References
- D.A. Abanin, S.V. Morozov, L.A. Ponomarenko, R.V. Gorbachev, A.S. Mayorov, M.I. Katsnelson, K. Watanabe, T. Taniguchi, K.S. Novoselov, L.S. Levito, and A.K. Geim, Giant nonlocality near the Dirac point in graphene, Science 332 (2011), pp. 328–330. doi: 10.1126/science.1199595
- A. Alvarez, Linearized Crank--Nicolson scheme for nonlinear Dirac equations, J. Comput. Phys. 99 (1992), pp. 348–350. doi: 10.1016/0021-9991(92)90214-J
- A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A 86 (1981), pp. 327–332. doi: 10.1016/0375-9601(81)90548-X
- A. Alvarez, P.Y. Kuo, and L. Vázquez, The numerical study of a nonlinear one-dimensional Dirac equation, Appl. Math. Comput. 13 (1983), pp. 1–15.
- C.D. Anderson, The positive electron, Phys. Rev. 43 (1933), pp. 491–498. doi: 10.1103/PhysRev.43.491
- W.Z. Bao, Y.Y. Cai, X.W. Jia, and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, Sci. China Math. 59 (2016), pp. 1461–1494. doi: 10.1007/s11425-016-0272-y
- W. Bao and X.G. Li, An efficient and stable numerical method for the Maxwell–Dirac system, J. Comput. Phys. 199 (2004), pp. 663–687. doi: 10.1016/j.jcp.2004.03.003
- Y. Cai and Y. Wang, A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime, ESAIM Math. Model. Numer. Anal.52 (2018), pp. 543–566. doi: 10.1051/m2an/2018015
- A.H. Castro Neto, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009), pp. 109–162. doi: 10.1103/RevModPhys.81.109
- T. Cazenave and L. Vázquez, Existence of localized solutions for a classical nonlinear Dirac field, Commun. Math. Phys. 105 (1986), pp. 35–47. doi: 10.1007/BF01212340
- J. Cuevas-Maraver, N. Boussaïd, A. Comech, R. Lan, P.G. Kevrekidis, and A. Saxena, Solitary waves in the nonlinear Dirac equation, in Nonlinear Systems, Mathematical Theory and Computational Methods, Vol. 1, Victoriano Carmona, Jesús Cuevas-Maraver, Fernando Fernández-Sánchez, Elisabeth García-Medina, eds., Springer International Publishing, Cham, 2018, pp. 89–143.
- 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, Phys. Rev. Lett. 116 (2016), pp214101. doi: 10.1103/PhysRevLett.116.214101
- P.A.M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A 117 (1928), pp. 610–624. doi: 10.1098/rspa.1928.0023
- P.A.M. Dirac, A theory of electrons and protons, Proc. R. Soc. Lond. A 126 (1930), pp. 360–365. doi: 10.1098/rspa.1930.0013
- M.J. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Commun. Math. Phys. 171 (1995), pp. 323–350. doi: 10.1007/BF02099273
- F. Fillion-Gourdeau, E. Lorin, and A.D. Bandrauk, Resonantly enhanced pair production in a simple diatomic model, Phys. Rev. Lett. 110 (2013), pp. 013002. doi: 10.1103/PhysRevLett.110.013002
- R. Finkelstein, C. Fronsdal, and P. Kaus, Nonlinear spinor field, Phys. Rev. 103 (1956), pp. 1571–1579. doi: 10.1103/PhysRev.103.1571
- R. Finkelstein, R. Lelevier, and M. Ruderman, Nonlinear spinor fields, Phys. Rev. 83 (1951), pp. 326–332. doi: 10.1103/PhysRev.83.326
- J.D. Frutos and J.M. Sanz-serna, Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys. 83 (1989), pp. 407–423. doi: 10.1016/0021-9991(89)90127-7
- L.H. Haddad and L.D. Carr, The nonlinear Dirac equation in Bose–Einstein condensates: superfluid fluctuations and emergent theories from relativistic linear stability equations, New J. Phys. 17 (2015), pp. 093037.
- W. Heisenberg, Quantum theory of fields and elementary particles, Rev. Mod. Phys. 29 (1957), pp. 269–278. doi: 10.1103/RevModPhys.29.269
- J. Hong and C. Li, Multi-symplectic Runge–Kutta methods for nonlinear Dirac equations, J. Comput. Phys. 211 (2006), pp. 448–472. doi: 10.1016/j.jcp.2005.06.001
- D.Y. Hua and X.G. Li, The finite element method for computing the ground states of the dipolar Bose–Einstein condensates, Appl. Math. Comput. 234 (2014), pp. 214–222.
- Z. Huang, J. Shi, P.A. Markowich, C. Sparber, and C. Zheng, A time-splitting spectral scheme for the Maxwell–Dirac system, J. Comput. Phys. 208 (2005), pp. 761–789. doi: 10.1016/j.jcp.2005.02.026
- D.D. Ivanenko, Notes to the theory of interaction via particles, Zhurn. Exp. Teoret. Fiz. 8 (1938), pp. 260–266.
- W. Jiang, H.Q. Wang, and X.G. Li, A numerical study of the ground state and dynamics of atomic molecular Bose--Einstein condensates, Comput. Phys. Commun. 184 (2013), pp. 2396–2407. doi: 10.1016/j.cpc.2013.05.024
- T. Lakoba, Numerical study of solitary wave stability in cubic nonlinear Dirac equations in 1D, Phys. Lett. A 382 (2018), pp. 300–308. doi: 10.1016/j.physleta.2017.11.032
- S.C. Li and X.G. Li, High-order compact methods for the nonlinear Dirac equation, Comput. Appl. Math. 37 (2018), pp. 6483–6498. doi: 10.1007/s40314-018-0705-4
- S.C. Li, X.G. Li, and F.Y. Shi, Time-splitting methods with charge conservation for the nonlinear Dirac equation, Numer. Meth. Part D E 33 (2017), pp. 1582–1602. doi: 10.1002/num.22154
- S.C. Li, X.G. Li, and F.Y. Shi, Numerical methods for the derivative nonlinear Schrödinger equation, Int. J. Nonlinear Sci. Numer. 19 (2018), pp. 239–249. doi: 10.1515/ijnsns-2016-0184
- X.G. Li, C. Chan, and Y. Hou, A numerical method with particle conservation for the Maxwell--Dirac system, Appl. Math. Comput. 216 (2010), pp. 1096–1108.
- X.G. Li, J. Zhu, R.P. Zhang, and S. Cao, A combined discontinuous Galerkin method for the dipolar Bose–Einstein condensation, J. Comput. Phys. 275 (2014), pp. 363–376. doi: 10.1016/j.jcp.2014.07.013
- H. Liang, J. Meng, and S.G. Zhou, Hidden pseudospin and spin symmetries and their origins in atomic nuclei, Phys. Rep. 570 (2015), pp. 1–84. doi: 10.1016/j.physrep.2014.12.005
- F. Merle, Existence of stationary states for nonlinear Dirac equations, J. Differ. Equ. 74 (1988), pp. 50–68. doi: 10.1016/0022-0396(88)90018-6
- K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438 (2005), pp. 197–200. doi: 10.1038/nature04233
- B. Saha, Nonlinear spinor fields and its role in cosmology, Int. J. Theor. Phys. 51 (2012), pp. 1812–1837. doi: 10.1007/s10773-011-1059-5
- D.W. Sciama, The physical structure of general relativity, Rev. Mod. Phys. 36 (1964), pp. 463–469. doi: 10.1103/RevModPhys.36.463
- S. Shao and H. Tang, Interaction for the solitary waves of a nonlinear Dirac model, Phys. Lett. A 345 (2005), pp. 119–128. doi: 10.1016/j.physleta.2005.07.007
- S. Shao and H. Tang, Interaction of solitary waves with a phase shift in a nonlinear Dirac model, Commun. Comput. Phys. 3 (2008), pp. 950–967.
- 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, Phys. Rev. E 90 (2014), pp. 032915. doi: 10.1103/PhysRevE.90.032915
- S. Shao and H. Tang, Higher-order accurate Runge–Kutta discontinuous Galerkin methods for a nonlinear Dirac model, Discrete Cont. Dyn. B 6 (2006), pp. 623–640. doi: 10.3934/dcdsb.2006.6.623
- M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D 1 (1970), pp. 2766–2769. doi: 10.1103/PhysRevD.1.2766
- G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), pp. 506–517. doi: 10.1137/0705041
- B. Thaller, The Dirac Equation, Springer, New York, 1992.
- W.E. Thirring, A soluble relativistic field theory, Ann. Phys. 3 (1958), pp. 91–112. doi: 10.1016/0003-4916(58)90015-0
- H. Wang and H. Tang, An efficient adaptive mesh redistribution method for a non-linear Dirac equation, J. Comput. Phys. 222 (2007), pp. 176–193. doi: 10.1016/j.jcp.2006.07.011
- T. Wang, B. Guo, and Q. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schödinger equation in two dimensions, J. Comput. Phys. 243 (2013), pp. 382–399. doi: 10.1016/j.jcp.2013.03.007
- Z.Q. Wang and B.Y. Guo, Modified Legendre rational spectral method for the whole line, J. Comput. Math. 22 (2004), pp. 457–474.
- J. Xu, S. Shao, and H. Tang, Numerical methods for nonlinear Dirac equation, J. Comput. Phys. 245 (2013), pp. 131–149. doi: 10.1016/j.jcp.2013.03.031
- J. Xu, S. Shao, H. Tang, and D. Wei, Multi-hump solitary waves of a nonlinear Dirac equation, Commun. Math. Sci. 13 (2015), pp. 1219–1242. doi: 10.4310/CMS.2015.v13.n5.a7
- J.J. Zhang, X.G. Li, and J.F. Shao, Implicit integration factor method for the nonlinear Dirac equation, Int. J. Model. Simul. Sci. Comput. 9 (2018), pp. 1850019.