After every sleep cycle, a chemical is released that eliminates our short term memories so they never get a chance to become long term memories. In short, you would need to notice the moment in which you wake up after every sleep cycle and make a note of the dreams from it then fall back asleep or completely change your brain chemistry. It is possible, but unlikely.

To comment about your dreams only lasting a second, that's very unlikely. The dream part of the sleep cycle is rather long, so your dream will last longer. However, some do think it possible to make a long dream occur in a short amount of time. I'm uncertain if it's merely the illusion of time or not, and am somewhat skeptical of it myself, but I had a month long LD once.