Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2017, Article ID 3634815, 8 pages
https://doi.org/10.1155/2017/3634815
Research Article

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

School of Applied Sciences, Beijing Information Science and Technology University, Beijing 100192, China

Correspondence should be addressed to Jing-Jing Zhang; moc.361@2111gnahzgnijgnij

Received 18 June 2017; Revised 7 September 2017; Accepted 19 September 2017; Published 18 October 2017

Academic Editor: Francisco R. Villatoro

Copyright © 2017 Jing-Jing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society of London A, vol. 117, no. 778, pp. 610–624, 1928. View at Google Scholar
  2. C. D. Anderson, “The positive electron,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 43, no. 6, pp. 491–494, 1933. View at Publisher · View at Google Scholar · View at Scopus
  3. K. S. Novoselov, A. K. Geim, S. V. Morozov et al., “Two-dimensional gas of massless Dirac fermions in graphene,” Nature, vol. 438, no. 7065, pp. 197–200, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D: Nonlinear Phenomena, vol. 238, no. 15, pp. 1413–1421, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Physical Review Letters, vol. 80, no. 23, pp. 5117–5120, 1998. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Xu, S. Shao, H. Tang, and D. Wei, “Multi-hump solitary waves of a nonlinear Dirac equation,” Communications in Mathematical Sciences, vol. 13, no. 5, pp. 1219–1242, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. Xu, S. Shao, and H. Tang, “Numerical methods for nonlinear Dirac equation,” Journal of Computational Physics, vol. 245, pp. 131–149, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. Y. Lee, T. K. Kuo, and A. Gavrielides, “Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions,” Physical Review D Particles & Fields, vol. 12, no. 8, pp. 2249–2253, 1975. View at Google Scholar
  9. P. Mathieu, “Soliton solutions for Dirac equations with homogeneous non-linearity in (1+1) dimensions,” Journal of Physics A: Mathematical and General, vol. 18, no. 16, pp. L1061–L1066, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. Shao, N. R. Quintero, F. G. Mertens, F. Cooper, A. Khare, and A. Saxena, “Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity,” Physical Review E, vol. 90, no. 3, Article ID 032915, 2014. View at Google Scholar
  11. A. Alvarez, P. Y. Kuo, and L. Vzquez, “The numerical study of a nonlinear one-dimensional Dirac equation,” Applied Mathematics and Computation, vol. 13, no. 1-2, pp. 1–15, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Alvarez, “Linearized CRAnk-NICholson scheme for nonlinear DIRac equations [CORrected title: Linearized CRAnk-NIColson scheme for nonlinear DIRac equations],” Journal of Computational Physics, vol. 99, no. 2, pp. 348–350, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. de Frutos and J. M. Sanz-Serna, “Split-step spectral schemes for nonlinear Dirac systems,” Journal of Computational Physics, vol. 83, no. 2, pp. 407–423, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. Z.-q. Wang and B.-y. Guo, “Modified Legendre rational spectral method for the whole line,” Journal of Computational Mathematics, vol. 22, no. 3, pp. 457–474, 2004. View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. Shao and H. Tang, “Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model,” Discrete and Continuous Dynamical Systems - Series B, vol. 6, no. 3, pp. 623–640, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. Bao, Y. Cai, X. Jia, and J. Yin, “Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime,” Science China Mathematics, vol. 59, no. 8, pp. 1461–1494, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. Hong and C. Li, “Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations,” Journal of Computational Physics, vol. 211, no. 2, pp. 448–472, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. H. Wang and H. Tang, “An efficient adaptive mesh redistribution method for a non-linear Dirac equation,” Journal of Computational Physics, vol. 222, no. 1, pp. 176–193, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. F. de la Hoz and F. Vadillo, “An integrating factor for nonlinear Dirac equations,” Computer Physics Communications, vol. 181, no. 7, pp. 1195–1203, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. S. C. Li, X. G. Li, and F. Y. Shi, “Time-splitting methods with charge conservation for the nonlinear Dirac equation,” Numerical Methods for Partial Differential Equations, vol. 33, no. 5, pp. 1582–1602, 2017. View at Google Scholar
  21. W. Bao and Y. Cai, “Ground states and dynamics of spin-orbit-coupled Bose-Einstein condensates,” SIAM Journal on Applied Mathematics, vol. 75, no. 2, pp. 492–517, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. W. Bao and X.-G. Li, “An efficient and stable numerical method for the Maxwell-Dirac system,” Journal of Computational Physics, vol. 199, no. 2, pp. 663–687, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X.-G. Li, C. K. Chan, and Y. Hou, “A numerical method with particle conservation for the Maxwell-Dirac system,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1096–1108, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Z. Huang, S. Jin, P. A. Markowich, C. Sparber, and C. Zheng, “A time-splitting spectral scheme for the Maxwell-Dirac system,” Journal of Computational Physics, vol. 208, no. 2, pp. 761–789, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. Q. Nie, Y.-T. Zhang, and R. Zhao, “Efficient semi-implicit schemes for stiff systems,” Journal of Computational Physics, vol. 214, no. 2, pp. 521–537, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. S. Chen and Y.-T. Zhang, “Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods,” Journal of Computational Physics, vol. 230, no. 11, pp. 4336–4352, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. T. Jiang and Y.-T. Zhang, “Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations,” Journal of Computational Physics, vol. 253, pp. 368–388, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Soler, “Classical, stable, Classical, stable, nonlinear spinor field with positive rest energy,” Physical Review D Particles & Fields, vol. 1, no. 10, pp. 2766–2769, 1970. View at Google Scholar
  29. A. Alvarez and B. Carreras, “Interaction dynamics for the solitary waves of a nonlinear Dirac model,” Physics Letters A, vol. 86, no. 6-7, pp. 327–332, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. S. Shao and H. Tang, “Interaction of solitary waves with a phase shift in a nonlinear Dirac model,” Communications in Computational Physics, vol. 3, no. 4, pp. 950–967, 2008. View at Google Scholar · View at MathSciNet · View at Scopus
  31. T. Wang, B. Guo, and Q. Xu, “Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions,” Journal of Computational Physics, vol. 243, pp. 382–399, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. X.-G. Li, J. Zhu, R.-P. Zhang, and S. Cao, “A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation,” Journal of Computational Physics, vol. 275, pp. 363–376, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. S. C. Li, X. G. Li, J. J. Cao, and W. B. Li, “High-order numerical method for the derivative nonlinear Schrödinger equation,” International Journal of Modeling, Simulation & Scientific Computing, vol. 8, Article ID 1750017, 2017. View at Google Scholar
  34. G. Berkolaiko and A. Comech, “On spectral stability of solitary waves of nonlinear dirac equation in 1D,” Mathematical Modelling of Natural Phenomena, vol. 7, no. 2, pp. 13–31, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  35. A. Contreras, D. E. Pelinovsky, and Y. Shimabukuro, “L2 orbital Stability of Dirac Solitons in the Massive Thirring Model,” Communications in Partial Differential Equations, vol. 41, no. 2, pp. 227–255, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  36. J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena, A. Comech, and R. Lan, “Stability of solitary waves and vortices in a 2D nonlinear DIRac model,” Physical Review Letters, vol. 116, no. 21, Article ID 214101, 214101, 6 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus