How was Alexanders army able to deal with the Persian archers?

[Ca Putt]

The Acheamenid persians made great use of Greek mercenary Hoplites and also employed similar troops in smaller numbers.

Concerning the reluctance to "switch" to heavy infantry on the side of the persians probably roots in very different circumstances:
(in no particular order)

being a heavy infantry man is not pleasant. - You have to carry a heavy armor on foot, engage your enemy in melee and in case the battle turns sour, it's unlikely you'll get away, because your slow. When you're going to war you'll either want a horse(to get away quicker) or a bow(to avoid melee), unless some code of honor prohibits such cowardly conduct.

Tradition - the Bulk of the Persian Empire had either equestian or Archery Tradition, which helps a lot with recruitment of either. And as explained above and "shown"* in many games, If you can get good cavalry and archers you'll not tell your sheperds and lower nobility to get a shield and off of that horse. And the results would be subpar at best. A noble who has fought from horseback for X generations will mostlikely not make a very good hoplite. In antiquity a great part of the military education was through the father(or uncle or grandfather) and like today a boy who went hunting with his dad is morelikely to become a good marksman/soldier than one who's dad went fishing(something that is not required in todays armed forces) with him. changes like this would take several generations to yield superior heavy infantry

Population structure - While incorporating plenty large cities the Predominant recruitment pool of persians probably was based on cattle-droving seminomads OR people who have just given up this Lifestyle one or two generations ago. Some historians may actually bring up the ethernal fight between Cain and Abel, Something I'd be carefull with however. Not to say Persians had less Cities, they just had more other stuff, whereas Greece basically consists of rocks, water and settlements that consider themselves cities,... well and Tessaly.
In essence city dwelling farmers make "good" Heavy infantry, seminomadic stockbreeders make "good" archers or Cavalry(depending on wealth and terrain).

Terrain - "Persia" is mostly arid or mountainous there also are fertile regions and in antiquity there also were some forests. but most of the time you have perfect conditions for Cavalry and archers. Greece on the other hand is mostly a coastal thing with valleys placed conveniently to fight a pitched battle or two, neat for archers and shielbearers, but terrible for cavalry.


*overly pronounced