One of the great weaknesses of totalitarian governments (and government wannabes such as ISIS) is their tendency to think that free societies are run as theirs are.
One of the Nork’s diplomats has gone on record saying that President Trump has declared war on them. While Rocket Man has the authority to declare war, the President of the United States does not. According to Art. I, Sec. 8, Cl. 11 of the Constitution, only Congress has the power to declare war.
The President hasn’t and can’t declare war or North Korea. However, he does have the general authority to order our forces to act in self-defense. Further, if presented with a violation of the 1953 Korean armistice agreement, he probably has the authority to carry forward the UN police action prosecuted by Presidents Truman and (briefly) Eisenhower.