Lie-Poisson structure (Subscript[E, 2])

Example from Bender, C. M., Dunne, G.V. and Mead, L. R., Underdetermined systems of partial differential equations. Journal of Mathematical Physics, vol 41 (2000), pp. 6388-6398.

Let us introduce the following Ore algebra A of partial differential operators (A is first Weyl algebra in three variables)

A=OreAlgebra[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]),Der[(Subscript[x, 1])], Der[(Subscript[x, 2])],Der[(Subscript[x, 3])]]

and the matrix R of partial differential operators, which defines the system

R0={{(Subscript[x, 1])Der[(Subscript[x, 3])],(Subscript[x, 2])Der[(Subscript[x, 3])],0},
{(Subscript[x, 2])Der[(Subscript[x, 1])]-(Subscript[x, 1])Der[(Subscript[x, 2])],-1,(Subscript[x, 2])Der[(Subscript[x, 3])]},
{-1,(Subscript[x, 1])Der[(Subscript[x, 2])]-(Subscript[x, 2])Der[(Subscript[x, 1])],(Subscript[x, 1])Der[(Subscript[x, 3])]}};
MatrixForm[R=ToOrePolynomial[R0,A]]

Let us compute the adjoint of R:

MatrixForm[Radj=Involution[R,A]]

Let us check whether or not M is torsion-free:

MatrixForm/@({Ann,Rp,Q}=Exti[Radj,A,1])

Since the first matrix is not identity, we deduce that M admits nontrivial torsion elements and thus the corresponding system admits autonomous elements τ[1], τ[2], τ[3], defined by:

aut=AutonomousElements[R, {u[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])],v[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])],w[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])]},τ,A,Relations->True]

The first list gives the definition of the autonomous elements T[1], τ[2], τ[3], the second list gives the equations they satisfy and the last list gives the relations between τ[1], τ[2], τ[3]. Note that τ[3] is zero.

Let us now introduce the second Weyl algebra B in three variables (Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])

B=OreAlgebra[Der[(Subscript[x, 1])], Der[(Subscript[x, 2])],Der[(Subscript[x, 3])]]

and the matrix R of partial differential operators in B, which defines the system

MatrixForm[R=ToOrePolynomial[R0,B]]

and the left B-module, finitely presented by R, defined by N = (B^(1×3))/(B^(1×3))R.

Let us compute the adjoint of R:

MatrixForm[Radj=Involution[R,B]]

Let us check whether or not N is torsion-free:

MatrixForm/@({Ann,Rp,Q}=Exti[Radj,B,1])

Since the first matrix is not identity, we deduce that N admits nontrivial torsion elements and thus the corresponding system admits autonomous elements τ[1], τ[2], defined by:

aut=AutonomousElements[R, {u[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])],v[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])],w[(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])]},τ,B,Relations->True]

The first list gives the definition of the autonomous elements τ[1], τ[2], the second list gives the equations they satisfy and the last list gives the relations between τ[1], τ[2]. Note that as we are now computing over the second Weyl algebra B, some denominators appear in the relations.