Jonas is known for being a serial dater in Hollywood. He had a 2006 fling with pop artist AJ Michalka and had a solid relationship with Taylor Swift two years later that ended badly when he dumped her over the phone. Then, he spent a year with Camilla Belle, who starred in a Jonas Brothers music video for the song Lovebug, and then there was last years romp with Demi Lovato, who starred with Jonas in Camp Rock and Camp Rock 2.

But Greene, who is the daughter of a U.S. Marine and attended Christian schools, knows what she wants in life and seems to have the musician smitten enough to follow her on the road.

Recently, the couple, who were introduced last summer in London by mutual friends, had a late night romantic dinner at Ichiban Japanese Grill and Sushi Bar in Baton Rouge, La., where Greene is currently filming, Breaking Dawn. They shared sushi and spent most of the meal cuddling and talking to each other.

"They were holding hands and seemed really happy to be together," says a source at the restaurant.

While Greene and Jonas live in Los Angeles, they understand the demands of each others busy careers. While Jonas tours the world with his Jonas Brothers concert tours, Greene is busy making movies stateside.

"He respects her as a professional, which could make the relationship go a long way," says the music source.

And Greene recently told Teen Vogue that she and Jonas have worked out a "traveling relationship" which is "crazy" because they both live in Los Angeles.

"We have already met each others parents," she told the magazine, "which I think is a good result of the kind of lifestyle we live. Hes sweet and really funny, which is huge for me."