There are several synonyms for the phrase "abandon on" which refer to the act of leaving or deserting someone or something in a particular place. These include "leave behind", "forsake", "cast off", "disown", "discard", and "give up on". These phrases can be used in various contexts such as abandoning a person in a time of need or leaving behind a project or goal that is no longer achievable. While the act of abandonment can be seen as negative, it may also lead to positive outcomes such as self-discovery or the realization of personal boundaries.