AI 中文总结
针对全次临界长程非线性薛定谔方程，通过非线性终态正规形与解析空间混合迭代构造修正波算子，获得定量L^q渐近性与唯一性。
AI 中文摘要
我们在全次临界长程情形$0<p<2/d$下，为非线性薛定谔方程$i\partial_tu+\frac12\Delta u=|u|^pu$构造了修正波算子，其中最终数据$W(x)$为小且非零的解析函数，且具有有界对数梯度。先前的结果仅针对特定类别的柯西数据建立了长时间渐近性。此外，对于$p<1/d$的情形，精确渐近展开仍属未知。当$1/d<p<2/d$时，我们的结果给出了近似式$$\frac{1}{(it)^{\frac{d}{2}}} e^{\frac{i|x|^2}{2t}} W\left(\frac{x}{t}\right) \exp\left[ -i\frac{t^{1-\frac{dp}{2}}-1}{1-\frac{dp}{2}} \left|W\left(\frac{x}{t}\right)\right|^p \right]$$波算子通过在半径递减的解析空间中进行迭代构造。当$p\le 1/d$时，我们通过Fuchsian方程的有限截断与输运方程耦合来构造轮廓。该轮廓仍留下一个长程三角耦合，其终端积分不保持所需的快速衰减类。该构造给出了$2\le q\le\infty$范围内的定量$L^q$渐近性，并在指定的解析渐近类中具有唯一性。核心新要素是一个非线性终态正规形，用于消除该长程耦合，以及在特定解析空间中的混合迭代。
英文摘要
We construct modified wave operators for the nonlinear Schrödinger equation $i\partial_tu+\frac12Δu=|u|^pu$ 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 $$\frac{1}{(it)^{\frac{d}{2}}} e^{\frac{i|x|^2}{2t}} W\left(\frac{x}{t}\right) \exp\left[ -i\frac{t^{1-\frac{dp}{2}}-1}{1-\frac{dp}{2}} \left|W\left(\frac{x}{t}\right)\right|^p \right]$$ The wave operator is constructed by an iteration in the analytic spaces with decreasing radius. When $p\le 1/d$, 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 $2\le q\le\infty$ 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.
Comments91 pages; comments are welcome!
