发表机构
Sorbonne Université; Beijing Normal University（索邦大学; 北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对6维能量临界聚焦非线性薛定谔方程，引入修正调制参数克服基态不属于$\dot H^{-1}$导致的标准方法失效问题，构造了双泡解并建立线性化能量的强制性估计。
AI 中文摘要
我们针对空间维数$N=6$的能量临界聚焦非线性薛定谔方程构造纯双泡解。这类解在至少一个时间方向上是全局的，且趋近于两个均以原点为中心的定态的叠加：其中一个泡的尺度为1，另一个泡的长度尺度以$e^{-|t|}$的速率收敛于0，两个泡的相位构成直角。此前这类解已在$N\geq7$的维数中构造过，而6维情形存在特定困难：基态不属于$\dot H^{-1}$，这使得能量泛函失去强制性，无法通过合适的正交条件消除调制方程中的线性项，无法使用标准方法。本工作的主要创新点是引入修正调制参数以克服该问题，这些参数可视为调制分析语境下的范式变换的类似物；我们还为线性化能量建立了新的强制性估计，其正的常数明确依赖于正交条件的选择。
英文摘要
We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N = 6$. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $e^{-|t|}$. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension $N \geq 7$. The six-dimension case presents specific difficulties, as the ground state does not belong to $\dot H^{-1}$. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.
Comments47 pages, Comments welcome!
