Non-localized data focusing lifespan conjecture

Let a one-dimensional dispersive flow on the real line have a cubic nonlinearity, and let its initial data have size $\epsilon$ in a suitable Sobolev norm. Non-localized data focusing lifespan conjecture. The solution remains of comparable size for at least an $\epsilon^{-8}$ time-scale. The paper proves this assertion under suitable assumptions, while the conjecture is intended as a broad statement for focusing flows with small, non-localized data.