Sorry, but in the 15 years I've dealt with cabinet making and installing I have never seen solid wood seperate. I have only seen laminates (wood, paper or plastic) separate or pull off. If you look at the picture here close enough you can clearly see what looks like a wood laminate, laminated to a substrate (usually mdf or particle board).
Hardly anything is built from solid wood anymore...just parts of it. No one wants to pay for it. Trust me.
If you could provide a better picture I am offering to provide a possible solution. I just need to see more to understand where it is separating.