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Small amplitude solitary waves in the Dirac-Maxwell system
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Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$
Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models
1. | Universite Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon CEDEX, France |
2. | Texas A & M University, College Station, TX 77843, USA |
3. | IITP, Moscow 127051, Russia |
4. | St. Petersburg State University, St. Petersburg 199178, Russia |
We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac-Klein-Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing.
We show the relation of $± 2ω\mathrm{i}$ eigenvalues of the linearization at a solitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.
References:
[1] |
A. Aceves, A. Auditore, M. Conforti and C. De Angelis, Discrete localized modes in binary waveguide arrays, in Nonlinear Photonics (NLP), 2013 IEEE 2nd International Workshop, 2013, 38–42. |
[2] |
A. Auditore, M. Conforti, C. De Angelis and A. B. Aceves,
Dark-antidark solitons in waveguide arrays with alternating positive-negative couplings, Optics Communications, 297 (2013), 125-128.
|
[3] |
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. |
[4] |
A. Betlej, S. Suntsov, K. G. Makris, L. Jankovic, D. N. Christodoulides, G. I. Stegeman, J. Fini, R. T. Bise and D. J. DiGiovanni,
All-optical switching and multifrequency generation in a dual-core photonic crystal fiber, Opt. Lett., 31 (2006), 1480-1482.
|
[5] |
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. |
[6] |
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. |
[7] |
N. Boussaïd and S. Cuccagna,
On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056.
doi: 10.1080/03605302.2012.665973. |
[8] |
N. J. Cerf, M. Bourennane, A. Karlsson and N. Gisin,
Security of quantum key distribution using d-level systems, Physical Review Letters, 88 (2002), 127902.
|
[9] |
J. M. Chadam and R. T. Glassey,
On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions, Arch. Rational Mech. Anal., 54 (1974), 223-237.
doi: 10.1007/BF00250789. |
[10] |
A. Comech, T. V. Phan and A. Stefanov,
Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincare 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. Poincare Anal. Non Linéaire, 31 (2014), 639-654.
doi: 10.1016/j.anihpc.2013.06.001. |
[12] |
J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena, F. Cooper, A. Khare, A. Comech and C. M. Bender,
Solitary waves of a $ \mathcal{PT}$-symmetric nonlinear Dirac equation, IEEE Journal of Selected Topics in Quantum Electronics, 22 (2016), 1-9.
|
[13] |
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. |
[14] |
T. Durt, D. Kaszlikowski, J.-L. Chen and L. C. Kwek,
Security of quantum key distributions with entangled qudits, Phys. Rev. A, 69 (2004), 032313.
doi: 10.1103/PhysRevA.69.032313. |
[15] |
B. J. Eggleton, C. M. De Sterke and R. E. Slusher,
Nonlinear pulse propagation in Bragg gratings, J. Opt. Soc. Am. B, 14 (1997), 2980-2993.
|
[16] |
A. Galindo,
A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.
doi: 10.1007/BF02785129. |
[17] |
D. J. Gross and A. Neveu,
Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D, 10 (1974), 3235-3253.
|
[18] |
D. D. Ivanenko,
Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz, 8 (1938), 260-266.
|
[19] |
M. Kues, C. Reimer, P. Roztocki, L. R. Cort, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Aza and R. Morandotti,
On-chip generation of high-dimensional entangled quantum states and their coherent control, Nature, 546 (2017), 622-626.
doi: 10.1038/nature22986. |
[20] |
N. Lazarides and G. P. Tsironis,
Gain-driven discrete breathers in $\mathcal{P}\mathcal{T}$-symmetric nonlinear metamaterials, Phys. Rev. Lett., 110 (2013), 053901.
doi: 10.1103/PhysRevLett.110.053901. |
[21] |
S. Y. Lee and A. Gavrielides,
Quantization of the localized solutions in two-dimensional field theories of massive fermions, Phys. Rev. D, 12 (1975), 3880-3886.
|
[22] |
A. Marini, S. Longhi and F. Biancalana,
Optical simulation of neutrino oscillations in binary waveguide arrays, Phys. Rev. Lett., 113 (2014), 150401.
doi: 10.1103/PhysRevLett.113.150401. |
[23] |
A. A. Melnikov and L. E. Fedichkin,
Quantum walks of interacting fermions on a cycle graph, Sci. Rep., 6 (2016), 34226.
doi: 10.1038/srep34226. |
[24] |
R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov and Y. S. Kivshar,
Observation of discrete gap solitons in binary waveguide arrays, Opt. Lett., 29 (2004), 2890-2892.
|
[25] |
T. Ozawa and K. Yamauchi,
Structure of Dirac matrices and invariants for nonlinear Dirac equations, Differential Integral Equations, 17 (2004), 971-982.
|
[26] |
D. E. Pelinovsky and A. Stefanov,
Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), 073705, 27.
doi: 10.1063/1.4731477. |
[27] |
J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis and T. Kottos,
$\mathcal{PT}$-symmetric electronics, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 444029.
|
[28] |
J. Schindler, A. Li, M. C. Zheng, F. M. Ellis and T. Kottos,
Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries, Phys. Rev. A, 84 (2011), 040101.
doi: 10.1103/PhysRevA.84.040101. |
[29] |
M. Soler,
Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.
|
[30] |
W. E. Thirring,
A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112.
doi: 10.1016/0003-4916(58)90015-0. |
[31] |
M. Wakano,
Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141.
|
show all references
References:
[1] |
A. Aceves, A. Auditore, M. Conforti and C. De Angelis, Discrete localized modes in binary waveguide arrays, in Nonlinear Photonics (NLP), 2013 IEEE 2nd International Workshop, 2013, 38–42. |
[2] |
A. Auditore, M. Conforti, C. De Angelis and A. B. Aceves,
Dark-antidark solitons in waveguide arrays with alternating positive-negative couplings, Optics Communications, 297 (2013), 125-128.
|
[3] |
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. |
[4] |
A. Betlej, S. Suntsov, K. G. Makris, L. Jankovic, D. N. Christodoulides, G. I. Stegeman, J. Fini, R. T. Bise and D. J. DiGiovanni,
All-optical switching and multifrequency generation in a dual-core photonic crystal fiber, Opt. Lett., 31 (2006), 1480-1482.
|
[5] |
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. |
[6] |
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. |
[7] |
N. Boussaïd and S. Cuccagna,
On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056.
doi: 10.1080/03605302.2012.665973. |
[8] |
N. J. Cerf, M. Bourennane, A. Karlsson and N. Gisin,
Security of quantum key distribution using d-level systems, Physical Review Letters, 88 (2002), 127902.
|
[9] |
J. M. Chadam and R. T. Glassey,
On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions, Arch. Rational Mech. Anal., 54 (1974), 223-237.
doi: 10.1007/BF00250789. |
[10] |
A. Comech, T. V. Phan and A. Stefanov,
Asymptotic stability of solitary waves in generalized Gross–Neveu model, Ann. Inst. H. Poincare 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. Poincare Anal. Non Linéaire, 31 (2014), 639-654.
doi: 10.1016/j.anihpc.2013.06.001. |
[12] |
J. Cuevas-Maraver, P. G. Kevrekidis, A. Saxena, F. Cooper, A. Khare, A. Comech and C. M. Bender,
Solitary waves of a $ \mathcal{PT}$-symmetric nonlinear Dirac equation, IEEE Journal of Selected Topics in Quantum Electronics, 22 (2016), 1-9.
|
[13] |
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. |
[14] |
T. Durt, D. Kaszlikowski, J.-L. Chen and L. C. Kwek,
Security of quantum key distributions with entangled qudits, Phys. Rev. A, 69 (2004), 032313.
doi: 10.1103/PhysRevA.69.032313. |
[15] |
B. J. Eggleton, C. M. De Sterke and R. E. Slusher,
Nonlinear pulse propagation in Bragg gratings, J. Opt. Soc. Am. B, 14 (1997), 2980-2993.
|
[16] |
A. Galindo,
A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.
doi: 10.1007/BF02785129. |
[17] |
D. J. Gross and A. Neveu,
Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D, 10 (1974), 3235-3253.
|
[18] |
D. D. Ivanenko,
Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz, 8 (1938), 260-266.
|
[19] |
M. Kues, C. Reimer, P. Roztocki, L. R. Cort, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Aza and R. Morandotti,
On-chip generation of high-dimensional entangled quantum states and their coherent control, Nature, 546 (2017), 622-626.
doi: 10.1038/nature22986. |
[20] |
N. Lazarides and G. P. Tsironis,
Gain-driven discrete breathers in $\mathcal{P}\mathcal{T}$-symmetric nonlinear metamaterials, Phys. Rev. Lett., 110 (2013), 053901.
doi: 10.1103/PhysRevLett.110.053901. |
[21] |
S. Y. Lee and A. Gavrielides,
Quantization of the localized solutions in two-dimensional field theories of massive fermions, Phys. Rev. D, 12 (1975), 3880-3886.
|
[22] |
A. Marini, S. Longhi and F. Biancalana,
Optical simulation of neutrino oscillations in binary waveguide arrays, Phys. Rev. Lett., 113 (2014), 150401.
doi: 10.1103/PhysRevLett.113.150401. |
[23] |
A. A. Melnikov and L. E. Fedichkin,
Quantum walks of interacting fermions on a cycle graph, Sci. Rep., 6 (2016), 34226.
doi: 10.1038/srep34226. |
[24] |
R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov and Y. S. Kivshar,
Observation of discrete gap solitons in binary waveguide arrays, Opt. Lett., 29 (2004), 2890-2892.
|
[25] |
T. Ozawa and K. Yamauchi,
Structure of Dirac matrices and invariants for nonlinear Dirac equations, Differential Integral Equations, 17 (2004), 971-982.
|
[26] |
D. E. Pelinovsky and A. Stefanov,
Asymptotic stability of small gap solitons in nonlinear Dirac equations, J. Math. Phys., 53 (2012), 073705, 27.
doi: 10.1063/1.4731477. |
[27] |
J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis and T. Kottos,
$\mathcal{PT}$-symmetric electronics, Journal of Physics A: Mathematical and Theoretical, 45 (2012), 444029.
|
[28] |
J. Schindler, A. Li, M. C. Zheng, F. M. Ellis and T. Kottos,
Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries, Phys. Rev. A, 84 (2011), 040101.
doi: 10.1103/PhysRevA.84.040101. |
[29] |
M. Soler,
Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.
|
[30] |
W. E. Thirring,
A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112.
doi: 10.1016/0003-4916(58)90015-0. |
[31] |
M. Wakano,
Intensely localized solutions of the classical Dirac-Maxwell field equations, Progr. Theoret. Phys., 35 (1966), 1117-1141.
|
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