American Institute of Mathematical Sciences

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July  2018, 17(4): 1331-1347. doi: 10.3934/cpaa.2018065

Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models

Nabile Boussïd 1, and  Andrew Comech 2,3,4,,

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

* Corresponding author: Andrew Comech

Received  November 2017 Revised  February 2018 Published  April 2018

Fund Project: The first author aknowledges the support of Region Bourgogne Franche-Comté through the project "Projet du LMB: Analyse mathematique et simulation numérique d'EDP issues de problèmes de contrôle et du trafic routier". The second author was supported by the Russian Foundation for Sciences (project 14-50-00150).

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.

Keywords: Bi-frequency solitary waves, Bogoliubov SU(1, 1) symmetry, Dirac-Klein-Gordon system, spectral stability, nonlinear Dirac equation, linear instability, Soler model, qubits, qudits, Yukawa model.
Mathematics Subject Classification: Primary: 35C08, 35Q41, 37K40, 81Q05; Secondary: 37N20.
Citation: Nabile Boussïd, Andrew Comech. Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1331-1347. doi: 10.3934/cpaa.2018065
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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.  doi: 10.1007/BF02785129.  Google Scholar

[17]

D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D, 10 (1974), 3235-3253.   Google Scholar

[18]

D. D. Ivanenko, Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz, 8 (1938), 260-266.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

T. Ozawa and K. Yamauchi, Structure of Dirac matrices and invariants for nonlinear Dirac equations, Differential Integral Equations, 17 (2004), 971-982.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.   Google Scholar

[30]

W. E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112.  doi: 10.1016/0003-4916(58)90015-0.  Google Scholar

[31]

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

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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

A. Galindo, A remarkable invariance of classical Dirac Lagrangians, Lett. Nuovo Cimento (2), 20 (1977), 210-212.  doi: 10.1007/BF02785129.  Google Scholar

[17]

D. J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D, 10 (1974), 3235-3253.   Google Scholar

[18]

D. D. Ivanenko, Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz, 8 (1938), 260-266.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[25]

T. Ozawa and K. Yamauchi, Structure of Dirac matrices and invariants for nonlinear Dirac equations, Differential Integral Equations, 17 (2004), 971-982.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D, 1 (1970), 2766-2769.   Google Scholar

[30]

W. E. Thirring, A soluble relativistic field theory, Ann. Physics, 3 (1958), 91-112.  doi: 10.1016/0003-4916(58)90015-0.  Google Scholar

[31]

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

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