Scattering Across the Long-Range Threshold for One-Dimensional Nonlinear Schrödinger Equations
We study small solutions to the one-dimensional nonlinear Schrödinger family \[ \mathrm{i}\partial_t u_δ=-\partial_x^2u_δ+κ|u_δ|^{2+δ}u_δ, \qquad 0\leqδ\leqδ_0, \] uniformly as $δ\to0^+$ and $t\to\infty$, across the threshold between cubic modified scattering and nearby-power ordinary scattering. For initial data small in $H^{0,1}(\mathbb{R})$, with one spatial weight and no physical derivative assumed, we prove a joint-limit asymptotic formula governed by an explicit nonlinear clock. The variable $ δ\log t$ gives three regimes according as it tends to zero, a positive finite limit, or infinity. Within the regime $δ\log t\to0$, successive Taylor terms of the clock become visible at an infinite hierarchy of time scales, each requiring finer absolute phase accuracy. These scales accumulate at the transition scale, where the exact clock describes them together. In the transition and saturated regimes, the clock is of order $δ^{-1}$, so the first-order variation of its coefficient contributes a finite phase correction.