Scattering Across the Long-Range Threshold for One-Dimensional Nonlinear Schrödinger Equations
Abstract
We study small solutions to the one-dimensional nonlinear Schrödinger family i∂t uδ=−∂x2uδ+κ|uδ|2+δuδ, 0≤δ≤δ0, uniformly as δ→0+ and t→∞, across the threshold between cubic modified scattering and nearby-power ordinary scattering. For initial data small in H0,1(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 δlog t→0, 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.
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BibTeX
@misc{cho2026scattering,
title = {{Scattering Across the Long-Range Threshold for One-Dimensional Nonlinear Schrödinger Equations}},
author = {Yonggeun Cho and Jinyeop Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12423},
eprint = {2609.12423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12423},
}
RIS
TY - UNPB
TI - Scattering Across the Long-Range Threshold for One-Dimensional Nonlinear Schrödinger Equations
AU - Yonggeun Cho
AU - Jinyeop Lee
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12423
UR - https://arxiv.org/abs/2609.12423
AB - We study small solutions to the one-dimensional nonlinear Schrödinger family i∂_t u_δ=−∂_x^2u_δ+κ|u_δ|^2+δu_δ, 0≤δ≤δ_0, uniformly as δ→0^+ and t→∞, across the threshold between cubic modified scattering and nearby-power ordinary scattering. For initial data small in H^0,1(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 δlog t→0, 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.
ER -