发表机构
Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University; Department of Applied Mathematics, Kyung Hee University（全北国立大学数学系与纯应用数学研究所; 庆熙大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维非线性薛定谔方程在临界幂次附近的小解散射，通过显式非线性时钟统一描述跨阈值行为，并揭示δlog t控制的三种渐近区域。
AI 中文摘要
我们研究一维非线性薛定谔族的小解 \\[ \mathrm{i}\partial_t u_\delta=-\partial_x^2u_\delta+\kappa|u_\delta|^{2+\delta}u_\delta, \qquad 0\leq\delta\leq\delta_0, \\] 在 $\delta\to0^+$ 和 $t\to\infty$ 时，跨越三次修正散射与邻近幂次普通散射之间的阈值的一致行为。对于在 $H^{0,1}(\mathbb{R})$ 中具有一个空间权重且不假设物理导数的小初值，我们证明了一个由显式非线性时钟控制的联合极限渐近公式。变量 $\delta\log t$ 根据其趋于零、趋于正有限极限或趋于无穷大而给出三种区域。在 $\delta\log t\to0$ 的区域内，时钟的逐次泰勒项在无限的时间尺度层级上变得可见，每一层都需要更精细的绝对相位精度。这些尺度在过渡尺度处累积，其中精确时钟将它们一起描述。在过渡和饱和区域内，时钟的量级为 $\delta^{-1}$，因此其系数的一阶变化贡献了有限的相位修正。
英文摘要
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.
