Critical Phenomena in Nonlinear Sigma Models

Steven L. Liebling and Eric W. Hirschmann
Southampton College
James Isenberg
University of Oregon

The Text

The text of the paper can be found posted at LANL as math-ph/9911020 and has been submitted to the Journal of Mathematical Physics.

Tuning of the Turok-Spergel Solution (Type I regime):

We consider initial data of the form Eq.(9) from our paper which represents the parameterized Turok-Spergel data. For epsilon=1, the evolution is known to be singular. For small epsilon, it is confirmed that the solution is non-singular, and certainly our evolutions support that conclusion. So I present a movie which shows the field chi(r,t) versus r. Four evolutions are shown corresponding (from bottom to top of the frame) of epsilon = 0.30, 0.35, 0.40, and 0.45.

One can see in this movie that for small epsilon, the solution appears perfectly regular. For large epsilon, the solution becomes singular.

If you continue with this tuning, you find an evolution which appears to "blow" off lots of energy, leaving behind an almost static solution. In fact, this static-looking solution looks remarkably like a solution found explicitly by assuming staticity. Here I show the results of one such tuning plotted along with the evolution of one of the static solutions. Shown is the field chi(r,t) (the tuned solution[magenta]; the explicitly static solution[light blue]) as well as the energy density as a function of r and t, rho(r,t) (the tuned solution[yellow]; the explicitly static solution[navy blue]).

One thing to note is that the static solution is remarkably static. That is, what you're seeing is the output from our code when given the static solution as initial data. In other words, this static solution looks static despite the inevitable "perturbations" provded by the numerics. This seems consistent with the fact that for "spread-out" static solutions, the instability is very slow-growing. More importantly, one sees the energy that is blown off, and that what remains appears to be the static solution.

Keep in mind here, that there is a continuous family of static solutions. As the tuning proceeds, it would seem that different static solutions are approached. Hence, it doesn't seem as if any of the static solutions represent any kind of attractor (e.g. an intermediate attractor).

Here's another recently made movie which I'm hoping will help explain at least the difference
that we're seeing between super- and sub- critical initial data. Shown is the energy density
for four different evolutions of the Turok-Spergel data:

which appear to bracket criticality. One note about this evolution is that you're seeing only radius in the range [0,300] whereas the computational domain consisted of [0,800] so that effects from the boundary can be somewhat isolated to large radius.

Explicitly Self-Similar Solutions:

Here I consider explicitly the self-similar solutions. I therefore take the self-similar solution for a given n and input it as initial data to the code and evolve. However, I can also add to the initial data a small amplitude Gaussian pulse which serves to (nonlinearly) perturb the self-similar solution.

For an ingoing Gaussian perturbation (r_0=0.5, del=0.2, amp=+/-0.01), I show the n=0, the n=1, and the n=2 solutions. Looking at just the n=0 and n=2 cases, one observes all three evolutions (amp=-0.01, amp=0, amp=+0.01) collapsing, suggesting that the solutions are not on threshold. However, the case of n=1 is unique in that the amp=+0.01 case collapses quite quickly while the amp=-0.01 disperses. These movies then suggest that only the n=1 solution sits on threshold, and presumably this would indicate the self-similar solution we are seeing in tuning of the Gaussian initial data is indeed the n=1 self-similar solution.

Last updated November 19, 1999.
Steve Liebling (home)