In Arrival, as I understood Louise's husband had abandoned her and her daughter. However, I feel like I've missed on the true reason why. We see that he is not with the family from the early flashbacks so what exactly happened there?
First of all, they weren't flashbacks. One of the very pivotal points of the movie is that somehow, by learning the language of the heptapods one is able to see the future. As an audience we start to see scenes with her daughter and we are meant to think they are flashbacks, but as the movie proceeds we learn along with Louise that she is having visions of the future. She is seeing that in the future she will have a daughter who will die at an early age.
So to your question. Why did the husband leave? I'm going to use someone else's wording here because I couldn't have explained it much better myself.
At the end of the movie, Louise says she realizes why her husband left her—or why he will leave her in the future. Why does he leave her?
Because she knew even before they became a couple that their daughter would eventually die of an unnamed disease. When Ian asks if Louise wants to make a baby, she agrees, despite the knowledge that she and Ian will both outlive their kid. Years later, Louise finally tells Ian that their daughter is going to die, and he isn’t ready to hear it: Louise explains to her daughter that the girl’s father left because he thinks she “made the wrong choice.”
Louise only had one husband - Ian Donnelly - the scientist she worked with in communicating with the aliens. What happens is that Louise ends up realizing that the visions she has of her daughter are in the future, not the past. After the incident with the aliens, she marries Ian, has a daughter, and then watches her daughter die from cancer. She doesn't tell Ian what's going to happen, and Ian resents it once he realizes what's going on. That leads to their divorce.