2003/09/30 by Wouter Kager, Bernard Nienhuis, Leo P. Kadanoff · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Nonlinear Dynamics and Pattern Formation #Nonlinear Partial Differential Equations #math-ph #math.MP
paper · pdf · doi:10.1023/b:joss.0000022380.93241.24
published as J. Stat. Phys. 115:805-822 (2004) · 20 pages, 7 figures, LaTeX, one minor correction, and improved hyperrefs
openalex publication_date 2004/04/03 · arxiv created 2004/07/26 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this note, we solve the Loewner equation in the upper half-plane with forcing function xi(t), for the cases in which xi(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If xi(t)=2*(kappa*t)1/2, the trace is a straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))1/2, the behavior of the trace as t approaches 1 depends on the coefficient kappa. Our calculations give an explicit solution in which for kappa<4 the trace spirals into a point in the upper half-plane, while for kappa>4 it intersects the real axis. We also show that for kappa=9/2 the trace becomes a half-circle. The third case with forcing xi(t)=t gives a trace that moves outward to infinity, but stays within fixed distance from the real axis. We also solve explicitly a more general version of the evolution equation, in which xi(t) is a superposition of the values +1 and -1.