Simply put, Jonathan's PDA borders on inappropriate.
Sometimes, it will be the typical holding hands, light touches, and occasional kiss or arm around your shoulder, but most of the time it isn't. His favorite game is seeing how far he can go without anyone else noticing. This has led you both to being kicked out of the Iceberg Lounge by Penguin on numerous occasions. The old bird always lets you guys back in though, mainly because you're valued customers.