October  2011, 4(5): 1095-1105. doi: 10.3934/dcdss.2011.4.1095

The dynamics of the kink in curved large area Josephson junction

1. 

Institute of Physics UP, Podchorążych 2, 30-084 Cracow, Poland

Received  August 2009 Revised  December 2009 Published  December 2010

A formalism that allows description of the kink motion in an arbitrarily curved large area Josephson junction is proposed. A general formula for the lagrangian density that describes the curved Josephson junction, in small curvature regime, is obtained. Examples of propagation of the kink along the curved Josephson junction are considered.
Citation: Tomasz Dobrowolski. The dynamics of the kink in curved large area Josephson junction. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1095-1105. doi: 10.3934/dcdss.2011.4.1095
References:
[1]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering,", London Mathematical Society Lecture Note Series, (1991). Google Scholar

[2]

P. W. Anderson and J. M. Rowell, Probable observation of the Josephson superconducting tunneling effect,, Phys. Rev. Lett., 10 (1963), 230. doi: 10.1103/PhysRevLett.10.230. Google Scholar

[3]

H. Arodź and R. Pełka, Evolution of interfaces and expansion in width,, Phys. Rev. E, 62 (2000), 6749. doi: 10.1103/PhysRevE.62.6749. Google Scholar

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems,", Cambridge Monographs on Mathematical Physics, (2003). Google Scholar

[5]

A. Barone and G. Paterno, "Physics and Applications of the Josephson Effect,", Wiley, (1982). doi: 10.1002/352760278X. Google Scholar

[6]

T. Dobrowolski, Construction of curved domain walls,, Phys. Rev. E, 77 (2008). Google Scholar

[7]

A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm,, Rev. Mod. Phys., 68 (1996), 733. doi: 10.1103/RevModPhys.68.733. Google Scholar

[8]

J. C. Fernandez, M. J. Goupil, O. Legrand and G. Reinisch, Relativistic dynamics of sine-Gordon solitons trapped in confining potentials,, Phys. Rev. B, 34 (1986), 6207. doi: 10.1103/PhysRevB.34.6207. Google Scholar

[9]

L. A. Ferreira, B. Piette and W. J. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model,, Phys. Rev. E, 77 (2008). Google Scholar

[10]

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Quantum superposition of distinct macroscopic states,, Nature, 406 (2000), 43. doi: 10.1038/35017505. Google Scholar

[11]

C. Gorria, Yu. B. Gaididei, M. P. Soerensen, P. L. Christiansen and J. G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction,, Phys. Rev. B, 69 (2004). Google Scholar

[12]

B. D. Josephson, Possible new effects in superconductive tunnelling,, Phys. Lett., 1 (1962), 251. doi: 10.1016/0031-9163(62)91369-0. Google Scholar

[13]

B. D. Josephson, Supercurrents through barriers,, Adv. Phys., 14 (1965), 419. doi: 10.1080/00018736500101091. Google Scholar

[14]

A. Kemp, A. Wallraff, A. V. Ustinov, Josephson vortex qubit: Design, preparation and read-out,, Phys. Stat. Sol., B 233 (2002), 472. doi: 10.1002/1521-3951(200210)233:3<472::AID-PSSB472>3.0.CO;2-J. Google Scholar

[15]

K. K. Kobayashi and M. Izutsu, Exact solution on the n-dimensional sine-Gordon equation,, J. Phys. Soc. Japan, 41 (1976), 1091. doi: 10.1143/JPSJ.41.1091. Google Scholar

[16]

G. Leibbrandt, New exact solutions of the classical sine-Gordon equation in 2 + 1 and 3 + 1 dimensions,, Phys. Rev. Lett., 41 (1978), 435. doi: 10.1103/PhysRevLett.41.435. Google Scholar

[17]

B. A. Malomed, Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model,, Physica D, 52 (1991), 157. doi: 10.1016/0167-2789(91)90118-S. Google Scholar

[18]

Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Coherent control of macroscopic quantum states in a single-Cooperpair box,, Nature, 398 (1999), 786. Google Scholar

[19]

M. J. Rice, Physical dynamics of solitons,, Phys. Rev. B, 28 (1983), 3587. doi: 10.1103/PhysRevB.28.3587. Google Scholar

[20]

E. Turlot, D. Esteve, C. Urbina and M. Devoret, Dynamical isoperimeter pattern in the square sine-Gordon system,, Phys. Rev. B, 42 (1990), 8418. doi: 10.1103/PhysRevB.42.8418. Google Scholar

[21]

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd and J. E. Mooij, Quantum superposition of macroscopic persistent-current states,, Science, 290 (2000), 773. doi: 10.1126/science.290.5492.773. Google Scholar

[22]

J. Zagrodziński, Particular solutions of the sine-Gordon equation in 2 + 1 dimensions,, Phys. Lett. A, 72 (1979), 284. doi: 10.1016/0375-9601(79)90469-9. Google Scholar

show all references

References:
[1]

M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering,", London Mathematical Society Lecture Note Series, (1991). Google Scholar

[2]

P. W. Anderson and J. M. Rowell, Probable observation of the Josephson superconducting tunneling effect,, Phys. Rev. Lett., 10 (1963), 230. doi: 10.1103/PhysRevLett.10.230. Google Scholar

[3]

H. Arodź and R. Pełka, Evolution of interfaces and expansion in width,, Phys. Rev. E, 62 (2000), 6749. doi: 10.1103/PhysRevE.62.6749. Google Scholar

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems,", Cambridge Monographs on Mathematical Physics, (2003). Google Scholar

[5]

A. Barone and G. Paterno, "Physics and Applications of the Josephson Effect,", Wiley, (1982). doi: 10.1002/352760278X. Google Scholar

[6]

T. Dobrowolski, Construction of curved domain walls,, Phys. Rev. E, 77 (2008). Google Scholar

[7]

A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm,, Rev. Mod. Phys., 68 (1996), 733. doi: 10.1103/RevModPhys.68.733. Google Scholar

[8]

J. C. Fernandez, M. J. Goupil, O. Legrand and G. Reinisch, Relativistic dynamics of sine-Gordon solitons trapped in confining potentials,, Phys. Rev. B, 34 (1986), 6207. doi: 10.1103/PhysRevB.34.6207. Google Scholar

[9]

L. A. Ferreira, B. Piette and W. J. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model,, Phys. Rev. E, 77 (2008). Google Scholar

[10]

J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo and J. E. Lukens, Quantum superposition of distinct macroscopic states,, Nature, 406 (2000), 43. doi: 10.1038/35017505. Google Scholar

[11]

C. Gorria, Yu. B. Gaididei, M. P. Soerensen, P. L. Christiansen and J. G. Caputo, Kink propagation and trapping in a two-dimensional curved Josephson junction,, Phys. Rev. B, 69 (2004). Google Scholar

[12]

B. D. Josephson, Possible new effects in superconductive tunnelling,, Phys. Lett., 1 (1962), 251. doi: 10.1016/0031-9163(62)91369-0. Google Scholar

[13]

B. D. Josephson, Supercurrents through barriers,, Adv. Phys., 14 (1965), 419. doi: 10.1080/00018736500101091. Google Scholar

[14]

A. Kemp, A. Wallraff, A. V. Ustinov, Josephson vortex qubit: Design, preparation and read-out,, Phys. Stat. Sol., B 233 (2002), 472. doi: 10.1002/1521-3951(200210)233:3<472::AID-PSSB472>3.0.CO;2-J. Google Scholar

[15]

K. K. Kobayashi and M. Izutsu, Exact solution on the n-dimensional sine-Gordon equation,, J. Phys. Soc. Japan, 41 (1976), 1091. doi: 10.1143/JPSJ.41.1091. Google Scholar

[16]

G. Leibbrandt, New exact solutions of the classical sine-Gordon equation in 2 + 1 and 3 + 1 dimensions,, Phys. Rev. Lett., 41 (1978), 435. doi: 10.1103/PhysRevLett.41.435. Google Scholar

[17]

B. A. Malomed, Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model,, Physica D, 52 (1991), 157. doi: 10.1016/0167-2789(91)90118-S. Google Scholar

[18]

Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Coherent control of macroscopic quantum states in a single-Cooperpair box,, Nature, 398 (1999), 786. Google Scholar

[19]

M. J. Rice, Physical dynamics of solitons,, Phys. Rev. B, 28 (1983), 3587. doi: 10.1103/PhysRevB.28.3587. Google Scholar

[20]

E. Turlot, D. Esteve, C. Urbina and M. Devoret, Dynamical isoperimeter pattern in the square sine-Gordon system,, Phys. Rev. B, 42 (1990), 8418. doi: 10.1103/PhysRevB.42.8418. Google Scholar

[21]

C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd and J. E. Mooij, Quantum superposition of macroscopic persistent-current states,, Science, 290 (2000), 773. doi: 10.1126/science.290.5492.773. Google Scholar

[22]

J. Zagrodziński, Particular solutions of the sine-Gordon equation in 2 + 1 dimensions,, Phys. Lett. A, 72 (1979), 284. doi: 10.1016/0375-9601(79)90469-9. Google Scholar

[1]

Rinaldo M. Colombo, Mauro Garavello. Comparison among different notions of solution for the $p$-system at a junction. Conference Publications, 2009, 2009 (Special) : 181-190. doi: 10.3934/proc.2009.2009.181

[2]

Roman Czapla, Vladimir V. Mityushev. A criterion of collective behavior of bacteria. Mathematical Biosciences & Engineering, 2017, 14 (1) : 277-287. doi: 10.3934/mbe.2017018

[3]

Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123

[4]

Gabrielle Demange. Collective attention and ranking methods. Journal of Dynamics & Games, 2014, 1 (1) : 17-43. doi: 10.3934/jdg.2014.1.17

[5]

Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573

[6]

Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1629-1645. doi: 10.3934/dcdss.2016067

[7]

Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001

[8]

Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096

[9]

Luca Consolini, Alessandro Costalunga, Manfredi Maggiore. A coordinate-free theory of virtual holonomic constraints. Journal of Geometric Mechanics, 2018, 10 (4) : 467-502. doi: 10.3934/jgm.2018018

[10]

Niclas Kruff, Sebastian Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020075

[11]

Michael Blank. Emergence of collective behavior in dynamical networks. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 313-329. doi: 10.3934/dcdsb.2013.18.313

[12]

Mauro Garavello. The LWR traffic model at a junction with multibuffers. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 463-482. doi: 10.3934/dcdss.2014.7.463

[13]

Claudio Muñoz. The Gardner equation and the stability of multi-kink solutions of the mKdV equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3811-3843. doi: 10.3934/dcds.2016.36.3811

[14]

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 851-879. doi: 10.3934/dcdsb.2018210

[15]

Rinaldo M. Colombo, Mauro Garavello. A Well Posed Riemann Problem for the $p$--System at a Junction. Networks & Heterogeneous Media, 2006, 1 (3) : 495-511. doi: 10.3934/nhm.2006.1.495

[16]

Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149

[17]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225

[18]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225

[19]

Kelum Gajamannage, Erik M. Bollt. Detecting phase transitions in collective behavior using manifold's curvature. Mathematical Biosciences & Engineering, 2017, 14 (2) : 437-453. doi: 10.3934/mbe.2017027

[20]

Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks & Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]