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Long-time asymptotics of 3-solitary waves for the damped nonlinear Klein-Gordon equation

Kenjiro Ishizuka

TL;DR

This work analyzes the long-time behavior of 3-soliton solutions to the damped nonlinear Klein-Gordon equation in energy-subcritical regimes, showing that their centers align on a straight line with alternating signs and logarithmic separation in time, up to $(\log\log t)/\log t$ corrections.A modulation framework around the sum of ground states is developed, together with a spectral decomposition that isolates unstable and translational modes, and a hierarchy of energy estimates that control the residual and inter-soliton interactions.The inter-soliton forces are encoded by a universal interaction function $\mathcal F(|z_i-z_j|)$ with exponential decay, driving the centers to escape along a common direction while preserving a line-like arrangement.The results advance the soliton-resolution program for damped dispersive equations by giving an explicit, universal asymptotic description for a 3-soliton configuration in all admissible dimensions, including precise decay rates and sign structure.

Abstract

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $α>0$, $1\leq d\leq 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that 3-solitary waves behave as if the three solitons are on a line. Furthermore, the solitary waves have alternative signs and their distances are of order $\log{t}$.

Long-time asymptotics of 3-solitary waves for the damped nonlinear Klein-Gordon equation

TL;DR

This work analyzes the long-time behavior of 3-soliton solutions to the damped nonlinear Klein-Gordon equation in energy-subcritical regimes, showing that their centers align on a straight line with alternating signs and logarithmic separation in time, up to $(\log\log t)/\log t$ corrections.A modulation framework around the sum of ground states is developed, together with a spectral decomposition that isolates unstable and translational modes, and a hierarchy of energy estimates that control the residual and inter-soliton interactions.The inter-soliton forces are encoded by a universal interaction function $\mathcal F(|z_i-z_j|)$ with exponential decay, driving the centers to escape along a common direction while preserving a line-like arrangement.The results advance the soliton-resolution program for damped dispersive equations by giving an explicit, universal asymptotic description for a 3-soliton configuration in all admissible dimensions, including precise decay rates and sign structure.

Abstract

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where , and energy sub-critical exponents . In this paper, we prove that 3-solitary waves behave as if the three solitons are on a line. Furthermore, the solitary waves have alternative signs and their distances are of order .

Paper Structure

This paper contains 18 sections, 23 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main result
  4. Previous results
  5. Basic properties of the ground state
  6. Dynamics of $K$-solitary waves
  7. Modulation of the center
  8. Decomposition close to the sum of solitary waves
  9. Spectral decomposition of the solution
  10. Energy estimates
  11. Estimate of the instability term
  12. Soliton repulsivity condition
  13. Estimates of the distance between each solitary waves
  14. Global dynamics of the center of each solitary wave
  15. Considerations on the location of the center
  16. ...and 3 more sections

Key Result

Theorem 1.1

If a solution $\vec{u}$ of DNKG is a $3$-solitary wave, there exist $\sigma=\pm 1$, $z_{\infty}\in \mathbb{R}^d$, $\omega_{\infty}\in S^{d-1}$, and a function $z_k:[0,\infty)\to \mathbb{R}^d$ for $k=1,2,3$ such that for all $t>0$, and for $k=1,2,3$, where $c_0$ is a constant depending only on $d,\ \alpha$, and $p$.

Theorems & Definitions (51)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Remark 3.1
  • proof
  • Lemma 3.2
  • ...and 41 more