Free Access
Issue
ESAIM: M2AN
Volume 46, Number 4, July-August 2012
Page(s) 841 - 874
DOI https://doi.org/10.1051/m2an/2011071
Published online 03 February 2012
  1. A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys. 99 (1992) 348–350. [CrossRef] [Google Scholar]
  2. A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model. Phys. Lett. A 86 (1981) 327–332. [CrossRef] [Google Scholar]
  3. 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. [CrossRef] [Google Scholar]
  4. N. Bournaveas, Local and global solutions for a nonlinear Dirac system. Advances Differential Equations 9 (2004) 677–698. [Google Scholar]
  5. N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension. Discrete Contin. Dyn. Syst. Ser. A 20 (2008) 605–616. [CrossRef] [Google Scholar]
  6. 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. [CrossRef] [Google Scholar]
  7. 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 Ingenieria 5 (1989) 185–202. [Google Scholar]
  8. J. De Frutos and J.M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys. 83 (1989) 407–423. [CrossRef] [Google Scholar]
  9. 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. [CrossRef] [MathSciNet] [Google Scholar]
  10. T. Dupont, Galerkin methods for first order hyperbolics : an example. SIAM J Numer. Anal. 10 (1973) 890–899. [CrossRef] [MathSciNet] [Google Scholar]
  11. R.T. Glassey, On one-dimensional coupled Dirac equations. Trans. Amer. Math. Soc. 231 (1977) 531–539. [CrossRef] [MathSciNet] [Google Scholar]
  12. 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. [CrossRef] [MathSciNet] [Google Scholar]
  13. J. Hong and C. Li, Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211 (2006) 448–472. [CrossRef] [Google Scholar]
  14. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer-Verlag (1997). [Google Scholar]
  15. S. Jiménez, Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput. 64 (1994) 13–45. [CrossRef] [Google Scholar]
  16. T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19 (1967) 508–520. [CrossRef] [MathSciNet] [Google Scholar]
  17. S. Machihara, One dimensional Dirac equation with quadratic nonlinearities. Discrete Contin. Dyn. Syst. Ser. A 13 (2005) 277–290. [CrossRef] [Google Scholar]
  18. S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension. Commun. Contemp. Math. 9 (2007) 421–435. [CrossRef] [MathSciNet] [Google Scholar]
  19. S. Machihara, M. Nakamura and T. Ozawa, Small global solutions for nonlinear Dirac equations. Differential Integral Equations 17 (2004) 623–636. [MathSciNet] [Google Scholar]
  20. 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. [CrossRef] [MathSciNet] [Google Scholar]
  21. S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50 (2010) 403–451. [CrossRef] [MathSciNet] [Google Scholar]
  22. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci. 53 (1984). [Google Scholar]
  23. E. Salusti and A. Tesei, On a semi-group approach to quantum field equations. Nuovo Cimento A 2 (1971) 122–138. [Google Scholar]
  24. I. E. Segal, Non-linear semi-groups. Ann. of Math. 78 (1963) 339–364. [CrossRef] [MathSciNet] [Google Scholar]
  25. S. Shao and H. Tang, Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 623–640. [CrossRef] [MathSciNet] [Google Scholar]
  26. B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, Heidelberg (2010). [Google Scholar]
  27. H. Wang and H. Tang, An efficient adaptive mesh redistribution method for a non-linear Dirac equation. J. Comput. Phys. 222 (2007) 176–193. [CrossRef] [Google Scholar]
  28. G.E. Zouraris, On the convergence of a linear conservative two-step finite element method for the nonlinear Schrödinger equation. ESAIM : M2AN 35 (2001) 389–405. [CrossRef] [EDP Sciences] [Google Scholar]

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