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Long-time asymptotics of (1,3)-sign solitary waves for the damped nonlinear Klein-Gordon equation

Kenjiro Ishizuka

TL;DR

This paper establishes the long-time behavior of a four-soliton configuration for the damped nonlinear Klein-Gordon equation in dimensions $2\le d\le 5$ with subcritical nonlinearity. By formulating a modulation framework and deriving a closed ODE system for inter-soliton geometry via inner products, the authors show that a $(1,3)$-sign state forces the three like-signed solitons to converge into an expanding equilateral-triangle configuration around the oppositely signed soliton, with centers following a refined logarithmic-in-time law $z_k(t)=z_{\infty}+(\log t-(d-1)/2\log\log t+c_0)\omega_k+\text{lower-order}$. A Lyapunov functional and a two-tier separation of time scales separate angular dynamics from radial distances, enabling a sharp equilateral-triangle decomposition and precise asymptotics for the soliton centers. The result advances soliton-resolution-type understanding for the damped NLKG in higher dimensions by showing a robust, symmetry-independent equilateral geometry for $(1,3)$-sign multi-solitons and providing a detailed dynamical description of their centers. Overall, the work combines modulation analysis, interaction estimates, and geometric reductions to deliver a complete long-time characterization of this four-soliton regime.

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 $α>0$, $2\leq d\leq 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that any solution which is asymptotic to a superposition of four solitons with exactly one soliton of opposite sign evolves so that the three like-signed solitons spread out in an equilateral-triangle configuration centered at the oppositely signed soliton.

Long-time asymptotics of (1,3)-sign solitary waves for the damped nonlinear Klein-Gordon equation

TL;DR

This paper establishes the long-time behavior of a four-soliton configuration for the damped nonlinear Klein-Gordon equation in dimensions with subcritical nonlinearity. By formulating a modulation framework and deriving a closed ODE system for inter-soliton geometry via inner products, the authors show that a -sign state forces the three like-signed solitons to converge into an expanding equilateral-triangle configuration around the oppositely signed soliton, with centers following a refined logarithmic-in-time law . A Lyapunov functional and a two-tier separation of time scales separate angular dynamics from radial distances, enabling a sharp equilateral-triangle decomposition and precise asymptotics for the soliton centers. The result advances soliton-resolution-type understanding for the damped NLKG in higher dimensions by showing a robust, symmetry-independent equilateral geometry for -sign multi-solitons and providing a detailed dynamical description of their centers. Overall, the work combines modulation analysis, interaction estimates, and geometric reductions to deliver a complete long-time characterization of this four-soliton regime.

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 any solution which is asymptotic to a superposition of four solitons with exactly one soliton of opposite sign evolves so that the three like-signed solitons spread out in an equilateral-triangle configuration centered at the oppositely signed soliton.
Paper Structure (25 sections, 31 theorems, 321 equations)

This paper contains 25 sections, 31 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Long-time asymptotics of multi solitary waves
  4. Main result
  5. Difficulties and Ideas for the main result
  6. Previous results
  7. Notation
  8. Preliminaries
  9. Basic properties of the ground state
  10. Basic properties of the dynamics of $K$-solitary waves
  11. Soliton interactions for $(1,3)$-sign solitary waves
  12. Estimates on interactions between like-signed solitons
  13. Basic properties of $(1,3)$-sign solitary waves
  14. Distance analysis I
  15. Distance analysis II
  16. ...and 10 more sections

Key Result

Theorem 1.1

If a solution $\vec{u}$ of DNKG is a $(1,3)$-sign solitary waves, there exist $z_{\infty}\in \mathbb{R}^d$, an equilateral triple $\omega_1,\omega_2,\omega_3\in S^{d-1}$, and a function $z_k:[0,\infty)\to \mathbb{R}^d$ for $k=1,2,3$ such that for all $t>2$, and for $k=1,2,3$, where $c_0$ is a constant depending only on $d,\ \alpha$, and $p$.

Theorems & Definitions (73)

  • Definition 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.3
  • Remark 1.5
  • Theorem 1.1
  • Remark 1.6
  • ...and 63 more