I think you need this:
...struck at dawn. When? 'at dawn'. 'at dawn' is an optional (you can delete it and nothing happens) time adverbial; 'at' is a preposition and it takes 'dawn' as its object.The prepositional phrase is an adverb.
...struck on the solution. Stuck where? 'on the solution'? No. Phrasal verb detected. 'on' is a preposition and not an adverb particle; you can't delete the prepositional phrase because there is a strong cohesion working between the verb and the particle.
Student or Learner