2012/04/17 by Victor M. Buchstaber, Buchstaber, Victor M., Elena Yu. Bunkova +1
Mathematics · Physics and Astronomy · #33E05 #34M55 #35K05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:33E05 #msc:34M55 #msc:35K05
paper · pdf · doi:10.48550/arxiv.1204.3784
20 pages
arxiv created 2012/04/17 · arxiv updated 2012/04/18
This paper is devoted to the one-dimensional heat equation and the non-linear ordinary differential equations associated to it. We consider homogeneous polynomial dynamical systems in the n-dimensional space, n = 0, 1, 2, .... For any such system our construction matches a non-linear ordinary differential equation. We describe the algorithm that brings the solution of such an equation to a solution of the heat equation. The classical fundamental solution of the heat equation corresponds to the case n=0 in terms of our construction. Solutions of the heat equation defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n=2. The group SL(2, C) acts on the space of solutions of the heat equation. We show this action for each n induces the action of SL(2, C) on the space of solutions of the corresponding ordinary differential equations. In the case n=2 this leads to the well-known action of this group on the space of solutions of the Chazy-3 equation. An explicit description of the family of ordinary differential equations arising in our approach is given.