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On soliton asymptotics for 2D Maxwell-Lorentz equations with rotating particle

Elena Kopylova

TL;DR

This work analyzes the 2D Maxwell-Lorentz system for an extended rotating charge, proving long-time stability of soliton solutions that travel with constant velocity and rotate with a fixed rate. The authors develop a modulation framework built on a symplectic orthogonal projection to the soliton manifold, derive a linearized transversal dynamics generator, and establish decay via resolvent analysis under a Wiener-type spectral condition (the M-condition). They show that solutions near a soliton decompose into a moving soliton with asymptotic parameters and dispersive radiation, with the transversal component decaying like $t^{-2}$ and the radiative field contributing a $t^{-1}$-decay in energy norm. The results extend prior work by treating nonzero rotation ($\omega$) and identify a regime (large inertia $I$) where the spectral condition holds, yielding rigorous asymptotic stability for a broad class of initial data. This advances understanding of soliton asymptotics and radiation damping in coupled field-particle systems in two dimensions.

Abstract

We consider 2D Maxwell-Lorentz equations with extended charged rotating particle. The system admits solitons which are solutions corresponding to a particle moving with a constant velocity and rotating with a constant angular velocity. Our main result is asymptotic stability of the solitons.

On soliton asymptotics for 2D Maxwell-Lorentz equations with rotating particle

TL;DR

This work analyzes the 2D Maxwell-Lorentz system for an extended rotating charge, proving long-time stability of soliton solutions that travel with constant velocity and rotate with a fixed rate. The authors develop a modulation framework built on a symplectic orthogonal projection to the soliton manifold, derive a linearized transversal dynamics generator, and establish decay via resolvent analysis under a Wiener-type spectral condition (the M-condition). They show that solutions near a soliton decompose into a moving soliton with asymptotic parameters and dispersive radiation, with the transversal component decaying like and the radiative field contributing a -decay in energy norm. The results extend prior work by treating nonzero rotation () and identify a regime (large inertia ) where the spectral condition holds, yielding rigorous asymptotic stability for a broad class of initial data. This advances understanding of soliton asymptotics and radiation damping in coupled field-particle systems in two dimensions.

Abstract

We consider 2D Maxwell-Lorentz equations with extended charged rotating particle. The system admits solitons which are solutions corresponding to a particle moving with a constant velocity and rotating with a constant angular velocity. Our main result is asymptotic stability of the solitons.
Paper Structure (19 sections, 20 theorems, 240 equations)

This paper contains 19 sections, 20 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian structure and well-posedness
  3. Solitons
  4. Main result
  5. Symplectic Structure
  6. Linearization on the solitary manifold
  7. The Linearized Equation
  8. Symplectic Decomposition of the Dynamics
  9. Frozen Transversal Dynamics
  10. Decay of linearized dynamics
  11. Decay of transversal componentI
  12. Soliton Asymptotics
  13. Linearized dynamics
  14. Solving the linearized equation
  15. Asymptotics of matrix $L^{-1}(\lambda)$
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let rosym holds, and let $Y_0=(A_0, \Pi_0, q_0, p_0)\in {\cal E}$. Then (i) there exists a unique solution $Y(t)\in C(\mathbb{R}, {\cal E})$ to the Cauchy problem for mls3; (ii) the energy conserves: $H(Y(t))=H(Y_0)$ for $t\in R$; (iii) the estimate holds,

Theorems & Definitions (37)

  • Proposition 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • Definition 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Corollary 5.4
  • Lemma 7.1
  • proof
  • ...and 27 more