One dimensional energy cascades in a fractional quasilinear NLS

The paper proves precisely the claimed growth-of-Sobolev-norms result for the 1D fractional quasilinear NLS ∂t u = −i|D|^α u + |u|^2 u_x, α∈(0,1): Theorem 1.1 asserts that for some s0>3/2 and any s>3 s0, one can start with arbitrarily small H^s data and obtain arbitrarily large H^s at a later time while keeping an H^{s0}-norm small, with the mechanism built via a paradifferential normal form producing an effective transport operator and a positive commutator (Mourre) estimate, and the growth occurring on a timescale ∼δ^{-2} log(δ^{-1}) . The candidate solution mirrors this strategy step-by-step (local theory, quasilinear normal form yielding OpBW(i⟨V⟩ ξ), two-mode reduction, Mourre positivity, exponential-in-time growth, and undoing near-identity transforms). Minor deviations are cosmetic (e.g., using modes ±n instead of the paper’s ±1, a typo in the L^2 calculation, and not explicitly mentioning the renormalization removing mass/momentum terms), but the substance matches the paper’s argument .