Sign-changing solutions for a kind of Klein–Gordon–Maxwell system

2021 ◽  
Vol 62 (9) ◽  
pp. 091507
Author(s):  
Qi Zhang
Keyword(s):  
Maxwell System ◽  
Klein Gordon ◽  
Sign Changing Solutions

scholarly journals ON THE ASYMPTOTIC ANALYSIS OF THE DIRAC–MAXWELL SYSTEM IN THE NONRELATIVISTIC LIMIT

2005 ◽  
Vol 02 (01) ◽  
pp. 129-182 ◽  
Author(s):  
PHILIPPE BECHOUCHE ◽  
NORBERT J. MAUSER ◽  
SIGMUND SELBERG
Keyword(s):  
Convergence Result ◽  
Nonrelativistic Limit ◽  
Dirac Spinor ◽  
Maxwell System ◽  
Poisson System ◽  
Pauli Equation ◽  
Existence Time ◽  
Self Consistent Field ◽  
Klein Gordon ◽  
Well Posed

We study the behavior of solutions of the Dirac–Maxwell system (DM) in the nonrelativistic limit c → ∞, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c → ∞, sometimes also called post-Newtonian, yields a Schrödinger–Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H1 data (for fixed c), and that as c → ∞ the existence time grows at least as fast as log(c), provided the data are uniformly bounded in H1. Moreover, if the datum for the Dirac spinor converges in H1, then the solution of DM converges, modulo a phase correction, in C([0,T];H1) to a solution of a Schrödinger–Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit. The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Klein–Gordon–Maxwell system.

scholarly journals Positive ground state solutions for the critical Klein–Gordon–Maxwell system with potentials

2012 ◽  
Vol 75 (10) ◽  
pp. 4068-4078 ◽  
Author(s):  
Paulo C. Carrião ◽  
Patrícia L. Cunha ◽  
Olímpio H. Miyagaki
Keyword(s):  
Ground State ◽  
Maxwell System ◽  
Ground State Solutions ◽  
Klein Gordon

scholarly journals Wave dispersion derived from the square-root Klein–Gordon–Poisson system

2013 ◽  
Vol 79 (4) ◽  
pp. 371-376 ◽  
Author(s):  
F. HAAS
Keyword(s):  
Relativistic Quantum ◽  
Relativistic Effects ◽  
Fundamental Aspect ◽  
Physical Parameters ◽  
Linear Wave ◽  
Relativistic Energy ◽  
Square Root ◽  
Maxwell System ◽  
Poisson System ◽  
Klein Gordon

AbstractRecently, there has been great interest around quantum relativistic models for plasmas. In particular, striking advances have been obtained by means of the Klein–Gordon–Maxwell system, which provides a first-order approach to the relativistic regimes of quantum plasmas. The Klein–Gordon–Maxwell system provides a reliable model as long as the plasma spin dynamics is not a fundamental aspect, to be addressed using more refined (and heavier) models involving the Pauli–Schrödinger or Dirac equations. In this work, a further simplification is considered, tracing back to the early days of relativistic quantum theory. Namely, we revisit the square-root Klein–Gordon–Poisson system, where the positive branch of the relativistic energy–momentum relation is mapped to a quantum wave equation. The associated linear wave propagation is analyzed and compared with the results in the literature. We determine physical parameters where the simultaneous quantum and relativistic effects can be noticeable in weakly coupled electrostatic plasmas.

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