- CONSTRUCTING THE BRYANT STEADY SOLITON
FIGURE 1. Phase portrait for system (l.48), n = 2, drawn
using Maple.Since :ft log w = x, we have w = ( e-^2 t + 1 )-^1 /^2. Moreover
r = j wdt = ln (et + y' 1 + e2t)
= arcsinh (et).
19Hence et = sinh r, so that w (r) = ( csch^2 r + 1 )-^1 /^2 = tanh r, and we have
the cigar soliton.
EXERCISE 1.31. For n = 1, what is the solution with x, y --+ 1 as t --+
-oo, and x (t) increasing?
We will establish the existence of a complete steady gradient Ricci soliton
on ~n+l by doing phase plane analysis on the system (1.48). Note that the
stationary solutions of (l.48) satisfy y = nx and x^2 - 1 = 0, so that the
stationary points are (x, y) = (1, n) and (x, y) = (-1, -n). These are both