On the NLS approximation for the nonlinear Klein-Gordon equation

TL;DR

This paper introduces a Fourier analysis-based method to establish a low regularity NLS approximation for the 1D cubic Klein-Gordon equation, providing convergence rates under regularity assumptions.

Contribution

It develops a new approach for NLS approximation of Klein-Gordon equations using Fourier analysis, achieving results for low regularity solutions.

Findings

Established low regularity NLS approximation for Klein-Gordon

Derived convergence rates with additional regularity

Extended results to energy class solutions

Abstract

In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class solutions which are formally asymptotically in $L^2(\mathbb{R})$. A precise rate of convergence is also obtained assuming more regularity.