Long-time behaviors of the two-component nonlinear Klein-Gordon equation: higher-order asymptotics

TL;DR

This paper analyzes the long-time behavior of solutions to a two-component nonlinear Klein-Gordon equation using inverse scattering and Riemann-Hilbert methods, providing higher-order asymptotics and validating results with numerical simulations.

Contribution

It introduces a detailed asymptotic analysis framework for the two-component nonlinear Klein-Gordon equation, including higher-order terms and numerical validation.

Findings

Derived higher-order asymptotics inside the light cone.

Established properties of reflection coefficients.

Validated asymptotic results with numerical simulations.

Abstract

This work investigates the long-time asymptotic behaviors of solutions to the initial value problem of the two-component nonlinear Klein-Gordon equation by inverse scattering transform and Riemann-Hilbert formulism. Two reflection coefficients are defined and their properties are analyzed in detail. The Riemann-Hilbert problem associated with the initial value problem is constructed in term of the two reflection coefficients. The Deift-Zhou nonlinear steepest descent method is then employed to analyze the Riemann-Hilbert problem, yielding the long-time asymptotics of the solution in different regions. Specifically, a higher-order asymptotic expansion of the solution inside the light cone is provided, and the leading term of this asymptotic solution is compared with results from direct numerical simulations, showing excellent agreement. This work not only provides a comprehensive analysis of the long-time behaviors of the two-component nonlinear Klein-Gordon equation but also offers a robust framework for future studies on similar nonlinear systems with third-order Lax pair.