Weak wave cascade (${\tau }_{\mathrm{cas}}\sim {\tau }_{\mathrm{nl}}^2/{\tau }_w$)

If a wave packet experiences a fractional nonlinear change $\chi ={\tau }_w/{\tau }_{\mathrm{nl}}\ll 1$ per interaction, and successive interaction phases are decorrelated, $N$ changes add as a random walk of size $\sqrt{N}\chi$. Order-one transfer needs $N\sim {\chi }^{-2}$ interactions, giving the displayed cascade time. Phase coherence or resonant constraints can alter this estimate; weakness alone does not justify random addition.