Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the motion of charged particles in constant electromagnetic field: the parallel case

Shuang Miao, Shiwu Yang, Pin Yu

Abstract

This paper is devoted to presenting a rigorous mathematical derivation for the classical phenomenon in Maxwell's theory that a charged particle moves along a straight line in a constant electromagnetic field if the initial velocity is parallel to the constant electromagnetic field. The particle is modeled by scaled solitons to a class of nonlinear Klein-Gordon equations and the nonlinear interaction between the charged particle and the electromagnetic field is governed by the Maxwell-Klein-Gordon system. We show that when the size and amplitude of the particle are sufficiently small, the solution to the coupled nonlinear system exists up to any given time and the energy of the particle concentrates along a straight line. The method relies on the modulation approach for the study of stability for solitons and weighted energy estimates for the Maxwell-Klein-Gordon equations.

On the motion of charged particles in constant electromagnetic field: the parallel case

Abstract

This paper is devoted to presenting a rigorous mathematical derivation for the classical phenomenon in Maxwell's theory that a charged particle moves along a straight line in a constant electromagnetic field if the initial velocity is parallel to the constant electromagnetic field. The particle is modeled by scaled solitons to a class of nonlinear Klein-Gordon equations and the nonlinear interaction between the charged particle and the electromagnetic field is governed by the Maxwell-Klein-Gordon system. We show that when the size and amplitude of the particle are sufficiently small, the solution to the coupled nonlinear system exists up to any given time and the energy of the particle concentrates along a straight line. The method relies on the modulation approach for the study of stability for solitons and weighted energy estimates for the Maxwell-Klein-Gordon equations.

Paper Structure

This paper contains 21 sections, 16 theorems, 327 equations.

Table of Contents

  1. Introduction
  2. The Maxwell-Klein-Gordon system and stable solitons
  3. Related works
  4. Comments on the proof
  5. Stable solitons in Minkowski space
  6. Stability of stable solitons on a fixed electromagnetic field
  7. Orthogonality condition and modulation equations
  8. Nondegeneracy of the Modulation Equations
  9. Initial Data
  10. Bootstrap assumptions
  11. Estimates for the Modulation Curve
  12. Energy Decomposition
  13. Weighted energy estimates in the exterior region
  14. Energy Estimates for the Full Solution
  15. Improving the Bootstrap Assumptions on the Remainder Terms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider the Cauchy problem to the system eq:MKG:scaled. Let the assumptions EM initial constraint, soliton initial, soliton initial weight, and EM initial hold. Let $T>0$ be any given time. Then for all $2\leq p < \frac{7}{3}$ and all $\lambda_0=(\omega_0, \theta_0, 0, u_0)\in\Lambda$ such that $u_ such that and the solution $\phi$ is close to the translated solitons under the Lorentz gauge con

Theorems & Definitions (25)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 15 more