Free Access
Volume 46, Number 4, July-August 2012
Page(s) 841 - 874
Published online 03 February 2012
  1. A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys. 99 (1992) 348–350. [CrossRef]
  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]
  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]
  4. N. Bournaveas, Local and global solutions for a nonlinear Dirac system. Advances Differential Equations 9 (2004) 677–698.
  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]
  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]
  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.
  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]
  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]
  10. T. Dupont, Galerkin methods for first order hyperbolics : an example. SIAM J Numer. Anal. 10 (1973) 890–899. [CrossRef] [MathSciNet]
  11. R.T. Glassey, On one-dimensional coupled Dirac equations. Trans. Amer. Math. Soc. 231 (1977) 531–539. [CrossRef] [MathSciNet]
  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]
  13. J. Hong and C. Li, Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211 (2006) 448–472. [CrossRef]
  14. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer-Verlag (1997).
  15. S. Jiménez, Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput. 64 (1994) 13–45. [CrossRef]
  16. T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19 (1967) 508–520. [CrossRef] [MathSciNet]
  17. S. Machihara, One dimensional Dirac equation with quadratic nonlinearities. Discrete Contin. Dyn. Syst. Ser. A 13 (2005) 277–290. [CrossRef]
  18. S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension. Commun. Contemp. Math. 9 (2007) 421–435. [CrossRef] [MathSciNet]
  19. S. Machihara, M. Nakamura and T. Ozawa, Small global solutions for nonlinear Dirac equations. Differential Integral Equations 17 (2004) 623–636. [MathSciNet]
  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]
  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]
  22. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci. 53 (1984).
  23. E. Salusti and A. Tesei, On a semi-group approach to quantum field equations. Nuovo Cimento A 2 (1971) 122–138. [CrossRef]
  24. I. E. Segal, Non-linear semi-groups. Ann. of Math. 78 (1963) 339–364. [CrossRef] [MathSciNet]
  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]
  26. B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, Heidelberg (2010).
  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]
  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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you