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On global attraction to solitons for 3D Maxwell-Lorentz equations

E. A. Kopylova, A. I. Komech

TL;DR

This work proves a global attraction to solitons for the 3D Maxwell–Lorentz system with a rotating charge at the origin. By reformulating in Maxwell potentials and exploiting a Kirchhoff representation, it couples orbital stability with a carefully constructed trajectory modification to obtain sharp large-time convergence to a particular soliton $Y_{\pm\omega}$ in any local energy region. The analysis relies on a Lyapunov-type functional built from the Hamiltonian and invariants (Casimirs), together with decay of the radiative part and a nonresonance condition on the bare mass. The results extend nonlinear stability and asymptotics to the rotating case, demonstrating that finite-energy dynamics disperse and settle to stationary solitons in the Maxwell field.

Abstract

We consider the Maxwell field coupled to a single rotating charge. This Hamiltonian system admits soliton-type solutions, where the field is static, while the charge rotates with constant angular velocity. We prove that any solution of finite energy converges, in suitable local energy seminorms, to the corresponding soliton in the long time limit.

On global attraction to solitons for 3D Maxwell-Lorentz equations

TL;DR

This work proves a global attraction to solitons for the 3D Maxwell–Lorentz system with a rotating charge at the origin. By reformulating in Maxwell potentials and exploiting a Kirchhoff representation, it couples orbital stability with a carefully constructed trajectory modification to obtain sharp large-time convergence to a particular soliton in any local energy region. The analysis relies on a Lyapunov-type functional built from the Hamiltonian and invariants (Casimirs), together with decay of the radiative part and a nonresonance condition on the bare mass. The results extend nonlinear stability and asymptotics to the rotating case, demonstrating that finite-energy dynamics disperse and settle to stationary solitons in the Maxwell field.

Abstract

We consider the Maxwell field coupled to a single rotating charge. This Hamiltonian system admits soliton-type solutions, where the field is static, while the charge rotates with constant angular velocity. We prove that any solution of finite energy converges, in suitable local energy seminorms, to the corresponding soliton in the long time limit.
Paper Structure (10 sections, 11 theorems, 80 equations)

This paper contains 10 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The Maxwell potentials
  3. Well posedness
  4. Kirchhoff representation of solutions
  5. Orbital stability of solitons
  6. Main result
  7. Modification of the trajectory
  8. Modified fields
  9. Asymptotics for large $t$
  10. Proof of Proposition \ref{['5.1']}

Key Result

Proposition 3.2

For any initial state $Y(0)=(A(x,0),\Pi(x,0),\omega(0))\in{\mathbb{Y}}$, the equation (eq-Y) admits a unique solution ii) The map $W(t):Y(0)\mapsto Y(t)$ is continuous in ${\mathbb{Y}}$ for every $t\in{\rm I R}$. iii) The energy is conserved: iii) The estimate holds,

Theorems & Definitions (18)

  • Definition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Proposition 5.1
  • Theorem 6.1
  • Proposition 6.2
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • ...and 8 more