Out of curiosity, we're always told that American options are like a melting/solidification problem because of the free boundary (although I think cuch pointed out that water gets mushy so not the melting/solidification of water)
Is there an analogous physical problem for Bermudan options?
Interesting question. Just speculating -- perhaps if one posited a one-dimensional water-ice interface in which the diffusion coefficient of the water cycled from 0 to D>0 repeatedly, one might get something analogous. So, nothing happens when the diffusion is zero, but then melting proceeds normally when the diffusion is D>0. (Maybe [$]D(t)[$] needs to be periodic with delta function spikes??). I know it sounds quite artificial, but then a one-dimensional water-ice interface (the standard solvable Stefan problem) is also arguably artificial/toy problem.
In my last book, I asked and answered somewhat of a reverse question: is there an exact
financial interpretation/analog of the standard 1D Stefan problem? American-style Black-Scholes puts are a rough analog but not an exact map. (Proof: the std Stefan problem is exactly solvable and the American put problem is not; plus the two free boundaries are well-known to be different). Anyway, I was able to come up with a financial analog that, IMO, wasn't too much of a stretch. It's all in Chapter 9: 'Back to basics: update on the discrete dividend problem'.