Is it permissible for a husband to travel alone for vacation after his wife refused to travel with him and their children?

The basic principle is that a wife resides where her husband resides, and she must obey him in traveling to his country unless she has a valid excuse. There is no harm in the husband traveling and leaving his wife if he is confident of her safety, for her residence in a country other than her own does not require the presence of a mahram for it to be permissible. Rather, it requires the assurance of safety from temptation and the absence of suspicion. The mahram, however, is only required for travel.