talk to admiral in astera. he should have blue exclamation point at the top of his head. after talking to him, go to gathering hub and talk to the quest counter. you should see the kulve taroth siege
I did Kulve Solo many times, even on pursuit lvl 5 and what I found out was, in Phase 4 I can stagger her much more easily if i just shoot the chest since the chest is a huge weak spot, the Horns are not. Sometimes I fail to break the horns if I just concentrate my fire on them. The last stage is all about dishing the most damage possible.