There is a process by which you can ask the Court for a termination of parental rights. On the the grounds for a termination is abandonment. Under the statute, "It shall be presumed that the parent intends to abandon the child who has been left by his parent without provision for his identification or who has been left by his parent in the care and custody of another without any provision for his support, or without communication from such parent for a period of 6 months." RSA 170-C:5, I.
The parent who's rights are being terminated has a right to an attorney and the Court will appoint a guardian ad litem. Most commonly a termination happens in the context of an adoption (including a step-parent adoption).
These can be complicated cases and I'd recommend speaking with a family law attorney how the law applies to your case and the best way to proceed. I've included a link to the applicable statute which can provide you with some more information.