They definatly can self-split. I had a fungia plate that nearly died and came back with 3-4 mouths; itself pinched off one of the extra mouths and is now about nickle sized. My other plate I bought shortly after my other one started recovering is also now showing signs of an upcoming split, usually starts similar to the way a mountain does, raising up into a ridge and then ultimatly sloughs off one side. Give it a few weeks you should have two.