Shape maps for second order partial differential equations
Differential Geometry
2023-07-20 v1 Analysis of PDEs
Abstract
We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example.
Cite
@article{arxiv.1512.04990,
title = {Shape maps for second order partial differential equations},
author = {O. Rossi and D. J. Saunders and G. E. Prince},
journal= {arXiv preprint arXiv:1512.04990},
year = {2023}
}