Maxwell equations (first set of equations)

Let us introduce the following Ore algebra A of partial differential operators in four variables t, (Subscript[x, 1]), (Subscript[x, 2]), (Subscript[x, 3]).

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

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

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

and the left A-module, finitely presented by R, defined by M = (A^(1×6))/(A^(1×4))R.

Let us write the equations of the system explicitly:

u={t,(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])};
EMF={B1@@u,B2@@u,B3@@u,ℰ1@@u,ℰ2@@u,ℰ3@@u}

sys=Thread[ApplyMatrix[R, EMF]==0]

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 identity, M is torsion-free and thus the first set of Maxwell equations is parametrized by Q, i.e:

Thread[EMF==ApplyMatrix[Q,{(Subscript[ξ, 1])@@u,(Subscript[ξ, 2])@@u,(Subscript[ξ, 3])@@u,(Subscript[ξ, 4])@@u}]]

We note the above parametrization involves four arbitrary functions (Subscript[ξ, 1]),...(Subscript[ξ, 4]). Let us now compute the rank of the system

OreRank[R,A]

From that we know that there exist parametrizations involving only three arbitrary functions. Let us compute some of them:

MatrixForm/@(paras=MinimalParametrizations[R,A])

For example, let us consider the first one.

sol=Thread[EMF==ApplyMatrix[paras[[1]],{(Subscript[ξ, 1])@@u,(Subscript[ξ, 2])@@u,(Subscript[ξ, 3])@@u}]]

ApplyMatrix[R,ApplyMatrix[paras[[1]],{(Subscript[ξ, 1])@@u,(Subscript[ξ, 2])@@u,(Subscript[ξ, 3])@@u}]]