If a mother leaves the country to care for a boyfriend for 10 to12 weeks and leaves her son with his dad, is that considered abandment in Arkansas? What can the father do?

Abandonment is when the child is left with no one to care for them, such as leaving child on street corner and walking away. As long as the child is left with an adult then it is not abandonment.

No its not. What does the father want to do? It MIGHT be grounds to change custody, but based on what little you have told us, I doubt it.

There is no criminal charge, it may be used to ask for a change in custody, so father can file in family court to get legal custody