(272h) Examples of Self-Similar Blowup: To Infinity and (Sometimes) Back

In a number of joint works with I.G. Kevrekidis, we have
explored the dynamics of systems that exhibit self-similarity.
The main perspective has been that of seeking to consider
self-similarity as a bifurcation problem whereby a frame
exists (co-exploding with the solution) where the solution
appears to be steady. Then, one can perform stability analysis
and comprehend the dynamical features of the solution.
This stability analysis also has some interesting implications, including
the potential shift of eigenvalues that relate to symmetries.

However, blowup is also an interesting dynamical problem. One
can envision case examples where the solution goes through infinity
and emerges on the other side. We have considered a few select
such examples to illustrate how to potentially compute with such
equations both at the ODE level and at the PDE level. A number
of open problems along these lines will be highlighted both at
the bifurcation level as well as at the dynamical evolution level.