Bibikov P. Spherization of 1-jet space J^1R^n and point classification of first order partial differential equations, talk at The Workshop on Geometry of PDEs and Integrability, 14-18 October 2013, Teplice nad Becvou, Czech Republic (abstract)

Speaker: Pavel Bibikov

Title: Spherization of 1-jet space $J^{1}\mathbb {R} ^{n}$ and point classification of first order partial differential equations

Abstract:
Let $J^{1}\mathbb {R} ^{n}$ and $J^{0}\mathbb {R} ^{n}$ be 1- and 0-jet spaces of functions $f\colon \mathbb {R} ^{n}\to \mathbb {R}$, and $\pi _{1,0}\colon J^{1}\mathbb {R} ^{n}\to J^{0}\mathbb {R} ^{n}$ be the canonic projection.

First order differential equation can be considered as smooth function from $C^{\infty }(J^{1}\mathbb {R} ^{n})$ defined up to the multiplication on non-zero smooth functions from $C^{\infty }(J^{0}\mathbb {R} ^{n})$. In our talk we provide point classification of first order differential equations, which are polynomial in all derivatives.

The main difficulty in this classification is that point transformations act on the fibers of bundle $\pi _{1,0}$ by projective transformations. To overcome this difficulty we consider spherization $\mathbb {S} (J^{1}\mathbb {R} ^{n})\simeq T^{*}(J^{0}\mathbb {R} ^{n})\setminus \{0\}$ of 1-jet space $J^{1}\mathbb {R} ^{n}$ and rewrite our problem as follows: classify smooth functions on $T^{*}(J^{0}\mathbb {R} ^{n})$, which are homogeneous in fiber coordinates of cotangent bundle $\tau ^{*}\colon T^{*}(J^{0}\mathbb {R} ^{n})\to J^{0}\mathbb {R} ^{n}$, with respect to diffeomorphisms of $J^{0}\mathbb {R} ^{n}$.

This problem can be solved with the help of the methods developed in our joint works with Valentin Lychagin on classification of homogeneous forms.

Slides: Bibikov P. Symplectization of 1-jet space J^1R^n and point classification of PDEs (presentation at The Workshop on Geometry of PDEs and Integrability, 14-18 October 2013, Teplice nad Becvou, Czech Republic).pdf