Modified wave operators for nonlinear Schrödinger equations in the full subcritical long range regime
Abstract
We construct modified wave operators for the nonlinear Schrödinger equation i∂tu+1/2Δu=|u|pu in the full subcritical long-range case 0<p<2/d, with small, nonvanishing, analytic final data W(x) with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for p<1/d remained unknown. When 1/d<p<2/d, our result gives the approximation 1/(it)d/2 ei|x|2/2t W(x/t) exp[ −i(t1−dp/2−1)/(1−dp/2) |W(x/t)|p ] The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When p≤ 1/d, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative Lq asymptotics for 2≤ q≤∞ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.
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BibTeX
@misc{shen2026modified,
title = {{Modified wave operators for nonlinear Schrödinger equations in the full subcritical long range regime}},
author = {Jia Shen and Yifei Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11095},
eprint = {2609.11095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11095},
}
RIS
TY - UNPB
TI - Modified wave operators for nonlinear Schrödinger equations in the full subcritical long range regime
AU - Jia Shen
AU - Yifei Wu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11095
UR - https://arxiv.org/abs/2609.11095
AB - We construct modified wave operators for the nonlinear Schrödinger equation i∂_tu+1/2Δu=|u|^pu in the full subcritical long-range case 0<p<2/d, with small, nonvanishing, analytic final data W(x) with bounded logarithmic gradients. Previous results established large-time asymptotics for selected classes of Cauchy data. Moreover, the exact asymptotic expansion for p<1/d remained unknown. When 1/d<p<2/d, our result gives the approximation 1/(it)^d/2 e^i|x|^2/2t W(x/t) exp[ −i(t^1−dp/2−1)/(1−dp/2) |W(x/t)|^p ] The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When p≤ 1/d, we construct the profile from a finite truncation of a Fuchsian equation coupled with a transport equation. This profile still leaves a long-range triangular coupling whose terminal integral does not preserve the required fast decay class. The construction yields quantitative L^q asymptotics for 2≤ q≤∞ and uniqueness in the prescribed analytic asymptotic classes. The central new ingredients are a nonlinear final-state normal form that removes this long-range coupling and a mixed iteration in particular analytic spaces.
ER -