2023.3.15 9:00 N613

We study the nonrelativistic limit of the cubic Klein-Gordon equations. We show the cubic Klein-Gordon equation converges to the cubic Schr\"odinger equation with a convergence rate of order $\epsilon^{2}$. In particular for the defocusing case, for `smooth' initial data, we show 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 show 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.