Exact Localized Solutions Of The Nonlinear Dirac Equation (original) (raw)

Survey on global existence in the nonlinear Dirac equations in one dimension

2010

We consider the nonlinear Dirac equations in one dimension and review various results on global existence of solutions in H 1. Depending on the character of the nonlinear terms, existence of the large-norm solutions can be extended for all times. Global existence of the small-norm solutions is proved for the most general nonlinear Dirac equations with cubic and higher-order nonlinear terms. Integrability of the massive Thirring model is used to find conditions that no solitons occur in the Cauchy problem with small initial data in a subspace of L 2 .

The 1-D Dirac equation with concentrated nonlinearity

2016

We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-Seba models and the concentrated versions of the Bragg Resonance, Gross-Neveu, and Soler type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable (the "charge").

The One-Dimensional Dirac Equation With Concentrated Nonlinearity

SIAM Journal on Mathematical Analysis

We define and study the Cauchy problem for a one-dimensional (1-D) nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-Šeba models and the concentrated versions of the Bragg resonance and 1-D Soler (also known as massive Gross-Neveu) type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable.

Localized states of Dirac equation

arXiv: Quantum Physics, 2019

In this paper, we introduce an extension of the Dirac equation, very similar to Dirac oscillator, that gives stationary localized wave packets as eigenstates of the equation. The extension to the Dirac equation is achieved through the replacement of the momentum operator by a PT-symmetric generalized momentum operator. In the 1D case, the solutions represent bound particles carrying spin having continuous energy spectrum, where the envelope parameter defines the width of the packet without affecting the dispersion relation of the original Dirac equation. In the 2D case, the solutions are localized wave packets and are eigenstates of the third component of total angular momentum and involve Bessel functions of integral order. In the 3D case, the solutions are localized spherical wave packets with definite total angular momentum.

Some Exact Solutions of the Dirac Equation

Hadron Physics 2000 - Topics on the Structure and Interaction of Hadronic Systems - Proceedings of the International Workshop, 2001

Exact analytic solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions.

Approximate analytic solutions to coupled nonlinear Dirac equations

Physics Letters A, 2017

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalarscalar self interactions g 2 1 2 (ψψ) 2 + g 2 2 2 (φφ) 2 + g 2 3 (ψψ)(φφ) as well as vector-vector interactions of the form g 2 1 2 (ψγµψ)(ψγ µ ψ) + g 2 2 2 (φγµφ)(φγ µ φ) + g 2 3 (ψγµψ)(φγ µ φ). Writing the two components of the assumed solitary wave solution of these equation in the form ψ = e −iω 1 t {R1 cos θ, R1 sin θ}, φ = e −iω 2 t {R2 cos η, R2 sin η}, and assuming that θ(x), η(x) have the same functional form they had when g3=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 2 3 /g 2 2 and g 2 3 /g 2 1. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.