Non-relativistic limit for the cubic nonlinear Klein-Gordon equations

TL;DR

This paper analyzes the non-relativistic limit of the cubic nonlinear Klein-Gordon equations, providing precise error bounds and demonstrating the effectiveness of modulated Schr"odinger and Schr"odinger-wave profiles as the light speed tends to infinity.

Contribution

It introduces new error estimates for the non-relativistic limit using advanced WKB expansion techniques and improves existing results on the Klein-Gordon equation's asymptotic behavior.

Findings

Error bounds of order c^{-2} in 2D with Schr"odinger-wave profiles

Error bounds of order c^{-2} + (c^{-2}t)^{eta} in 2D and 3D with Schr"odinger profiles

Sharpness of bounds and minimal regularity conditions established

Abstract

We investigate the non-relativistic limit of the Cauchy problem for the defocusing cubic nonlinear Klein-Gordon equations whose initial velocity contains a factor of $c^2$, with $c$ being the light speed. While the classical WKB expansion is applied to approximate these solutions, the modulated profiles can be chosen as solutions to either a Schr\"odinger-wave equation or a Schr\"odinger equation. We show that, as the light speed tends to infinity, the error function is bounded by, (1) in the case of 2D and modulated Schr\"odinger-wave profiles, $Cc^{-2}$ with $C$ being a generic constant uniformly for all time, under $H^2$ initial data; (2) in the case of both 2D and 3D and modulated Schr\"odinger profiles, $c^{-2} +(c^{-2}t)^{\alpha/4}$ multiplied by a generic constant uniformly for all time, under $H^\alpha$ initial data with $2 \leq \alpha \leq 4$. We also show the sharpness of the upper bounds in (1) and (2), and the required minimal regularity on the initial data in (2). One of the main tools is an improvement of the well-known result of Machihara, Nakanishi, and Ozawa in \cite{MaNaOz-KG-Limits} which may be of interest by itself. The proof also relies on \textit{a fantastic complex expansion} of the Klein-Gordon equation, \textit{introducing the leftward wave and exploring its enhanced performance} and a \textit{regularity gain mechanism} through a high-low decomposition.