Time-splitting methods with charge conservation for the nonlinear Dirac equation

2017 ◽  
Vol 33 (5) ◽  
pp. 1582-1602 ◽  
Author(s):  
Shu-Cun Li ◽  
Xiang-Gui Li ◽  
Fang-Yuan Shi
Keyword(s):  
Dirac Equation ◽  
Splitting Methods ◽  
Charge Conservation ◽  
Nonlinear Dirac Equation ◽  
Time Splitting

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

2014 ◽  
Vol 6 (4) ◽  
pp. 494-514 ◽  
Author(s):  
Yaming Chen ◽  
Songhe Song ◽  
Huajun Zhu
Keyword(s):  
Dirac Equation ◽  
Hamiltonian System ◽  
Numerical Experiments ◽  
Pseudospectral Method ◽  
Splitting Methods ◽  
Fourier Pseudospectral Method ◽  
Composition Method ◽  
Nonlinear Dirac Equation ◽  
Symplectic Scheme ◽  
Symplectic System

AbstractIn this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

scholarly journals On the nonlinear Dirac equation on noncompact metric graphs

2021 ◽  
Vol 278 ◽  
pp. 326-357
Author(s):  
William Borrelli ◽  
Raffaele Carlone ◽  
Lorenzo Tentarelli
Keyword(s):  
Dirac Equation ◽  
Metric Graphs ◽  
Nonlinear Dirac Equation

scholarly journals Solitary waves in the nonlinear Dirac equation in the presence of external driving forces

2016 ◽  
Vol 49 (6) ◽  
pp. 065402 ◽  
Author(s):  
Franz G Mertens ◽  
Fred Cooper ◽  
Niurka R Quintero ◽  
Sihong Shao ◽  
Avinash Khare ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Solitary Waves ◽  
Driving Forces ◽  
Nonlinear Dirac Equation ◽  
External Driving

Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

2021 ◽  
pp. 1-26
Author(s):  
Tianfang Wang ◽  
Wen Zhang ◽  
Jian Zhang
Keyword(s):  
Dirac Equation ◽  
Coulomb Potential ◽  
Ground State ◽  
Continuous Dependence ◽  
Energy Functional ◽  
Ground State Solution ◽  
State Solution ◽  
Nonlinear Dirac Equation ◽  
Relationship Of ◽  
Least Energy

In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ), α i and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0.

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