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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 & 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. Google Scholar

[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. Google Scholar

[3]

G. BerkolaikoA. 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. Google Scholar

[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. Google Scholar

[5]

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[9]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. Google Scholar

[10]

A. ComechT. 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. Google Scholar

[11]

A. ComechM. 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. Google Scholar

[12]

J. Cuevas-MaraverP. G. KevrekidisA. SaxenaA. 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. Google Scholar

[13]

P. A. M. Dirac, An extensible model of the electron, Proc. Roy. Soc. Ser. A, 268 (1962), 57-67. Google Scholar

[14]

P. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A, 117 (1928), 610-624. Google Scholar

[15]

M. J. EstebanV. 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. Google Scholar

[16]

M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e-prints, arXiv: 0812.2273.Google Scholar

[17]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories. Google Scholar

[18]

H. Kikuchi, Existence and Orbital Stability of Standing Waves for Nonlinear SChrÖdinger Equations via the Variational Method (Doctoral Thesis), Kyoto University, Kyoto, 2008.Google Scholar

[19]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. Google Scholar

[20]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. Google Scholar

[21]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[22]

A. G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A, 28 (1995), 5385-5392. Google Scholar

[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. Google Scholar

[24]

H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Differential Integral Equations, 13 (2000), 707-720. Google Scholar

[25]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, vol. 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

J. Sakurai, Advanced Quantum Mechanics, A-W series in advanced physics, Addison-Wesley, 1967.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

M. Wakano, Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

G. BerkolaikoA. 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. Google Scholar

[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. Google Scholar

[5]

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[9]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. Google Scholar

[10]

A. ComechT. 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. Google Scholar

[11]

A. ComechM. 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. Google Scholar

[12]

J. Cuevas-MaraverP. G. KevrekidisA. SaxenaA. 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. Google Scholar

[13]

P. A. M. Dirac, An extensible model of the electron, Proc. Roy. Soc. Ser. A, 268 (1962), 57-67. Google Scholar

[14]

P. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A, 117 (1928), 610-624. Google Scholar

[15]

M. J. EstebanV. 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. Google Scholar

[16]

M. Guan, Solitary wave solutions for the nonlinear Dirac equations, ArXiv e-prints, arXiv: 0812.2273.Google Scholar

[17]

A. Jaffe and C. Taubes, Vortices and Monopoles, vol. 2 of Progress in Physics, Birkhäuser Boston, Mass., 1980, Structure of static gauge theories. Google Scholar

[18]

H. Kikuchi, Existence and Orbital Stability of Standing Waves for Nonlinear SChrÖdinger Equations via the Variational Method (Doctoral Thesis), Kyoto University, Kyoto, 2008.Google Scholar

[19]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. Google Scholar

[20]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. Google Scholar

[21]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[22]

A. G. Lisi, A solitary wave solution of the Maxwell-Dirac equations, J. Phys. A, 28 (1995), 5385-5392. Google Scholar

[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. Google Scholar

[24]

H. Ounaies, Perturbation method for a class of nonlinear Dirac equations, Differential Integral Equations, 13 (2000), 707-720. Google Scholar

[25]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, vol. 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

J. Sakurai, Advanced Quantum Mechanics, A-W series in advanced physics, Addison-Wesley, 1967.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

M. Wakano, Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141. Google Scholar

[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. Google Scholar

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