scholarly journals Nonrelativistic limit of solitary waves for nonlinear Maxwell–Klein–Gordon equations

2021 ◽  
Vol 60 (5) ◽  
Author(s):  
Sangdon Jin ◽  
Jinmyoung Seok
Keyword(s):  
Solitary Waves ◽  
Nonrelativistic Limit ◽  
Klein Gordon

scholarly journals Multi-Solitary Waves for the Nonlinear Klein-Gordon Equation

2014 ◽  
Vol 39 (8) ◽  
pp. 1479-1522 ◽  
Author(s):  
Jacopo Bellazzini ◽  
Marco Ghimenti ◽  
Stefan Le Coz
Keyword(s):  
Solitary Waves ◽  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

scholarly journals ANY l-STATE ANALYTICAL SOLUTIONS OF THE KLEIN–GORDON EQUATION FOR THE WOODS–SAXON POTENTIAL

2010 ◽  
Vol 19 (07) ◽  
pp. 1463-1475 ◽  
Author(s):  
V. H. BADALOV ◽  
H. I. AHMADOV ◽  
S. V. BADALOV
Keyword(s):  
State Energy ◽  
Gordon Equation ◽  
Bound State ◽  
Nonrelativistic Limit ◽  
Saxon Potential ◽  
Bound State Energy ◽  
Klein Gordon ◽  
Uvarov Method ◽  
Pekeris Approximation ◽  
Klein Gordon Equation

The radial part of the Klein–Gordon equation for the Woods–Saxon potential is solved. In our calculations, we have applied the Nikiforov–Uvarov method by using the Pekeris approximation to the centrifugal potential for any l-states. The exact bound state energy eigenvalues and the corresponding eigenfunctions are obtained on the various values of the quantum numbers n and l. The nonrelativistic limit of the bound state energy spectrum was also found.

SOLITARY WAVES OF THE NONLINEAR KLEIN-GORDON EQUATION COUPLED WITH THE MAXWELL EQUATIONS

2002 ◽  
Vol 14 (04) ◽  
pp. 409-420 ◽  
Author(s):  
VIERI BENCI ◽  
DONATO FORTUNATO FORTUNATO
Keyword(s):  
Electromagnetic Field ◽  
Solitary Wave ◽  
Electric Field ◽  
Solitary Waves ◽  
Maxwell Equations ◽  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

This paper is divided in two parts. In the first part we construct a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field. In the second part we study the electrostatic case. We prove the existence of infinitely many pairs (ψ, E), where ψ is a solitary wave for the nonlinear Klein-Gordon equation and E is the electric field related to ψ.

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.

Relativistic quantum bouncing particles in a homogeneous gravitational field

2021 ◽  
pp. 2150098
Author(s):  
Ar Rohim ◽  
Kazushige Ueda ◽  
Kazuhiro Yamamoto ◽  
Shih-Yuin Lin
Keyword(s):  
Gravitational Field ◽  
Relativistic Effect ◽  
Relativistic Quantum ◽  
Energy Levels ◽  
Nonrelativistic Limit ◽  
Wave Functions ◽  
Dirac Equations ◽  
Rindler Coordinates ◽  
Klein Gordon ◽  
Homogeneous Gravitational Field

In this paper, we study the relativistic effect on the wave functions for a bouncing particle in a gravitational field. Motivated by the equivalence principle, we investigate the Klein–Gordon and Dirac equations in Rindler coordinates with the boundary conditions mimicking a uniformly accelerated mirror in Minkowski space. In the nonrelativistic limit, all these models in the comoving frame reduce to the familiar eigenvalue problem for the Schrödinger equation with a fixed floor in a linear gravitational potential, as expected. We find that the transition frequency between two energy levels of a bouncing Dirac particle is greater than the counterpart of a Klein–Gordon particle, while both are greater than their nonrelativistic limit. The different corrections to eigen-energies of particles of different nature are associated with the different behaviors of their wave functions around the mirror boundary.

scholarly journals Instability of solitary waves for the generalized Klein–Gordon–Hartree equation

2019 ◽  
Author(s):  
Boyan Kostadinov ◽  
Mirko Tarulli ◽  
George Venkov
Keyword(s):  
Solitary Waves ◽  
Hartree Equation ◽  
Klein Gordon
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