Don't get overwhelmed, add a new moderator or two to help you maintain your forum. I've experienced bursts of activity on my forum, and it's slowed down after. It usually happens when/if I run a new promotion.
Make sure to add new content to your forum. Keep those people coming back for more, think of unique ways to keep them coming back. If it's a community based around a specific niche, perhaps adding content not seen on other platforms could help. This might be a bit difficult, but if you need some help brainstorming you can always hit up the Brainstorming section of the forum here or on other platforms to get some ideas.
To keep the momentum going, it’s important to stay engaged with the new members and make them feel welcomed. Responding to their posts, starting fun discussions, and asking for their opinions helps a lot. Also, keep the content fresh and relevant, maybe even run a small challenge or giveaway to keep people coming back
Give them a reason to come back time and time again. Keep the topics fresh and interesting, make sure you try and connect with them to create a sense of community. Most importantly ride the wave of activity, make the most out of it it can be a great time to launch a competition or contest when lots of members are around to hear about it which in turn should help keep them coming back.
I’d engage with new members by welcoming them personally and encouraging discussions on their interests. I’d also introduce fun topics that invite participation, making the forum feel vibrant and inclusive.
I'd just do as I normally would: reply to everything with a reply that encouraged more conversation.
One liner post here and there are fine and appropriate at times but the biggest thing you want to do is keep the conversation going. Ask questions or request some clarification. Post stuff that's valuable. Keep the original party and other parties interested in the content.
I think it's ideal to always have a goal (or at least 99.999% of the time) to create a thread with the goal of it turning into several hundred pages of ongoing discussions.