Planar 2-homogeneous commutative rational vector fields
Date
2018-07-03
Authors
Alkauskas, Giedrius
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of 1-homogeneous rational functions W as curves W(x,y)=const. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high.
Description
Keywords
Translation equation, Flow, Rational vector fields, Linear ODEs, Autonomous non-linear ODEs, First order linear PDEs, Algebraic functions, Lie bracket, Commuting flows, Cremona groups, Wronskian
Citation
Alkauskas, G. (2018). Planar 2-homogeneous commutative rational vector fields. <i>Electronic Journal of Differential Equations, 2018</i>(138), pp. 1-21.
Rights
Attribution 4.0 International