She will either need the father's written, signed and notarized permission to take them out of Canada, a court order from a judge granting her permission if the father refuses to give it. Somehow, with the details that you have given, I don't think a court will have an issue granting the mother permission to leave Canada with the children. The courts will normally do what is in the best interests of the children. If the father is violent and a danger to the children, it should be a no-brainer. If it isn't that bad, the father may get permission to have the children during the summers and some holidays and such. If there is some urgency to this situation an emergency motion could be filed with the court.
As far as getting into the U.S., most of the time, Canadians do not have a problem entering the U.S. using just their passports, but technically, if they are going to stay, they should process an immigrant visa in Canada before coming to the U.S. The problem is that those take 6 to 9 months or more to process. So it may be worth the risk to bring them (with father's or a court's permission) into the U.S. with just their Canadian passports and have the mother then file for Adjustment of Status while they are in the U.S.
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