Systems of bi-linear algebraic equations with solutions

Supersymmetric evolution equations

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:

N=1, 1 fermion field f
weight(s): -> 13/225/237/24 9/2511/2613/27
weight(t): 1/2
1
3/2
2 x x
5/2
3 x x x
7/2
4
9/2
5 x


N=1, 1 boson field f
weight(s): -> 13/225/237/24 9/2511/2613/27
weight(t): 1/2
1
3/2
2 x
5/2
3 x
7/2
4
9/2
5


N=2, 1 fermion field f
weight(s): -> 13/225/237/24 9/2511/2613/27
weight(t): 1/2
1
3/2
2 x
5/2
3 x
7/2
4
9/2
5


N=2, 1 boson field b
weight(s): -> 13/225/237/24 9/2511/2613/27
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)

For weight(f) = 1 the 2nd order equation contains only the term f_xx and for weight(f) = 2 both, the equation and the symmetry, contain only the linear term.

weight(f) 1/2 3/2
# of unknowns in the equation42
# of unknowns in the symmetry73
total # of unknowns115
# of conditions164
total # of terms in all conditions536
average # of terms in a condition3.31.5
# of solutions20
time to solve conditions2.5s0.1s


N=1, 1 fermion field f, weight(t)= 2, weight(s)= 4

For weight(f) = 1 the 2nd order equation contains only the term f_xx and for weight(f) = 2 both, the equation and the symmetry, contain only the linear term.

weight(f) 1/2 3/2
# of unknowns in the equation42
# of unknowns in the symmetry135
total # of unknowns177
# of conditions309
total # of terms in all conditions12917
average # of terms in a condition4.31.9
# of solutions40
time to solve conditions10.4s0.8s


N=1, 1 fermion field f, weight(t)= 3, weight(s)= 5

For weight(f) = 2 the 3rd order equation contains only the term f_xxx.

weight(f) 1/2 1 3/2 5/2
# of unknowns in the equation7232
# of unknowns in the symmetry21674
total # of unknowns288106
# of conditions7914158
total # of terms in all conditions745375617
average # of terms in a condition9.52.73.72.1
# of solutions9130
time to solve conditions541s1.8s3.1s0.7s


N=1, 1 fermion field f, weight(t)= 3, weight(s)= 6

For weight(f) = 2 the ansatz for the equation is linear and for weight(f) = 3 the ansatz for the equation + symmetry is linear.

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

For weight(f) = 2 the 3rd order equation contains only the term f_xx.

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

For weight(f)=5/2, 7/2, 9/2 the ansatz for the equation + symmetry is still non-trivial but no non-trivial solutions exit. For integer weights >2 the ansatz is already trivial.

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

For weight(b) = 3/2, 2, 3 the ansatz is non-trivial but no non-trivial solution exists. For weight(b) = 5/2 the ansatz for equation and symmetry is linear.

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

For weight(b) = 3/2 no non-trivial combination of equation and symmetry exists. For weight(b) = 5/2 the ansatz for the equation is already linear and no non-trivial solution of equation + symmetry exists. For weight(b) = 4, 5 the ansatz for the equation is linear, the ansatz for the symmetry non-inear but no non-trivial solutions exist.

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
E-mail:
twolf@brocku.ca