After eating, what specific part of the nervous system controls the digestion of the
food after swallowing the food?

Once you begin swallowing, the process becomes automatic. Your brain signals the muscles of the esophagus and peristalsis begins. Lower esophageal sphincter. When food reaches the end of your esophagus, a ringlike muscle—called the lower esophageal sphincter —relaxes and lets food pass into your stomach.

showing a photograph would make this a hell of alot easier

a "theory" a suggestion that is not 100 % accurate, or a figure. some people say its a theory because they of all the passed theory's about one subject. it depends sometimes theory might be accurate and sometimes they are.