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Peer-reviewedCC BY 4.0Open Communications in Nonlinear Mathematical Physics

Integration of PDEs via C∞-structures

  • A J Pan-Collantes
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Abstract

Following classical method of differential constraints, interpreted in jet space, we present a constructive procedure for the integration of partial differential equations (PDEs). The approach consists of adjoining compatible auxiliary equations to reduce the PDE to a finitetype system in which the Vessiot distribution becomes involutive on an open set. Rather than classical solvable structures, we employ C∞-structures of distributions, a generalization with relaxed bracket conditions that are systematically easier to construct, yet still guarantee integration through successive Pfaffian equations. Three explicit examples illustrate the method: the linear wave equation with affine constraints, the Liouville equation, and a non-trivial rational nonlinear PDE. In every case the procedure yields an explicit family of solutions.

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Cite

BibTeXRIS
BibTeX
@article{pancollantes2026integration,
  title = {{Integration of PDEs via C∞-structures}},
  author = {A J Pan-Collantes},
  journal = {Open Communications in Nonlinear Mathematical Physics},
  year = {2026},
  month = jun,
  doi = {10.46298/ocnmp.18587},
  url = {https://hal.science/hal-05665940v2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Integration of PDEs via C∞-structures
AU  - A J Pan-Collantes
PY  - 2026
DA  - 2026/06/22
JO  - Open Communications in Nonlinear Mathematical Physics
DO  - 10.46298/ocnmp.18587
UR  - https://hal.science/hal-05665940v2
AB  - Following classical method of differential constraints, interpreted in jet space, we present a constructive procedure for the integration of partial differential equations (PDEs). The approach consists of adjoining compatible auxiliary equations to reduce the PDE to a finitetype system in which the Vessiot distribution becomes involutive on an open set. Rather than classical solvable structures, we employ C∞-structures of distributions, a generalization with relaxed bracket conditions that are systematically easier to construct, yet still guarantee integration through successive Pfaffian equations. Three explicit examples illustrate the method: the linear wave equation with affine constraints, the Liouville equation, and a non-trivial rational nonlinear PDE. In every case the procedure yields an explicit family of solutions.
SN  - 2802-9356
ER  -