Mower dies and needs rocked to restart

Hello. I have a John Deere STX38 with the yellow deck and 12hp Kohler engine. It will run for a bit then will just die. Doesn't sputter or make any sounds. Just shuts off. I put it in neutral and turn the key and it's complete silence. I have to get off and rock it either forward or backwards. Then turn the key and it will usually start back up. This happens often enough now that it's really becoming a problem. Any ideas?

Jason