomas Wolf
E-mail: twolf@brocku.ca
and was last modified on 25 Oct 2003.