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Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models
July  2018, 17(4): 1349-1370. doi: 10.3934/cpaa.2018066

## Small amplitude solitary waves in the Dirac-Maxwell system

 1 Texas A & M University, College Station, TX 77843, USA 2 IITP, Moscow 127051, Russia 3 St. Petersburg State University, St. Petersburg 199178, Russia 4 University of Cambridge, Cambridge CB3 0WA, UK

* Corresponding author: Andrew Comech

Received  December 2016 Revised  August 2017 Published  April 2018

Fund Project: The first author was supported by the Russian Foundation for Sciences (project 14-50-00150).

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form $φ(x,ω)e^{-iω t}$, with $ω∈(-m,ω_ *)$ for some $ω_ *>-m$. The solutions satisfy $φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$, and are small amplitude in the sense that ${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$ and ${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$. The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.

Citation: Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066
##### References:
 [1] S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. H. Poincaré Phys. Théor., 68 (1998), 229-244. [2] G. Berkolaiko and A. Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom., 7 (2012), 13-31.  doi: 10.1051/mmnp/20127202. [3] G. Berkolaiko, A. Comech and A. Sukhtayev, Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields, Nonlinearity, 28 (2015), 577-592.  doi: 10.1088/0951-7715/28/3/577. [4] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-electron Atoms, Plenum Publishing Corp., New York, 1977, Reprint of the 1957 original. [5] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964. [6] N. Boussaïd and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524.  doi: 10.1016/j.jfa.2016.04.013. [7] N. Boussaïd and A. Comech, Nonrelativistic asymptotics of solitary waves in the Dirac equation with soler-type nonlinearity, SIAM J. Math. Anal., 49 (2017), 2527-2572.  doi: 10.1137/16M1081385. [8] N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, ArXiv e-prints, arXiv: 1705.05481. [9] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [10] A. Comech, T. V. Phan and A. Stefanov, Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 157-196.  doi: 10.1016/j.anihpc.2015.11.001. [11] A. Comech, M. Guan and S. Gustafson, On linear instability of solitary waves for the nonlinear Dirac equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 639-654.  doi: 10.1016/j.anihpc.2013.06.001. [12] 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), 214101.  doi: 10.1103/PhysRevLett.116.214101. [13] P. A. M. Dirac, An extensible model of the electron, Proc. Roy. Soc. Ser. A, 268 (1962), 57-67. [14] P. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A, 117 (1928), 610-624. [15] M. J. Esteban, V. Georgiev and É. Séré, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations, 4 (1996), 265-281.  doi: 10.1007/BF01254347. [16] M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e-prints, arXiv: 0812.2273. [17] A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories. [18] H. Kikuchi, Existence and Orbital Stability of Standing Waves for Nonlinear SChrÖdinger Equations via the Variational Method (Doctoral Thesis), Kyoto University, Kyoto, 2008. [19] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1. [20] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [22] A. G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A, 28 (1995), 5385-5392. [23] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [24] H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Differential Integral Equations, 13 (2000), 707-720. [25] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, vol. 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987. [26] S. Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac equations, Ann. Henri Poincaré, 10 (2010), 1377-1393.  doi: 10.1007/s00023-009-0015-x. [27] S. Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac-Maxwell equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 791-794.  doi: 10.1016/j.crma.2010.06.003. [28] J. Sakurai, Advanced Quantum Mechanics, A-W series in advanced physics, Addison-Wesley, 1967. [29] D. M. A. Stuart, Periodic solutions of the abelian Higgs model and rigid rotation of vortices, Geom. Funct. Anal., 9 (1999), 568-595.  doi: 10.1007/s000390050096. [30] D. Stuart, Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system, J. Math. Phys., 51 (2010), 032501,13.  doi: 10.1063/1.3294085. [31] N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783-789. [32] M. Wakano, Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141. [33] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

show all references

##### References:
 [1] S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models, Ann. Inst. H. Poincaré Phys. Théor., 68 (1998), 229-244. [2] G. Berkolaiko and A. Comech, On spectral stability of solitary waves of nonlinear Dirac equation in 1D, Math. Model. Nat. Phenom., 7 (2012), 13-31.  doi: 10.1051/mmnp/20127202. [3] G. Berkolaiko, A. Comech and A. Sukhtayev, Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields, Nonlinearity, 28 (2015), 577-592.  doi: 10.1088/0951-7715/28/3/577. [4] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-electron Atoms, Plenum Publishing Corp., New York, 1977, Reprint of the 1957 original. [5] J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964. [6] N. Boussaïd and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal., 271 (2016), 1462-1524.  doi: 10.1016/j.jfa.2016.04.013. [7] N. Boussaïd and A. Comech, Nonrelativistic asymptotics of solitary waves in the Dirac equation with soler-type nonlinearity, SIAM J. Math. Anal., 49 (2017), 2527-2572.  doi: 10.1137/16M1081385. [8] N. Boussaïd and A. Comech, Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity, ArXiv e-prints, arXiv: 1705.05481. [9] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [10] A. Comech, T. V. Phan and A. Stefanov, Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 157-196.  doi: 10.1016/j.anihpc.2015.11.001. [11] A. Comech, M. Guan and S. Gustafson, On linear instability of solitary waves for the nonlinear Dirac equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 639-654.  doi: 10.1016/j.anihpc.2013.06.001. [12] 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), 214101.  doi: 10.1103/PhysRevLett.116.214101. [13] P. A. M. Dirac, An extensible model of the electron, Proc. Roy. Soc. Ser. A, 268 (1962), 57-67. [14] P. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A, 117 (1928), 610-624. [15] M. J. Esteban, V. Georgiev and É. Séré, Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations, 4 (1996), 265-281.  doi: 10.1007/BF01254347. [16] M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e-prints, arXiv: 0812.2273. [17] A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories. [18] H. Kikuchi, Existence and Orbital Stability of Standing Waves for Nonlinear SChrÖdinger Equations via the Variational Method (Doctoral Thesis), Kyoto University, Kyoto, 2008. [19] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1. [20] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. [21] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [22] A. G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A, 28 (1995), 5385-5392. [23] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [24] H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Differential Integral Equations, 13 (2000), 707-720. [25] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, vol. 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987. [26] S. Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac equations, Ann. Henri Poincaré, 10 (2010), 1377-1393.  doi: 10.1007/s00023-009-0015-x. [27] S. Rota Nodari, Perturbation method for particle-like solutions of the Einstein-Dirac-Maxwell equations, C. R. Math. Acad. Sci. Paris, 348 (2010), 791-794.  doi: 10.1016/j.crma.2010.06.003. [28] J. Sakurai, Advanced Quantum Mechanics, A-W series in advanced physics, Addison-Wesley, 1967. [29] D. M. A. Stuart, Periodic solutions of the abelian Higgs model and rigid rotation of vortices, Geom. Funct. Anal., 9 (1999), 568-595.  doi: 10.1007/s000390050096. [30] D. Stuart, Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system, J. Math. Phys., 51 (2010), 032501,13.  doi: 10.1063/1.3294085. [31] N. G. Vakhitov and A. A. Kolokolov, Stationary solutions of the wave equation in the medium with nonlinearity saturation, Radiophys. Quantum Electron., 16 (1973), 783-789. [32] M. Wakano, Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141. [33] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.
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