# Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas; Georgios E. Zouraris

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 4, page 841-874
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBournaveas, Nikolaos, and Zouraris, Georgios E.. "Theory and numerical approximations for a nonlinear 1 + 1 Dirac system." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 841-874. <http://eudml.org/doc/276381>.

@article{Bournaveas2012,

abstract = {We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.},

author = {Bournaveas, Nikolaos, Zouraris, Georgios E.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Existence; uniqueness; finite difference methods; error estimates; implicit-explicit finite difference method; nonlinear Dirac system},

language = {eng},

month = {2},

number = {4},

pages = {841-874},

publisher = {EDP Sciences},

title = {Theory and numerical approximations for a nonlinear 1 + 1 Dirac system},

url = {http://eudml.org/doc/276381},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Bournaveas, Nikolaos

AU - Zouraris, Georgios E.

TI - Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 4

SP - 841

EP - 874

AB - We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

LA - eng

KW - Existence; uniqueness; finite difference methods; error estimates; implicit-explicit finite difference method; nonlinear Dirac system

UR - http://eudml.org/doc/276381

ER -

## References

top- A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys.99 (1992) 348–350. Zbl0746.65090
- A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model. Phys. Lett. A86 (1981) 327–332.
- A. Alvarez, Pen-Yu Kuo and L. Vazquez, The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput.13 (1983) 1–15. Zbl0525.65071
- N. Bournaveas, Local and global solutions for a nonlinear Dirac system. Advances Differential Equations9 (2004) 677–698. Zbl1103.35087
- N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension. Discrete Contin. Dyn. Syst. Ser. A20 (2008) 605–616. Zbl1144.35306
- N. Boussaid, P. D’Ancona and L. Fanelli, Virial identity and weak dispersion for the magnetic Dirac equation. J. Math. Pures Appl.95 (2011) 137–150. Zbl1213.35149
- J. De Frutos, Estabilidad y convergencia de esquemas numericos para sistemas de Dirac no lineales. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingenieria5 (1989) 185–202.
- J. De Frutos and J.M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys.83 (1989) 407–423. Zbl0675.65131
- V. Delgado, Global solutions of the Cauchy problem for the classical Coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension. Proc. Amer. Math. Soc.69 (1978) 289–296. Zbl0351.35003
- T. Dupont, Galerkin methods for first order hyperbolics : an example. SIAM J Numer. Anal.10 (1973) 890–899. Zbl0237.65070
- R.T. Glassey, On one-dimensional coupled Dirac equations. Trans. Amer. Math. Soc.231 (1977) 531–539. Zbl0373.35009
- B.-Y. Guo, J. Shen and C.-L. Xu, Spectral and pseudospectral approximations using Hermite functions : application to the Dirac equation. Adv. Comput. Math.19 (2003) 35–55. Zbl1032.33004
- J. Hong and C. Li, Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys.211 (2006) 448–472. Zbl1120.65341
- L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer-Verlag (1997). Zbl0881.35001
- S. Jiménez, Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput.64 (1994) 13–45. Zbl0806.65081
- T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan19 (1967) 508–520. Zbl0163.38303
- S. Machihara, One dimensional Dirac equation with quadratic nonlinearities. Discrete Contin. Dyn. Syst. Ser. A13 (2005) 277–290. Zbl1077.35001
- S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension. Commun. Contemp. Math.9 (2007) 421–435. Zbl1119.35034
- S. Machihara, M. Nakamura and T. Ozawa, Small global solutions for nonlinear Dirac equations. Differential Integral Equations17 (2004) 623–636. Zbl1174.35452
- S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal.219 (2005) 1–20. Zbl1060.35025
- S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math.50 (2010) 403–451. Zbl1248.35170
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci.53 (1984). Zbl0537.76001
- E. Salusti and A. Tesei, On a semi-group approach to quantum field equations. Nuovo Cimento A2 (1971) 122–138.
- I. E. Segal, Non-linear semi-groups. Ann. of Math.78 (1963) 339–364. Zbl0204.16004
- S. Shao and H. Tang, Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete Contin. Dyn. Syst. Ser. B6 (2006) 623–640. Zbl1113.65095
- B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, Heidelberg (2010).
- H. Wang and H. Tang, An efficient adaptive mesh redistribution method for a non-linear Dirac equation. J. Comput. Phys.222 (2007) 176–193. Zbl1110.65085
- G.E. Zouraris, On the convergence of a linear conservative two-step finite element method for the nonlinear Schrödinger equation. ESAIM : M2AN35 (2001) 389–405.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.