Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation

In this paper, we study the nonrelativistic limit of the cubic nonlinear Klein-Gordon equation in ${\mathbb{R}}^3$ with a small parameter $0<\epsilon \ll 1$, which is inversely proportional to the speed of light. We show that the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schrödinger equation with a convergence rate of order ${\epsilon }^2$. In particular, for the defocusing case and smooth initial data, we prove error estimates of the form $(1+t){\epsilon }^2$ at time $t$ which is valid up to long time of order ${\epsilon }^{-1}$; while for nonsmooth initial data, we prove error estimates of the form $(1+t)\epsilon$ at time $t$ which is valid up to long time of order ${\epsilon }^{-\frac{1}{2}}$. These specific forms of error estimates coincide with the numerical results obtained in \cite{BZ19,SZ20}.