Child Abandonment - READ FIRST!
Child Abandonment is an often misused term in Family Law situations. Under Criminal law, Child Abandonment refers to leaving a child without adult supervision for a period. So a child left in a car while the mother runs into a store, could get the mother charged with child abandonment.
But under Family Law there is no charge for Child Abandonment nor can one "file" for Child Abandonment (except in a very few states, more on that below). Most often Child Abandonment MAY be used as grounds for something else. If you are seeking to modify custody or visitation you may be able to use Child Abandonment as grounds for the modifications.
So please don't ask how you can file or charge the other parent with abandonment. Please describe your ultimate goal so we can give you the best answer.
Note: A very few states do use Child Abandonment as the charge against a "deadbeat" parent who doesn't support their child. If such a parent is not providing support, either court ordered or voluntary, they could be charged with Child Abandonment. But even in those cases your ultimate goal is to get or collect child support payments. And that's what your question should be about.