Exact Solutions for Loewner Evolutions
| dc.creator | Kager, Wouter | |
| dc.creator | Nienhuis, Bernard | |
| dc.creator | Kadanoff, Leo P. | |
| dc.date | 2003-09-02 | |
| dc.date | 2004-07-26 | |
| dc.date.accessioned | 2026-07-07T04:30:31Z | |
| dc.date.available | 2026-07-07T04:30:31Z | |
| dc.description | 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. | |
| dc.description | 20 pages, 7 figures, LaTeX, one minor correction, and improved hyperrefs | |
| dc.identifier | https://arxiv.org/abs/math-ph/0309006 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0309006 | |
| dc.identifier | J. Stat. Phys. 115:805-822 (2004) | |
| dc.identifier | doi:10.1023/B:JOSS.0000022380.93241.24 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57490 | |
| dc.subject | Mathematical Physics | |
| dc.title | Exact Solutions for Loewner Evolutions | |
| dc.type | text |