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.
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 | 5 | 10 | 15 | 10 | 13 |
# of unknowns in the symmetry | 6 | 21 | 36 | 24 | 22 |
total # of unknowns | 11 | 31 | 51 | 34 | 35 |
# of conditions | 13 | 66 | 149 | 102 | 114 |
total # of terms in all conditions | 34 | 341 | 1093 | 529 | 694 |
average # of terms in a condition | 2.6 | 5.1 | 7.3 | 5.2 | 6.1 |
time to formulate alg. conditions | 0.5 s | 1.8 s | 8 s | 3.2 s | 6.3 s |
time to solve conditions | 0.5 s | 29 s | 29 s | 45 s | 22 s |
# of solutions | 0 | 3 | 0 | 0 | 1 |
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 | 5 | 10 | 15 | 10 | 13 |
# of unknowns in the symmetry | 12 | 39 | 79 | 54 | 66 |
total # of unknowns | 17 | 49 | 94 | 64 | 79 |
# of conditions | 26 | 123 | 313 | 215 | 276 |
total # of terms in all conditions | 77 | 770 | 3096 | 1462 | 2435 |
average # of terms in a condition | 2.9 | 6.3 | 9.9 | 6.8 | 8.8 |
time to formulate alg. conditions | 1 s | 5 s | 48 s | 13 s | 48 s |
time to solve conditions | 0.4 s | 1m 58s | 3m 44s | 1m 23s | 3m 40s |
# of solutions | 0 | 4 | 0 | 0 | 2 |
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 |
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.
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 |
This page is maintained by
Thomas Wolf