Preliminary Comments
All weights mentioned in the tables below are based on a weight of x equal 1.
If casses are labeled as having weight(t)=n then this means that each term of the right hand side of f_t=... has the same weight as f_nx. The highest x derivative may be lower than n but each term has to have exactly this weight. In the evolutionary equations the weight is called weight(t) whereas in the symmetry it is called weight(s).
Only evolution equations and their symmetries are computed which satisfy the following non-triviality conditions:
If entries in tables are empty then the corresponding computations have not been completed yet.
So far the following cases have been considered:
weight(s): -> | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 7 | |
weight(t): 1/2 | ||||||||||||||
1 | ||||||||||||||
3/2 | ||||||||||||||
2 | x | x | ||||||||||||
5/2 | ||||||||||||||
3 | x | x | x | |||||||||||
7/2 | ||||||||||||||
4 | ||||||||||||||
9/2 | ||||||||||||||
5 | x |
weight(s): -> | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 7 | |
weight(t): 1/2 | ||||||||||||||
1 | ||||||||||||||
3/2 | ||||||||||||||
2 | x | |||||||||||||
5/2 | ||||||||||||||
3 | x | |||||||||||||
7/2 | ||||||||||||||
4 | ||||||||||||||
9/2 | ||||||||||||||
5 |
weight(s): -> | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 7 | |
weight(t): 1/2 | ||||||||||||||
1 | ||||||||||||||
3/2 | ||||||||||||||
2 | x | |||||||||||||
5/2 | ||||||||||||||
3 | x | |||||||||||||
7/2 | ||||||||||||||
4 | ||||||||||||||
9/2 | ||||||||||||||
5 |
weight(s): -> | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 7 | |
weight(t): 1/2 | ||||||||||||||
1 | ||||||||||||||
3/2 | ||||||||||||||
2 | x | |||||||||||||
5/2 | ||||||||||||||
3 | x | |||||||||||||
7/2 | ||||||||||||||
4 | ||||||||||||||
9/2 | ||||||||||||||
5 |
N=1, 1 fermion field f, weight(t)= 2, weight(s)= 3 (i.e. up to 2nd order equations with up to 3rd order symmetries)
weight(f) | 1/2 | 3/2 |
# of unknowns in the equation | 4 | 2 |
# of unknowns in the symmetry | 7 | 3 |
total # of unknowns | 11 | 5 |
# of conditions | 16 | 4 |
total # of terms in all conditions | 53 | 6 |
average # of terms in a condition | 3.3 | 1.5 |
# of solutions | 2 | 0 |
time to solve conditions | 2.5s | 0.1s |
N=1, 1 fermion field f, weight(t)= 2, weight(s)= 4
weight(f) | 1/2 | 3/2 |
# of unknowns in the equation | 4 | 2 |
# of unknowns in the symmetry | 13 | 5 |
total # of unknowns | 17 | 7 |
# of conditions | 30 | 9 |
total # of terms in all conditions | 129 | 17 |
average # of terms in a condition | 4.3 | 1.9 |
# of solutions | 4 | 0 |
time to solve conditions | 10.4s | 0.8s |
N=1, 1 fermion field f, weight(t)= 3, weight(s)= 5
weight(f) | 1/2 | 1 | 3/2 | 5/2 |
# of unknowns in the equation | 7 | 2 | 3 | 2 |
# of unknowns in the symmetry | 21 | 6 | 7 | 4 |
total # of unknowns | 28 | 8 | 10 | 6 |
# of conditions | 79 | 14 | 15 | 8 |
total # of terms in all conditions | 745 | 37 | 56 | 17 |
average # of terms in a condition | 9.5 | 2.7 | 3.7 | 2.1 |
# of solutions | 9 | 1 | 3 | 0 |
time to solve conditions | 541s | 1.8s | 3.1s | 0.7s |
N=1, 1 fermion field f, weight(t)= 3, weight(s)= 6
weight(f) | 1/2 | 1 | 3/2 | 5/2 |
# of unknowns in the equation | 7 | 2 | 3 | 2 |
# of unknowns in the symmetry | 35 | 9 | 12 | 6 |
total # of unknowns | 42 | 11 | 15 | 8 |
# of conditions | 123 | 23 | 28 | 13 |
total # of terms in all conditions | 1495 | 75 | 119 | 30 |
average # of terms in a condition | 12.2 | 3.3 | 4.2 | 2.3 |
# of solutions | 5 | 0 | 1 | 0 |
time to solve conditions | 24m16s | 4.1s | 10.8 | 1.4s |
N=1, 1 fermion field f, weight(t)= 3, weight(s)= 7
weight(f) | 1/2 | 1 | 3/2 | 5/2 |
# of unknowns in the equation | 7 | 2 | 3 | 2 |
# of unknowns in the symmetry | 55 | 14 | 16 | 8 |
total # of unknowns | 62 | 8 | 19 | 10 |
# of conditions | 189 | 37 | 39 | 19 |
total # of terms in all conditions | 2829 | 143 | 193 | 49 |
average # of terms in a condition | 15 | 3.8 | 4.9 | 2.6 |
# of solutions | 10 | 0 | 4 | 0 |
time to solve conditions | 1h 21m 54s | 9.8s | 17.3s | 2.7s |
N=1, 1 fermion field f, weight(t)= 5, weight(s)= 7
weight(f) | 1/2 | 1 | 3/2 | 2 |
# of unknowns in the equation | 21 | 6 | 7 | 2 |
# of unknowns in the symmetry | 55 | 14 | 16 | 6 |
total # of unknowns | 76 | 20 | 23 | 8 |
# of conditions | 409 | 78 | 87 | 22 |
total # of terms in all conditions | 15955 | 781 | 896 | 72 |
average # of terms in a condition | 39 | 10 | 10.3 | 3.3 |
# of solutions | 2 | 8 | 1 | |
time to solve conditions | 7m 36s | 12m 34s | 4s |
N=1, 1 boson field b, weight(t)= 2, weight(s)= 3
For weight(b) = 1/2 and = 1 the ansatz for the equation is classical, for the symmetry non-classical but no non-classical and non-trivial solutions exist.
For weight(b) = 3/2 the ansatz for the 2nd order equation is already linear and the ansatz for equation and symmetry are classical.
N=1, 1 boson field b, weight(t)= 3, weight(s)= 5
For weight(b) = 3/2 and = 2 the ansatz for the equation is classical, for the symmetry non-classical but no non-classical and non-trivial solutions exist.
For weight(b) = 5/2 the ansatz for the equation is linear.
weight(b) | 1/2 | 1 |
# of unknowns in the equation | 6 | 6 |
# of unknowns in the symmetry | 19 | 16 |
total # of unknowns | 25 | 22 |
# of conditions | 72 | 51 |
total # of terms in all conditions | 422 | 243 |
average # of terms in a condition | 5.8 | 4.7 |
# of solutions | 2 | 3 |
time to solve conditions | 142s | 44s |
N=2, 1 fermion field f, weight(t)= 2, weight(s)= 3
For weight(f) = 1 no non-trivial combination of equation and symmetry exists. For weight(f) = 2 already the ansatz for both, the equation and the symmetry is linear. For weight(f) = 5/2 the ansatz is non-trivial but no non-trivial solutions exist.
For weight(f) = 3/2 solution 1 in the table shows a linear 2nd order PDE having among others a nonlinear 1st order symmetry. For this solution is f_ts=0=f_st.
weight(f) | 1/2 | 3/2 |
# of unknowns in the equation | 11 | 4 |
# of unknowns in the symmetry | 27 | 8 |
total # of unknowns | 38 | 12 |
# of conditions | 21 | |
total # of terms in all conditions | 66 | |
average # of terms in a condition | 2.9 | |
# of solutions | 2 | |
time to solve conditions | 4.6s |
N=2, 1 fermion field f, weight(t)= 3, weight(s)= 5
For weight(f) = 2 the ansatz is non-trivial but no non-trivial solutions exit. For weight(f) = 5/2 we have 2 solutions. For weight(f) = 3, 4, .. the ansatz for equation + symmetry is linear. For weight(f) = 7/2 we have 2 solutions. For weight(f) = 9/2 we have no non-trivial solutions.
weight(f) | 1/2 | 1 | 3/2 | 5/2 | 7/2 |
# of unknowns in the equation | 5 | 8 | 4 | 2 | |
# of unknowns in the symmetry | 27 | 27 | 12 | 8 | |
total # of unknowns | 32 | 35 | 16 | 10 | |
# of conditions | 150 | 127 | 42 | 14 | |
total # of terms in all conditions | 1263 | 1182 | 186 | 40 | |
average # of terms in a condition | 8.4 | 9.3 | 4.4 | 2.8 | |
# of solutions | 3 | 2 | 2 | ||
time to solve conditions | 47m 25s | 16.4s | 1.7s |
N=2, 1 boson field b, weight(t)= 2, weight(s)= 3
weight(b) | 1/2 | 1 |
# of unknowns in the equation | 6 | 6 |
# of unknowns in the symmetry | 17 | 15 |
total # of unknowns | 23 | 21 |
# of conditions | 76 | 53 |
total # of terms in all conditions | 467 | 268 |
average # of terms in a condition | 6.1 | 5 |
# of solutions | 2 | 5 |
time to solve conditions | 139s | 41s |
N=2, 1 boson field b, weight(t)= 3, weight(s)= 5
weight(b) | 1 | 2 | 3 |
# of unknowns in the equation | 5 | 3 | |
# of unknowns in the symmetry | 16 | 11 | |
total # of unknowns | 21 | 14 | |
# of conditions | 69 | 27 | |
total # of terms in all conditions | 415 | 98 | |
average # of terms in a condition | 6 | 3.6 | |
# of solutions | 3 | 2 | |
time to solve conditions | 84s | 6s |
This page is maintained by
Thomas Wolf