A PE firm can incentivize management to stay by getting management to put up some equity for the acquisition, such that management owns 5-10% of the company.
Another common method to incentivize management is with options. Options give management a percentage of the upside. For example, if management is given 7.5% of the upside and the value of the company increases, then management will get 7.5% of that value increase when the company is sold.
Finally, earn-outs can be used as well. An earnout is a payment to management for achieving a certain goal. For example, an earnout could stipulate that management needs to achieve 10% growth in gross profit in the next year in order to receive an earnout payment. This is important because it incentives management to grow the business, it aligns interests, and it also motivates management to stay with the company.