joint custody, domiciliary parent move

My ex and I live in New Orleans, LA, and have joint custody of our two kids, 13 & 16 yrs old. My ex is the domiciliary parent, and I get the kids roughly one week out of four. She is contemplating a move to another state, at the behest of her employer. Can I prevent her from taking the kids? I can think of several ways to go at this, all expensive (neither of us is in a position to spend megabucks on lawyers). The simplest thing would be if the threat of some legal action which has a good chance of succeeding stopped her from moving.
I should add that she doesn't want to go, but is being pressured by her employer. If I can make a credible threat, it may give her the ammunition she needs to say no.