After you beat Darkrai do some guild jobs for about 2 or 3 days then Manaphy will come back and do another 2 days and Manaphy will discover a island paradise you will be able to go there then you completed the side story
Chat with our AI personalities
Mostly, anything you want! You can look for treasure, become rich, try to acquire all special items, evolve pokemon, recruite pokemon, get IQ up, etc. But if you want to evolve, you have to let Manaphy join your team. After awhile, let Manaphy tell you about Marine Resort and the Ursarings will tell you something had happened at Luminous Spring. (This is for PMD Explorers of Sky, but i think it would work for the other too.)
you can either A: restart the game or B: complete the adventure log by collecting all the ledgendaries (mewtwo is in darkness and celibi in time) and also try to gain the master rank
Get your rank up and use Spinda's Cafe to get new dungeons and explore them. There is no more storyline though...