While the Mother does not exercise her visitation there is no such thing as filing for abandonment. You could file an OSC to modify the custody and visitation since she has had no contact for a period of time. You would file the OSC asking for sole physical and legal custody and for visitation to be terminated until she files with the court to modify the order.
If you are remarried and the Mother has had no contact for a period of at least one year your new spouse could request a step parent adoption and terminate the Mother parental rights.
Take a look at Family Code 7822 http://law.onecle.com/california/family/7822.html which states in part, Abandonment of Children - " (a) A proceeding under this part may be brought if any of the
(1) The child has been left without provision for the child's identification by the child's parent or parents.
(2) The child has been left by both parents or the sole parent in the care and custody of another person for a period of six months without any provision for the child's support, or without communication from the parent or parents, with the intent on the part of the parent or parents to abandon the child.
(3) One parent has left the child in the care and custody of the other parent for a period of one year without any provision for the child's support, or without communication from the parent, with the intent on the part of the parent to abandon the child.
(b) The failure to provide identification, failure to provide support, or failure to communicate is presumptive evidence of the intent to abandon. If the parent or parents have made only token efforts to support or communicate with the child, the court may
declare the child abandoned by the parent or parents. In the event that a guardian has been appointed for the child, the court may still declare the child abandoned if the parent or parents have failed to communicate with or support the child within the meaning of this
Note this answer does not constitute legal advice, and should not be relied on. Each situation is fact specific and court specific, and it is not possible to evaluate a legal problem without a comprehensive consultation and review of all the facts and court pleadings filed in the case. This answer does not create an attorney-client relationship
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