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

作者：Bizon Piotr^*; Chmaj Tadeusz; Szpak Nikodem

来源：Journal of Mathematical Physics, 2011, 52(10): 103703.

DOI：10.1063/1.3645363

• axiomatic field theory
• convergence of numerical methods
• nonlinear equations
• nonlinear field theory
• relativistic quantum field theory

We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation u(tt) - u(xx) + u - vertical bar u vertical bar(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 ["Global dynamics above from the ground state energy for the one-dimensional NLKG equation," preprint arXiv:1011.1776 [math.AP]].

• 出版日期2011-10