the leaves are turning brown and dead, should i cut the stalk, or leave the stalk, so it might grow next year?
You'll more likely find an answer to this question over on the Vegetable Gardening forum. The focus here is canning, freezing, and preserving food.
You harvest jerusalem artichokes just like potatoes.
One exception...do NOT mix the dirt into your compost or regular garden area.......they WILL come back.
Even as much a small piece of root or tiny 'choke will grow a new plant (think mint!!). Can be VERY invasive.
Otherwise, dig up the knobby little roots and boil, roast or fry.
Don't know if you've heard the warnings, but can cause stomach upset and severe ....um.....flatulation in some folks.
You've been warned!! :-)
The stalk is dead, might as well cut it. It comes back from the roots, not from a dead stalk.