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arXiv 1012.1033math-phmath.APmath.MP
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一维聚焦非线性Klein-Gordon方程爆破阈值附近的动力学

Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation

Piotr Bizoń, Tadeusz Chmaj, Nikodem Szpak

备注：
15 pages, 8 figures; one reference added, two removed, revised final remark 3
发表信息：
J. Math. Phys. 52, 103703 (2011)
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摘要：

研究了线上聚焦的非线性Klein-Gordon方程$u_{tt}-u_{xx} + u - |u|^{2α} u =0$的爆破阈值附近的动力学。使用混合的数值和分析方法，我们发现，解决方案，甚至从初始数据，微调到阈值，被困的静态解决方案$S$的中间时间。捕获的细节取决于电源$α$，即，我们观察到$α>1$的$S$的快速收敛，$α=1$的缓慢收敛，和非常缓慢（如果有的话）的收敛$0<α<1$。我们的研究结果是互补的，最近严格的分析相同的问题（$α>2$）的Krieger，Nakanishi，和Schlag \cite{kns}。

英文摘要：

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2α} u =0$ on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution $S$ for intermediate times. The details of trapping are shown to depend on the power $α$, namely, we observe fast convergence to $S$ for $α>1$, slow convergence for $α=1$, and very slow (if any) convergence for $0<α<1$. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for $α>2$) by Krieger, Nakanishi, and Schlag \cite{kns}.

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