find the (possible) minimal differential order that a LHPDEs system is in involution
a LHPDE object that is in rif-reduced form with respect to a total degree ranking (see IsTotalDegreeRanking, IsRifReduced)
The OrderOfInvolution method returns the order at which a LHPDE object is involutive, or a bound for this order. Note that the LHPDE object must be rif-reduced, with respect to a total degree ranking (see IsTotalDegreeRanking).
So far, the only implementation in this method is the Mansfield bound (ref: E. Mansfield. A Simple Criterion for Involutivity. Journal of the London Mathematics Society 54: 323-345,1996). This gives an upper bound for the order of involutivity. It is often -- but not always -- exact.
This method is associated with the LHPDE object. For more detail, see Overview of the LHPDE object.
Create an LHPDE object (these are the determining equations for the Euclidean group E(2))...
E2 ≔ LHPDE⁡∂2∂y2⁢ξ⁡x,y=0,∂∂x⁢η⁡x,y=−∂∂y⁢ξ⁡x,y,∂∂y⁢η⁡x,y=0,∂∂x⁢ξ⁡x,y=0,indep=x,y,dep=ξ,η
Create another LHPDE object that is rif-reduced with respect to a total degree ranking....
E2red ≔ RifReduce⁡E2,ξ,η
Now this can be checked for the order at which it becomes involutive....
The OrderOfInvolution command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
LHPDE (Object overview)
Download Help Document