The first 3 tables below describe algebraic systems that result as
conditions for a single vector equation to have a
higher order symmetry that also has the form of one vector equation.
One table deals with systems of 2 vector equations.
For more details see:
V.V. Sokolov and T. Wolf, Classification of integrable polynomial vector
evolution equations, J. Phys. A: Math. Gen. 34 (2001) 11139-11148.
Computations reported in all tables were performed with the computer algebra system REDUCE running with 120 MB on Linux on a 1.7 GHz Pentium 4 PC.
Conditions for a 2nd order vector equation with a 3rd order symmetry
The case lambda=2 can only have a linear 2nd and linear 3rd order equation and is therefore ommitted.
| lambda | 1 | 1/2 |
| # of unknowns in the equation | 2 | 4 |
| # of unknowns in the symmetry | 3 | 8 |
| total # of unknowns | 5 | 12 |
| # of conditions | 5 | 21 |
| total # of terms in all conditions | 9 | 80 |
| average # of terms in a condition | 1.8 | 3.8 |
| # of solutions | 0 | 0 |
| time to formulate alg. conditions | 0s | 1s |
| time to solve conditions | 0s | 0.4s |
Conditions for a 3rd order vector equation with a 5th order symmetry
The case lambda=2 can only have a linear 3rd order equation and is therefore ommitted.
| lambda | 1 | 1/2 |
| # of unknowns in the equation | 3 | 8 |
| # of unknowns in the symmetry | 9 | 27 |
| total # of unknowns | 12 | 35 |
| # of conditions | 26 | 129 |
| total # of terms in all conditions | 121 | 1603 |
| average # of terms in a condition | 4.6 | 12.4 |
| # of solutions | 2 | 1 |
| time to formulate alg. conditions | 1.3s | 17s |
| time to solve conditions | 0.5s | 32s |
| lambda | 2 | 1 | 1/2 |
| # of unknowns in the equation | 3 | 9 | 27 |
| # of unknowns in the symmetry | 7 | 24 | 82 |
| total # of unknowns | 10 | 33 | 109 |
| # of conditions | 34 | 198 | 927 |
| total # of terms in all conditions | 162 | 3125 | 52677 |
| average # of terms in a condition | 4.7 | 15.8 | 56.8 |
| # of solutions | 0 | 2 | 1 |
| time to formulate alg. conditions | 1.7s | 1m 1s | 13h 12m |
| time to solve conditions | 0.7s | 1m 15s | 2 days |
The following table describes the algebraic conditions for systems of 2 vector equations of 2nd order to have a 3rd order symmetry that has also the form of a system of 2 vector equations.
Conditions for a 2nd order 2-vector system with a 3rd order symmetry
The case lambda=2 does not allow non-linear 2nd order systems and is therefore ommitted.
| lambda | 1 | 1/2 |
| # of unknowns in the equation | 9 | 53 |
| # of unknowns in the symmetry | 15 | 155 |
| total # of unknowns | 24 | 208 |
| # of conditions | 78 | 1206 |
| total # of terms in all conditions | 242 | 28768 |
| average # of terms in a condition | 3.1 | 23.8 |
| time to formulate alg. conditions | 2.3s | 1h |
| time to solve conditions | 2.9s | 26m 15s |
| # of solutions | 2 | 7 |
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Thomas Wolf