Bjugstad was injured during a 2-1 loss at the New Jersey Devils on Nov. 15. The center has one goal in 10 games.
Bjugstad is the second Penguins center to have core muscle surgery this month; Sidney Crosby had the procedure Nov. 14 and is expected to be out at least another five weeks.
It was thought Schultz was injured during practice Wednesday, but Sullivan couldn't confirm if it was something more cumulative. The defenseman has eight points (two goals, six assists) in 21 games.
Pittsburgh (11-7-3) is already without defenseman Kris Letang, who hasn't played since Nov. 4 because of a lower-body injury. There is no time frame for his return.
"I think we have capable guys, and that's how we look at it," Sullivan said. "Once again, it's just the adversity, it's the reality of the game, and we're just going to have to learn to deal with it and find ways to win.
"We believe we've got capable people. We've got a deep organization, we feel as though we've got a deep roster. We've got capable guys that we can plug in, and it's a great opportunity for players to step up."
Penguins defenseman Chad Ruhwedel is expected to replace Schultz against the Islanders, and forward Patric Hornqvist will play after missing seven games because of a lower-body injury.
"We've got great leadership, we have good people, we have a deep roster," Sullivan said. "That's how we look at it. I give our players a lot of credit because they've never looked for an excuse, and our expectation is that we're going to win regardless of what team we play against. We believe we have what it takes, we just have to make sure we control what we can."
