In Chuck Yeager's autobiography, he went into a little bit of detail as to why the X-1 was able to break the sound barrier safely where as the P-51D he flew during WWII could not have safely broken the sound barrier. He mentioned it had to do with the controls... Something about flow separation, as I recall, and the elevator not working as a result in most aircraft before the X-1.

I was wondering if someone could explain this though? What advances where made during X-1 development that made it stable at Mach 1?