Solitary Waves in the Nonlinear Dirac Equation

  • Jesús Cuevas-MaraverEmail author
  • Nabile Boussaïd
  • Andrew Comech
  • Ruomeng Lan
  • Panayotis G. Kevrekidis
  • Avadh Saxena
Part of the Understanding Complex Systems book series (UCS)


In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schrödinger equation.


Solitons Solitary waves Vortices Nonlinear Dirac equation Stability Soler model 



The research of Andrew Comech was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Foundation for Sciences (project 14-50-00150). J.C.-M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). P.G.K. gratefully acknowledges the support of NSF-PHY-1602994, the Alexander von Humboldt Foundation, the Stavros Niarchos Foundation via the Greek Diaspora Fellowship Program, and the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096). This work was supported in part by the U.S. Department of Energy.


  1. 1.
    Ablowitz, M.J., Nixon, S.D., Zhu, Y.: Conical diffraction in honeycomb lattices. Phys. Rev. A 79, 053830 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  3. 3.
    Ablowitz, M.J., Zhu, Y.: Evolution of Bloch-mode envelopes in two-dimensional generalized honeycomb lattices. Phys. Rev. A 82, 013840 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Achilleos, V., Frantzeskakis, D.J., Kevrekidis, P.G., Pelinovsky, D.E.: Matter-wave bright solitons in spin-orbit coupled Bose–Einstein condensates. Phys. Rev. Lett. 110, 264101 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    Achilleos, V., Stockhofe, J., Kevrekidis, P.G., Frantzeskakis, D.J., Schmelcher, P.: Matter-wave dark solitons and their excitation spectra in spin-orbit coupled Bose–Einstein condensates. Europhys. Lett. 103(2), 20002 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of traveling waves. J. Reine Angew. Math. 410, 167–212 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Alfimov, G.L., Kevrekidis, P.G., Konotop, V.V., Salerno, M.: Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. Phys. Rev. E 66, 046608 (2002)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Alvarez, A., Soler, M.: Energetic stability criterion for a nonlinear spinorial model. Phys. Rev. Lett. 50, 1230–1233 (1983)ADSCrossRefGoogle Scholar
  9. 9.
    Alvarez, A., Soler, M.: Stability of the minimum solitary wave of a nonlinear spinorial model. Phys. Rev. D 34, 644–645 (1986)ADSCrossRefGoogle Scholar
  10. 10.
    Aubry, S.: Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Physica D 216, 1–30 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Barashenkov, I.V., Pelinovsky, D.E., Zemlyanaya, E.V.: Vibrations and oscillatory instabilities of gap solitons. Phys. Rev. Lett. 80, 5117–5120 (1998)ADSCrossRefGoogle Scholar
  12. 12.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bender, C.M., Berntson, B.K., Parker, D., Samuel, E.: Observation of PT phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Bender, C.M., Fring, A., Günther, U., Jones, H.: Special issue: quantum physics with non-hermitian operators. J. Phys. A Math. Theory 45(44), 020201 (2012)CrossRefGoogle Scholar
  15. 15.
    Bender, C.M., Jones, H.F., Rivers, R.J.: Dual PT-symmetric quantum field theories. Phys. Lett. B 625, 333–340 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Berkolaiko, G., Comech, A.: On spectral stability of solitary waves of nonlinear dirac equation in 1D. Math. Model. Nat. Phenom. 7, 13–31 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Berkolaiko, G., Comech, A., Sukhtayev, A.: Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields. Nonlinearity 28(3), 577–592 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Berthier, A., Georgescu, V.: On the point spectrum of Dirac operators. J. Funct. Anal. 71(2), 309–338 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bjorken, J., Drell, S.: Relativistic Quantum Mechanics. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  21. 21.
    Blanchard, P., Stubbe, J., Vázquez, L.: Stability of nonlinear spinor fields with application to the Gross–Neveu model. Phys. Rev. D 36, 2422–2428 (1987)ADSCrossRefGoogle Scholar
  22. 22.
    Bogolubsky, I.L.: On spinor soliton stability. Phys. Lett. A 73, 87–90 (1979)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Bournaveas, N.: Local existence for the Maxwell–Dirac equations in three space dimensions. Commun. Partial Differ. Equ. 21(5–6), 693–720 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Boussaïd, N.: Stable directions for small nonlinear Dirac standing waves. Commun. Math. Phys. 268(3), 757–817 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Boussaïd, N.: On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case. SIAM J. Math. Anal. 40(4), 1621–1670 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Boussaïd, N., Comech, A.: On spectral stability of the nonlinear Dirac equation. J. Funct. Anal. 271(6), 1462–1524 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Boussaïd, N., Comech, A.: Spectral stability of weakly relativistic solitary waves of the Dirac equation with the Soler-type nonlinearity (2016)Google Scholar
  28. 28.
    Boussaïd, N., Comech, A.: Spectral stability of weakly relativistic solitary waves of the Dirac equation with the Soler-type nonlinearity (2017). To appearGoogle Scholar
  29. 29.
    Boussaïd, N., Cuccagna, S.: On stability of standing waves of nonlinear Dirac equations. Commun. Part. Diff. Equ. 37, 1001–1056 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001)zbMATHGoogle Scholar
  31. 31.
    Braun, O.M., Kivshar, Y.S.: The Frenkel–Kontorova Model. Springer Nature (2004)Google Scholar
  32. 32.
    Buslaev, V.S., Perel\(^{\prime }\)man, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear Evolution Equations. Am. Math. Soc. Trans. Ser. (Am. Math. Soc., Providence, RI.) 164(2), 75–98 (1995)Google Scholar
  33. 33.
    Candy, T.: Global existence for an \(L^2\) critical nonlinear Dirac equation in one dimension. Adv. Differ. Equ. 16(7–8), 643–666 (2011)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Carretero-González, R., Talley, J.D., Chong, C., Malomed, B.A.: Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation. Physica D 216, 77–89 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Cazenave, T., Vazquez, L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys. 105, 35–47 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Chugunova, M., Pelinovsky, D.: Block-diagonalization of the symmetric first-order coupled-mode system. SIAM J. Appl. Dyn. Syst. 5(1), 66–83 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D 11, 2088–2097 (1975)ADSCrossRefGoogle Scholar
  39. 39.
    Comech, A., Guan, M., Gustafson, S.: On linear instability of solitary waves for the nonlinear Dirac equation. Ann. Inst. H. Poincaré - AN 31, 639–654 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Comech, A., Phan, T.V., Stefanov, A.: Asymptotic stability of solitary waves in generalized Gross-Neveu model. Ann. Inst. H. Poincaré - AN 34, 157–196 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Comech, A., Stuart, D.: Small solitary waves in the Dirac–Maxwell system (2012). ArXiv:1210.7261
  42. 42.
    Conduit, G.J.: Line of Dirac monopoles embedded in a Bose–Einstein condensate. Phys. Rev. A 86, 021605(R) (2012)ADSCrossRefGoogle Scholar
  43. 43.
    Contreras, A., Pelinovsky, D.E., Shimabukuro, Y.: L\(^2\) orbital stability of Dirac solitons in the massive Thirring model. Commun. Partial Differ. Equ. 41, 227–255 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Cooper, F., Khare, A., Mihaila, B., Saxena, A.: Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity. Phys. Rev. E 82, 036604 (2010)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Cuccagna, S., Tarulli, M.: On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential. J. Math. Anal. Appl. 436(2), 1332–1368 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Cuevas-Maraver, J., Kevrekidis, P., Saxena, A., Cooper, F., Mertens, F.: Solitary waves in the nonlinear Dirac equation at the continuum limit: stability and dynamics. In: Ordinary and Partial Differential Equations. Nova Science Publishers, New York (2015)Google Scholar
  47. 47.
    Cuevas-Maraver, J., Kevrekidis, P.G., Saxena, A.: Solitary waves in a discrete nonlinear Dirac equation. J. Phys. A: Math. Theory 48, 055204 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Cuevas-Maraver, J., Kevrekidis, P.G., Saxena, A., Comech, A., Lan, R.: Stability of solitary waves and vortices in a 2D nonlinear Dirac model. Phys. Rev. Lett. 116, 214101 (2016)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Cuevas-Maraver, J., Kevrekidis, P.G., Saxena, A., Cooper, F., Khare, A., Comech, A., Bender, C.M.: Solitary waves of a PT-symmetric nonlinear Dirac equation. IEEE J. Sel. Top. Quantum Electron. 22, 5000109 (2016)CrossRefGoogle Scholar
  50. 50.
    Cuevas-Maraver, J., Kevrekidis, P.G., Williams, F. (eds.): The Sine-Gordon Model and its Applications. Springer International Publishing (2014)Google Scholar
  51. 51.
    Dalibard, J., Gerbier, F., Juzeliunas, G., Öhberg, P.: Colloquium: artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    Darby, D., Ruijgrok, T.W.: A noncompact gauge group for the Dirac equation. Acta Phys. Polon. B 10, 959–973 (1979)MathSciNetGoogle Scholar
  53. 53.
    Dauxois, T., Peyrard, M.: Physics of Solitons. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  54. 54.
    De Wit, B., Smith, J.: Field Theory in Particle Physics. North Holland Physics Publishing, New York (1986)Google Scholar
  55. 55.
    Degasperis, A., Wabnitz, S., Aceves, A.: Bragg grating rogue wave. Phys. Lett. A 379, 1067–1070 (2015)zbMATHCrossRefGoogle Scholar
  56. 56.
    Derrick, G.H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Dirac, P.: The quantum theory of the electron. I. Proc. R. Soc. Lond. A 117, 610–624 (1928)ADSzbMATHCrossRefGoogle Scholar
  58. 58.
    Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Escobedo, M., Vega, L.: A semilinear Dirac equation in \(H^s({ R}^3)\) for \(s>1\). SIAM J. Math. Anal. 28(2), 338–362 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Esteban, M.J., Georgiev, V., Séré, É.: Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations. Calc. Var. Partial Differ. Equ. 4(3), 265–281 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Esteban, M.J., Lewin, M., Séré, É.: Variational methods in relativistic quantum mechanics. Bull. Amer. Math. Soc. (N.S.) 45(4), 535–593 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Esteban, M.J., Séré, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Esteban, M.J., Séré, É.: Solutions of the Dirac–Fock equations for atoms and molecules. Commun. Math. Phys. 203(3), 499–530 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Esteban, M.J., Séré, E.: Nonrelativistic limit of the Dirac–Fock equations. Ann. Henri Poincaré 2(5), 941–961 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Esteban, M.J., Séré, É.: Dirac–Fock models for atoms and molecules and related topics. In: XIVth International Congress on Mathematical Physics, pp. 21–28. World Scientific Publishing, Hackensack, NJ (2005)Google Scholar
  66. 66.
    Evans, J.: Nerve axon equations, I: Linear approximations. Indiana U. Math. J. 21, 877–955 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Evans, J.: Nerve axon equations, II: Stability at rest. Indiana U. Math. J. 22, 75–90 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Evans, J.: Nerve axon equations, III: Stability of the nerve impulse. Indiana U. Math. J. 22, 577–594 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Evans, J.: Nerve axon equations, IV: The stable and unstable impulse. Indiana U. Math. J. 24, 1169–1190 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Fedosov, B.V.: Index theorems. In: Partial Differential Equations, VIII Encyclopaedia Mathematical Sciences, vol. 65. Springer-Verlag, Berlin (1996)CrossRefGoogle Scholar
  71. 71.
    Feng, B., Sugino, O., Liu, R.Y., Zhang, J., Yukawa, R., Kawamura, M., Iimori, T., Kim, H., Hasegawa, Y., Li, H., Chen, L., Wu, K., Kumigashira, H., Komori, F., Chiang, T.C., Meng, S., Matsuda, I.: Dirac fermions in borophene. Phys. Rev. Lett. 118, 096401 (2017)ADSCrossRefGoogle Scholar
  72. 72.
    Fialko, O., Brand, J., Zülicke, U.: Hidden long-range order in a two-dimensional spin-orbit coupled bose gas. Phys. Rev. A 85, 051605(R) (2012)ADSCrossRefGoogle Scholar
  73. 73.
    Finkelstein, R., Lelevier, R., Ruderman, M.: Nonlinear spinor fields. Phys. Rev. 83, 326–332 (1951)ADSzbMATHCrossRefGoogle Scholar
  74. 74.
    Fring, A., Jones, H., Znojil, M.: Papers dedicated to the subject of the 6th international workshop on pseudo-Hermitian Hamiltonians in quantum physics (PHHQPVI). J. Phys. A: Math. Theory 41(44) (2008)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    Galindo, A.: A remarkable invariance of classical Dirac Lagrangians. Lett. Nuovo Cimento 20, 210–212 (1977)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Georgiev, V., Ohta, M.: Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations. J. Math. Soc. Jpn. 64(2), 533–548 (2012)zbMATHCrossRefGoogle Scholar
  77. 77.
    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Gross, D.J., Neveu, A.: Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D 10, 3235–3253 (1974)ADSCrossRefGoogle Scholar
  79. 79.
    Gross, L.: The Cauchy problem for the coupled Maxwell and Dirac equations. Commun. Pure Appl. Math. 19, 1–15 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)ADSCrossRefGoogle Scholar
  81. 81.
    Haddad, L.H., Carr, L.D.: The nonlinear Dirac equation in Bose–Einstein condensates: vortex solutions and spectra in a weak harmonic trap. New J. Phys. 17, 113011 (2015)ADSCrossRefGoogle Scholar
  82. 82.
    Haddad, L.H., O’Hara, K.M., Carr, L.D.: Nonlinear Dirac equation in Bose–Einstein condensates: preparation and stability of relativistic vortices. Phys. Rev. A 91, 043609 (2015)ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    Haddad, L.H., Weaver, C.M., Carr, L.D.: The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices. New J. Phys. 17, 063044 (2015)MathSciNetGoogle Scholar
  84. 84.
    Hadzievski, L., Maluckov, A., Stepić, M., Kip, D.: Power controlled soliton stability and steering in lattices with saturable nonlinearity. Phys. Rev. Lett. 93, 033901 (2004)ADSCrossRefGoogle Scholar
  85. 85.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993)zbMATHGoogle Scholar
  86. 86.
    Hamner, C., Zhang, Y., Khamehchi, M.A., Davis, M.J., Engels, P.: In a one-dimensional optical lattice, spin-orbit-coupled Bose–Einstein condensates. Phys. Rev. Lett. 114, 070401 (2014)CrossRefGoogle Scholar
  87. 87.
    Heisenberg, W.: Quantum theory of fields and elementary particles. Rev. Mod. Phys. 29, 269–278 (1957)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Herring, G., Carr, L.D., Carretero-González, R., Kevrekidis, P.G., Frantzeskakis, D.J.: Radially symmetric nonlinear states of harmonically trapped Bose–Einstein condensates. Phys. Rev. A 77, 023625 (2008)ADSCrossRefGoogle Scholar
  89. 89.
    Huh, H.: Global solutions to Gross–Neveu equation. Lett. Math. Phys. 103(8), 927–931 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Ivanenko, D.D.: Notes to the theory of interaction via particles. Sov. Phys. JETP 13, 141 (1938)MathSciNetGoogle Scholar
  91. 91.
    Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583–611 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Johansson, M., Kivshar, Y.S.: Discreteness-induced oscillatory instabilities of dark solitons. Phys. Rev. Lett. 82, 85–88 (1999)ADSCrossRefGoogle Scholar
  93. 93.
    Jones, C.: Stability of the travelling wave solutions of the Fitzhugh–Nagumo system. Trans. AMS 286(2), 431–469 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Kapitula, T., Sandstede, B.: Edge bifurcations for near integrable systems via Evans function techniques. SIAM J. Math. Anal. 33(5), 1117–1143 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Kartashov, Y.V., Konotop, V.V., Abdullaev, F.K.: Gap solitons in a spin-orbit-coupled Bose–Einstein condensate. Phys. Rev. Lett. 111, 060402 (2013)ADSCrossRefGoogle Scholar
  96. 96.
    Kawakami, T., Mizushima, T., Nitta, M., Machida, K.: Stable skyrmions in SU(2) gauged Bose–Einstein condensates. Phys. Rev. Lett. 109, 015301 (2012)ADSCrossRefGoogle Scholar
  97. 97.
    Kestelman, H.: Anticommuting linear transformations. Canad. J. Math. 13, 614–624 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Kevrekidis, P.G.: The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232. Springer-Verlag, Heidelberg (2009)zbMATHGoogle Scholar
  99. 99.
    Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, R.: The defocusing nonlinear Schrödinger equation: from dark solitons, to vortices and vortex rings. SIAM, Philadelphia (2015)zbMATHCrossRefGoogle Scholar
  100. 100.
    Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean. Springer-Verlag, Berlin (2009)zbMATHGoogle Scholar
  101. 101.
    Kivshar, Y.S., Agrawal, G.P.: Optical solitons: from fibers to photonic crystals. Academic Press, San Diego (2003)Google Scholar
  102. 102.
    Klaiman, S., Günther, U., Moiseyev, N.: Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    LeBlanc, L.J., Beeler, M.C., Jiménez-García, K., Perry, A.R., Sugawa, S., Williams, R.A., Spielman, I.B.: Direct observation of zitterbewegung in a Bose–Einstein condensate. New J. Phys. 15, 073011 (2013)CrossRefGoogle Scholar
  104. 104.
    Lee, S.Y., Kuo, T.K., Gavrielides, A.: Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions. Phys. Rev. D 12, 2249–2253 (1975)ADSCrossRefGoogle Scholar
  105. 105.
    Lee, Y.S., McLean, A.D.: Relativistic effects on \({R}_e\) and \({D}_e\) in AgH and AuH from all-electron Dirac–Hartree–Fock calculations. J. Chem. Phys. 76(1), 735–736 (1982)Google Scholar
  106. 106.
    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1977)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    Lin, Y.J., Jiménez-García, K., Spielman, I.B.: Spin-orbit-coupled Bose–Einstein condensates. Nature 471, 83–86 (2011)ADSCrossRefGoogle Scholar
  108. 108.
    Machihara, S., Nakamura, M., Nakanishi, K., Ozawa, T.: Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219(1), 1–20 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    Machihara, S., Nakanishi, K., Tsugawa, K.: Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50(2), 403–451 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Mak, K.F., Lee, C., Hone, J., Shan, J., Heinz, T.F.: Atomically thin MoS\(_2\): a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010)ADSCrossRefGoogle Scholar
  112. 112.
    Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in PT-symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)ADSCrossRefGoogle Scholar
  113. 113.
    Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT-symmetric periodic optical potentials. Int. J. Theor. Phys. 50, 1019–1041 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Marini, A., Longhi, S., Biancalana, F.: Optical simulation of neutrino oscillations in binary waveguide arrays. Phys. Rev. Lett. 113, 150401 (2014)Google Scholar
  115. 115.
    Mathieu, P., Morris, T.F.: Charged spinor solitons. Can. J. Phys. 64(3), 232–238 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Melvin, T.R.O., Champneys, A.R., Kevrekidis, P.G., Cuevas, J.: Radiationless traveling vaves in saturable nonlinear Schrödinger lattices. Phys. Rev. Lett. 97, 124101 (2006)ADSCrossRefGoogle Scholar
  117. 117.
    Merkl, M., Jacob, A., Zimmer, F.E., Öhberg, P., Santos, L.: Chiral confinement in quasirelativistic Bose–Einstein condensates. Phys. Rev. Lett. 104, 073603 (2010)ADSCrossRefGoogle Scholar
  118. 118.
    Merle, F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Equ. 74, 50–68 (1988)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Commun. Math. Phys. 129, 223–240 (1990)ADSzbMATHCrossRefGoogle Scholar
  120. 120.
    Mertens, F.G., Quintero, N.R., Cooper, F., Khare, A., Saxena, A.: Nonlinear dirac equation solitary waves in external fields. Phys. Rev. E 86, 046602 (2012)ADSCrossRefGoogle Scholar
  121. 121.
    Ng, W., Parwani, R.: Nonlinear Dirac equations. SIGMA 3, 023 (2009)MathSciNetzbMATHGoogle Scholar
  122. 122.
    Pauli, W.: Contributions mathématiques à la théorie des matrices de Dirac. Ann. Inst. H. Poincaré 6, 109–136 (1936)MathSciNetzbMATHGoogle Scholar
  123. 123.
    Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. Lond. Ser. A 340(1656), 47–94 (1992)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Peleg, O., Bartal, G., Freedman, B., Manela, O., Segev, M., Christodoulides, D.N.: Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett. 98, 103901 (2007)ADSCrossRefGoogle Scholar
  125. 125.
    Pelinovsky, D.: Survey on global existence in the nonlinear Dirac equations in one spatial dimension. In: Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, B26, pp. 37–50. Res. Inst. Math. Sci. (RIMS), Kyoto (2011)Google Scholar
  126. 126.
    Pelinovsky, D., Shimabukuro, Y.: Transverse instability of line solitons in massive Dirac equations. J. Nonlinear Sci. 26, 365–403 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    Pelinovsky, D.E., Shimabukuro, Y.: Orbital stability of Dirac solitons. Lett. Math. Phys. 104, 21–41 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Pelinovsky, D.E., Stefanov, A.: Asymptotic stability of small gap solitons in nonlinear Dirac equations. J. Math. Phys. 53, 073705 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Peng, B., Özdemir, S.K., Lei, F., Monifi, F., Gianfreda, M., Long, G.L., Fan, S., Nori, F., Bender, C.M., Yang, L.: Parity-time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014)CrossRefGoogle Scholar
  130. 130.
    Pethick, C.J., Smith, H.: Bose–Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2002)Google Scholar
  131. 131.
    Pitaevskii, L.P., Stringari, S.: Bose–Einstein condensation. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  132. 132.
    Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
  133. 133.
    Qu, C., Hamner, C., Gong, M., Zhang, C., Engels, P.: Observation of zitterbewegung in a spin-orbit-coupled Bose–Einstein condensate. Phys. Rev. A 88, 021064(R) (2013)CrossRefGoogle Scholar
  134. 134.
    Quiney, H.M., Glushkov, V.N., Wilson, S.: The Dirac equation in the algebraic approximation. IX. Matrix Dirac–Hartree–Fock calculations for the HeH and BeH ground states using distributed gaussian basis sets. Int. J. Quantum Chem. 99(6), 950–962 (2004)CrossRefGoogle Scholar
  135. 135.
    Rañada, A.F., Rañada, M.F., Soler, M., Vázquez, L.: Classical electrodynamics of a nonlinear Dirac field with anomalous magnetic moment. Phys. Rev. D 10(2), 517–525 (1974)ADSCrossRefGoogle Scholar
  136. 136.
    Radić, J., Sedrakyan, T.A., Spielman, I.B., Galitski, V.: Vortices in spin-orbit-coupled Bose–Einstein condensates. Phys. Rev. A 84, 063604 (2011)ADSCrossRefGoogle Scholar
  137. 137.
    Ramachandhran, B., Opanchuk, B., Liu, X.J., Pu, H., Drummond, P.D., Hu, H.: Half-quantum vortex state in a spin-orbit-coupled Bose–Einstein condensate. Phys. Rev. A 85, 023606 (2012)ADSCrossRefGoogle Scholar
  138. 138.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)Google Scholar
  139. 139.
    Regensburger, A., Bersch, C., Miri, M.A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488, 167–171 (2012)ADSCrossRefGoogle Scholar
  140. 140.
    Rota Nodari, S.: Perturbation method for particle-like solutions of the Einstein–Dirac equations. Ann. Henri Poincaré 10(7), 1377–1393 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    Rota Nodari, S.: Perturbation method for particle-like solutions of the Einstein–Dirac–Maxwell equations. C. R. Math. Acad. Sci. Paris 348(13–14), 791–794 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Ruschhaupt, A., Delgado, F., Muga, J.G.: Physical realization of PT-symmetric potential scattering in a planar slab waveguide. J. Phys. A: Math. Gen. 38, L171–L176 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Rüter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192–195 (2010)CrossRefGoogle Scholar
  144. 144.
    Sakaguchi, H., Li, B., Malomed, B.A.: Creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose–Einstein condensates in free space. Phys. Rev. E 89, 032920 (2014)ADSCrossRefGoogle Scholar
  145. 145.
    Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Experimental study of active LRC circuits with PT-symmetries. Phys. Rev. A 84, 040101 (2011)ADSCrossRefGoogle Scholar
  146. 146.
    Schindler, J., Lin, Z., Lee, J.M., Ramezani, H., Ellis, F.M., Kottos, T.: PT-symmetric electronics. J. Phys. A: Math. Theory 45, 444029 (2012)ADSzbMATHCrossRefGoogle Scholar
  147. 147.
    Selberg, S., Tesfahun, A.: Low regularity well-posedness for some nonlinear Dirac equations in one space dimension. Differ. Integr. Equ. 23(3–4), 265–278 (2010)MathSciNetzbMATHGoogle Scholar
  148. 148.
    Shampine, L.F., Hosea, M.E.: Analysis and implementation of TR-BDF2. Appl. Num. Math. 20, 21–37 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  149. 149.
    Shao, S., Quintero, N.R., Mertens, F.G., Cooper, F., Khare, A., Saxena, A.: Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity. Phys. Rev. E 90, 032915 (2014)ADSCrossRefGoogle Scholar
  150. 150.
    Sigal, I.M.: Nonlinear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2), 297–320 (1993)ADSzbMATHCrossRefGoogle Scholar
  151. 151.
    Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1), 9–74 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    Soler, M.: Classical, stable, nonlinear spinor field with positive rest energy. Phys. Rev. D 1, 2766–2769 (1970)ADSCrossRefGoogle Scholar
  153. 153.
    Soler, M.: Classical electrodynamics for a nonlinear spinor field: perturbative and exact approaches. Phys. Rev. D 8, 3424–3429 (1973)ADSCrossRefGoogle Scholar
  154. 154.
    Strauss, W.A., Vázquez, L.: Stability under dilations of nonlinear spinor fields. Phys. Rev. D 34(2), 641–643 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  155. 155.
    Stuart, D.: Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system. J. Math. Phys. 51(3), 032501 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  156. 156.
    Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation. Springer-Verlag, New York (1999)zbMATHGoogle Scholar
  157. 157.
    Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, Berlin (1992)Google Scholar
  158. 158.
    Thirring, W.E.: A soluble relativistic field theory. Ann. Phys. 3, 91–112 (1958)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    Tran, T.X., Longhi, S., Biancalana, F.: Optical analogue of relativistic Dirac solitons in binary waveguide arrays. Ann. Phys. 340, 179–187 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  160. 160.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  161. 161.
    Vakhitov, N.G., Kolokolov, A.A.: Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16, 783–789 (1973)ADSCrossRefGoogle Scholar
  162. 162.
    Vázquez, L.: Localised solutions of a non-linear spinor field. J. Phys. A: Math. Gen. 10, 1361–1368 (1977)ADSMathSciNetCrossRefGoogle Scholar
  163. 163.
    Vicencio, R.A., Johansson, M.: Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity. Phys. Rev. E 73, 046602 (2006)ADSCrossRefGoogle Scholar
  164. 164.
    Visscher, L., Dyall, K.: Dirac–Fock atomic enectronic structore calculations using different nuclear charge distributions. At. Data Nucl. Data Tables 67(2), 207–224 (1997)ADSCrossRefGoogle Scholar
  165. 165.
    van der Waerden, B.: Group Theory and Quantum Mechanics. Springer-Verlag, New York (1974)zbMATHCrossRefGoogle Scholar
  166. 166.
    Wakano, M.: Intensely localized solutions of the classical Dirac–Maxwell field equations. Prog. Theory Phys. 35, 1117–1141 (1966)ADSCrossRefGoogle Scholar
  167. 167.
    Wehling, T.O., Black-Schaffer, A.M., Balatsky, A.V.: Dirac materials. Adv. Phys. 63, 1–76 (2014)ADSCrossRefGoogle Scholar
  168. 168.
    Xu, J., Shao, S., Tang, H.: Numerical methods for nonlinear Dirac equation. J. Comput. Phys. 245, 131–149 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  169. 169.
    Xu, X.Q., Han, J.H.: Spin-orbit coupled Bose–Einstein condensate under rotation. Phys. Rev. Lett. 107, 200401 (2011)ADSCrossRefGoogle Scholar
  170. 170.
    Xu, Y., Zhang, Y., Wu, B.: Bright solitons in spin-orbit-coupled Bose–Einstein condensates. Phys. Rev. A 87, 013614 (2013)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jesús Cuevas-Maraver
    • 1
    • 2
    Email author
  • Nabile Boussaïd
    • 3
  • Andrew Comech
    • 4
    • 5
  • Ruomeng Lan
    • 4
  • Panayotis G. Kevrekidis
    • 6
  • Avadh Saxena
    • 7
  1. 1.Grupo de Física No Lineal, Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de SevillaSevillaSpain
  2. 2.Instituto de Matemáticas de la Universidad de Sevilla (IMUS)SevillaSpain
  3. 3.Université Bourgogne Franche-ComtéBesançon CEDEXFrance
  4. 4.Department of MathematicsTexas A&M UniversityCollege StationUSA
  5. 5.IITPMoscowRussia
  6. 6.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  7. 7.Los Alamos National LaboratoryCenter for Nonlinear Studies and Theoretical DivisionLos AlamosUSA

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