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

Deng-Shan Wang, Yingmin Yang, Liming Zang

TL;DR

This work analyzes the long-time dynamics of the two-component nonlinear Klein-Gordon equation using an inverse scattering framework based on a third-order Lax pair. By formulating a $3\times3$ Riemann-Hilbert problem in terms of two reflection coefficients and applying the Deift-Zhou nonlinear steepest descent, the authors derive high-order asymptotics, including a detailed inside-the-light-cone expansion and outside-cone decay rates, with rigorous error estimates. The reconstruction formulas connect the RH solution to $u(x,t)$ and $v(x,t)$, and numerical simulations corroborate the leading-order and higher-order terms. The developed approach provides a robust, generalizable framework for third-order Lax-pair integrable systems and their long-time behavior.

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.

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

TL;DR

This work analyzes the long-time dynamics of the two-component nonlinear Klein-Gordon equation using an inverse scattering framework based on a third-order Lax pair. By formulating a Riemann-Hilbert problem in terms of two reflection coefficients and applying the Deift-Zhou nonlinear steepest descent, the authors derive high-order asymptotics, including a detailed inside-the-light-cone expansion and outside-cone decay rates, with rigorous error estimates. The reconstruction formulas connect the RH solution to and , and numerical simulations corroborate the leading-order and higher-order terms. The developed approach provides a robust, generalizable framework for third-order Lax-pair integrable systems and their long-time behavior.

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.
Paper Structure (22 sections, 30 theorems, 208 equations, 8 figures)

This paper contains 22 sections, 30 theorems, 208 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Spectral analysis
  4. Direct scattering problem
  5. The eigenfunctions $X_{\pm}(x,\lambda)$
  6. Asymptotics of the eigenfunctions $X_+(x,\lambda)$ and $X_-(x,\lambda)$ as $\lambda\to \infty$
  7. Asymptotics of the eigenfunctions $X_+(x,\lambda)$ and $X_-(x,\lambda)$ as $\lambda\to 0$
  8. The scattering matrix
  9. The cofactor matrix
  10. The eigenfunctions $M_n(x, \lambda)$
  11. The reconstruction formula
  12. Higher-order asymptotics of the problem \ref{['3']}
  13. Long-time asymptotic behavior outside the light cone $\vert \frac{x}{t}\vert>1$
  14. The situation of $\xi>1$
  15. The situation of $\xi < -1$
  16. ...and 7 more sections

Key Result

Theorem 2.1

Assuming that the initial data $u_0$, $v_0\in \mathcal{S}(\mathbb{R})$, the reflection coefficients $r_1(\lambda)$ and $r_2(\lambda)$ are rigorously defined for $\lambda\in (0,\infty)$ and $\lambda \in (-\infty,0)$, respectively and exhibit the subsequent properties:

Figures (8)

  • Figure 1: Division of asymptotic regions in the upper $(x,t)$-half plane
  • Figure 2: The comparisons between the theoretical results given by Theorem \ref{['region']} and the full numerical simulations of the two-component nonlinear KG equation \ref{['3']} under the initial value condition \ref{['10']} for $t=50$ (left) and $t=100$ (right)
  • Figure 3: The jump contour $\Sigma$ decomposes the complex $\lambda$-plane into six parts
  • Figure 4: The signature of the real part of $\theta_{21}(\lambda)$ for $\vert\xi\vert<1$. In the shaded regions, ${\rm{Re}}\,\theta_{21}>0$; in the white regions, ${\rm{Re}}\,\theta_{21}<0$
  • Figure 5: The jump contour $\Sigma^{(1)}$ in the complex $\lambda$-plane
  • ...and 3 more figures

Theorems & Definitions (43)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Theorem 2.3
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 33 more