Does anyone know what's going on for this potato plant? The plants are still green (no indications of yellowing) but the stem is soft, the leaves are rolled and wilting while the surrounding plants look perfectly fine. Full disclosure, I am using urine to fertilize this container of potatoes as an experiment. This began 5 days ago after I had poured straight urine into the container. I've done this many times before with even more and there were no ill effects.

That's technically called "phytotoxicity": broadcast fertilization with very nitrogen-rich products will produce leaf scorch, plant stunting, necrosis and a host of other ill effects. If you relieve yourself in a bucket seeded with potatoes, you are basically drenching them in an urea-rich solution.
Different crops have different susceptibility to high nitrogen levels at different growth stages, for example corn gets far more susceptible as it grows and nitrogen fertilization should be carefully administered after V3. Also urea moves in the ground according to moisture "bands" in the potting medium (in your case) and will naturally concentrate causing scorch when applied in excess, with high draining soils such as sand being the worst.
So stop relieving yourself in the potato bucket.

PS: very broadly speaking (soil, weather etc conditions need to be taken into account) potatoes require NPK at a 3-5-5 ratio. Being tubers too much nitrogen can be counterproductive by stimulating too much foliar growth.

PPS: according to Jenny Uglow in her Little History of Gardening, in old European country estates gardeners and other workers were expected to relieve themselves in the compost heap, not in the flower beds.

Thanks. I'm surprised that they are suffering now. I've been giving them urine for the last 2 months, sometimes even more concentrated and more often but they have never reacted like this before. I'll hold off on fertilizing them for the time being as they should be ready to harvest in 30 days.