- ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 27
on JRn+l, we get a system of two second-order ODES for f and w as functions
of r:J" +A = nw" /w, ww' f' + Aw^2 = ww" + (n - l)((w')^2 - 1).
Again, Proposition 1.15, together with equation (1.25), provides a first in-
tegral for this syste:rn:f" + n(w' f' /w) - (!')^2 - 2Af = C.
We can reduce the order of the system by again introducing variables
that are invariant under the symmetries of translating r and translating f:X=TW, • I y7nw. I -w !'.
Because simultaneous scaling of r and w is no longer a symmetry of the
system, we must also retain w as a variable, yielding the following first-
order system:(1.59)dw/dt = xw,
dx / dt = x^2 - xy + Aw^2 + ~ - 1,
dy/dt = x(y - nx) + Aw^2 ,where the parameter tis related tor by dt = w-^1 dr. (Notice that the first
integral is not invariant under the translation symmetries, so it does not
give an invariant manifold for this system.)
Clearly, solutions of the steady soliton system (1.48) are also solutions of
(1.59) for which w = 0. (Of course, in the steady case, the warping function
is not identically zero, but it is recovered by the quadrature d(log w) = x dt.)
To get a metric that closes up smoothly at the origin, we need a trajectory
that emerges from the singular point P = (0, 1,n) as r increases. (We take
w, x, y, in that order, as coordinates on the phase space JR^3 .)
Linearization at P shows that there is a 2-dimensional unstable manifold
passing through P, whose tangent space at P is spanned by the vectors
(1, O, 0) and (0, -1, n).^12 The trajectory corresponding to the steady soliton
lies in this surface and is tangent to (0, -1, n) at P. Of course, we are only
(^1) 2The linearization of (1.59) at P is
dw _
dt=w,
dx dt = ( (^2) -n ) x-y,
dy - -
dt = -nx+y.
Note that the last two equations appeared in the linearization for the steady soliton
equation and the eigenvalues of the matrix
(
2 - n -l )
-n 1
are 2 and 1 - n, with corresponding eigenvectors (-1, n) and (1, 1), respectively.