solvePDE(Module) -- solve linear systems of PDE with constant coefficients

Synopsis

Function: solvePDE
Usage:

solvePDE U
Inputs:
- U, a module, a submodule of a free module, or a matrix, or an ideal
Outputs:
- a list,

Description

A homogeneous system of $l$ linear partial differential equations for a function $\psi \colon \RR^n \to \CC^k$ is encoded by a $(k \times l)$ matrix with entries in a polynomial ring. For example, if $n=4, k=2, l=3$, the PDE system $$ \frac{\partial^2 \psi_1}{\partial z_1 \partial z_3} + \frac{\partial^2 \psi_2}{\partial z_1^2} = \frac{\partial^2 \psi_1}{\partial z_1 \partial z_2} + \frac{\partial^2 \psi_2}{\partial z_2^2} = \frac{\partial^3 \psi_1}{\partial z_1^2 \partial z_2} + \frac{\partial^3 \psi_2}{\partial z_1^2 \partial z_4} =0 $$ is encoded by the matrix $$ M = \begin{bmatrix} \partial_{1} \partial_{3} & \partial_{1} \partial_{2} & \partial_{1}^2 \partial_{2}\\ \partial_{1}^2 & \partial_{2}^2 & \partial_{1}^2 \partial_{4} \end{bmatrix}, $$ or more precisely, by the submodule $U$ of $(K[\partial_1,\partial_2,\partial_3,\partial_4])^2$ generated by the columns of $M$.

By the Ehrenpreis-Palamodov fundamental principle, solutions to such PDE are of the form $$ \phi(\mathbf z) = \sum_{i = 1}^s \sum_{j=1}^{m_i} \int_{V_i} B_{i,j}(\mathbf{x}, \mathbf{z}) e^{\mathbf{x}^t \mathbf{z}} \mu_{i,j}(\mathbb{x}) $$ where $\mu_{i,j}$ are complex valued measures.

The function solvePDE computes the algebraic varieties $V_i$ and Noetherian multipliers $B_{i,j}(\mathbf x, \mathbf z)$. The input is either a matrix $M$ or a module $U$, where the $\partial_i$ is replaced by $x_i$. The output is a list of $s$ pairs. For the $i$th pair, the first entry is the prime ideal of $V_i$. The second entry is the list $B_{i,1}, \dotsc, B_{i,m_j}$ of vectors of polynomials in $2n$ variables, where the symbol $\mathbf{z}$ is replaced by the symbol $\mathbf{\mathtt{d}x}$.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : M = matrix{{x_1*x_3, x_1*x_2, x_1^2*x_2}, {x_1^2, x_2^2, x_1^2*x_4}}

o2 = | x_1x_3 x_1x_2 x_1^2x_2 |
     | x_1^2  x_2^2  x_1^2x_4 |

             2      3
o2 : Matrix R  <-- R

i3 : U = image M

o3 = image | x_1x_3 x_1x_2 x_1^2x_2 |
           | x_1^2  x_2^2  x_1^2x_4 |

                             2
o3 : R-module, submodule of R

i4 : sols = solvePDE M

                                                                            
o4 = {{ideal x , {| 1 |}}, {ideal (x , x ), {| -x_1 |}}, {ideal (x , x ), {|
              1   | 0 |             4   2    |  x_3 |             3   2    |
     ------------------------------------------------------------------------
                                                                            
     1 |}}, {ideal (x , x ), {| x_2dx_1 |}}, {ideal (x , x ), {| 0 |, |   0 
     0 |             3   1    |    -1   |             2   1    | 1 |  | dx_1
     ------------------------------------------------------------------------
                                           2                       2
     |, |   0  |, |     0    |}}, {ideal (x  - x x , x x  - x x , x  - x x ),
     |  | dx_2 |  | dx_1dx_2 |             2    1 4   1 2    3 4   1    2 3  
     ------------------------------------------------------------------------
     {| -x_4 |}}}
      |  x_2 |

o4 : List

i5 : netList sols

     +-----------------------------------------+-----------------------------------------+
o5 = |ideal x                                  |{| 1 |}                                  |
     |       1                                 | | 0 |                                   |
     +-----------------------------------------+-----------------------------------------+
     |ideal (x , x )                           |{| -x_1 |}                               |
     |        4   2                            | |  x_3 |                                |
     +-----------------------------------------+-----------------------------------------+
     |ideal (x , x )                           |{| 1 |}                                  |
     |        3   2                            | | 0 |                                   |
     +-----------------------------------------+-----------------------------------------+
     |ideal (x , x )                           |{| x_2dx_1 |}                            |
     |        3   1                            | |    -1   |                             |
     +-----------------------------------------+-----------------------------------------+
     |ideal (x , x )                           |{| 0 |, |   0  |, |   0  |, |     0    |}|
     |        2   1                            | | 1 |  | dx_1 |  | dx_2 |  | dx_1dx_2 | |
     +-----------------------------------------+-----------------------------------------+
     |        2                       2        |                                         |
     |ideal (x  - x x , x x  - x x , x  - x x )|{| -x_4 |}                               |
     |        2    1 4   1 2    3 4   1    2 3 | |  x_2 |                                |
     +-----------------------------------------+-----------------------------------------+

This output reveals that the general solution to the example system above consists of nine summands, one of which is $$ \phi(\mathbb z) = \int_{V(x_3,x_1)} \begin{bmatrix} z_1 x_2 \\ -1 \end{bmatrix} e^{x_1 z_1 + x_2 z_2 + x_3 z_3 + x_4 z_4} \, d\mu(x_1,x_2,x_3,x_4) $$

The total number of Noetherian multipliers is equal to the arithmetic multiplicity of the module $U$.

i6 : amult U == sum(sols / last / (l -> #l))

o6 = true

Note that the output of solvePDE can be interpreted as a differential primary decomposition.

References

Ait El Manssour, R., Härkönen, M., Sturmfels, B. (2021). Linear PDE with constant coefficients

Ways to use this method: