The name is a diminutive form of the Russian word соловей, "nightingale", since the Soloveitchiks are a family of Levites, who are commanded by the Torah to sing in the Beit Hamikdash.
After leaving school, Aba learned in yeshiva in Kapellen, Belgium, and then in Luzern, Switzerland, where studied under Rabbi Moshe Soloveitchik, with whom he maintained contact throughout the remainder of his life.
Rabbi Avrohom Yehoshua Soloveitchik (sometimes Avraham) (born 1949) is the rosh yeshiva of Yeshivas Brisk, one of the Brisk yeshivas in Jerusalem, Israel that admits only a select few elite Talmudic students from the United States.
Brisk tradition and Soloveitchik dynasty, a school of Jewish thought originated by the Soloveitchik family of Brest (see also Brisker method)
However, as Rabbi Joseph Soloveitchik suggested in his eulogy for the Brisker Rov, the full, true "Brisk approach" as we know it today was not developed until Rabbi Chaim Soloveitchik had been rabbi of Brisk for many years.