Falkner is the first Gym Leader in PTD 2, just like in Pokémon HeartGold and SoulSilver. The badge given for defeating him is the Zephyr Badge. Falkner is the leader of Violet City.
Route 31: Falkner is seen standing next to the guard, complaining about how Violet City is blocked off. He then challenges you to a battle and after defeating him, you will get the Zephyr Badge.
Violet City: Falkner appears again in Violet City in front of the Gym. Here he says he doesn't understand why he couldn't enter the city first and later challenges the player for a battle again, but this time much harder.