发表机构
University of Basel（巴塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了二维环面上聚焦三次薛定谔方程的一个全局光滑解，其在正无穷时集中于一点，质量等于基态质量、能量为零，浓度尺度满足对数增长，并证明了该渐近类中的唯一性。
AI 中文摘要
我们在方形二维环面上构造了聚焦三次非线性薛定谔方程的一个光滑解，该解在时间正负两个方向都是全局的，并且当时间趋于正无穷时集中于一点。其质量等于欧几里得基态的质量，能量为零。若以$\lambda(t)$表示其浓度尺度，$L$表示周期，则$L/\lambda(t)=2\log t+\frac{11}{2}\log\log t+O(1)$。该解趋近于一个带相位的周期化基态，误差在任意固定的Sobolev空间中趋于零。特别地，对于每个$s>0$，其$H^s$范数渐近于一个显式常数乘以$(\log t)^s$。我们还证明了在构造中所用的定量单轮廓渐近类内，解在常数相位和时间平移意义下的唯一性。其机制是轮廓与其周期像之间的指数小相互作用。精确的质量和能量约束选取了浓度分支，而移动约束空间上的正能量控制了每个固定正则阶的终值问题。
英文摘要
We construct a smooth solution of the focusing cubic nonlinear Schrödinger equation on a square two-dimensional torus which is global in both time directions and concentrates at one point as time tends to positive infinity. Its mass is the mass of the Euclidean ground state and its energy is zero. If $λ(t)$ denotes its concentration scale and $L$ the period, then $L/λ(t)=2\log t+\frac{11}{2}\log\log t+O(1)$. The solution approaches a phased, periodized ground state with an error tending to zero in every fixed Sobolev space. In particular, its $H^s$ norm is asymptotic to an explicit constant times $(\log t)^s$ for every $s>0$. We also prove uniqueness, up to constant phase and time translation, within the quantitative single-profile asymptotic class used in the construction. The mechanism is the exponentially small interaction of the profile with its periodic images. Exact mass and energy constraints select the concentrating branch, while a positive energy on the moving constrained space controls the terminal-value problem at every fixed regularity order.
