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

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

2010-12-05

arxiv.org/abs/1012.1033v2

J. Math. Phys. 52, 103703 (2011)

Physics

Mathematical Physics

15 pages, 8 figures; one reference added, two removed, revised final remark 3

Scientific paper

10.1063/1.3645363

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2\alpha} u =0$ 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 $\alpha$, namely, we observe fast convergence to $S$ for $\alpha>1$, slow convergence for $\alpha=1$, and very slow (if any) convergence for $0<\alpha<1$. Our findings are complementary with respect to the recent rigorous analysis of the same problem (for $\alpha>2$) by Krieger, Nakanishi, and Schlag \cite{kns}.

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