Abstract

We analyze a mode-locked regime in Kerr frequency combs generated in nonlinear microresonators. Using damped driven nonlinear Schrödinger equations we show that the combs can produce subpicosecond optical pulses when the resonators are characterized with a small enough anomalous group velocity dispersion. We provide an analytical solution of the problem for the case of small damping.

© 2011 Optical Society of America

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  1. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
    [CrossRef] [PubMed]
  2. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
    [CrossRef] [PubMed]
  3. P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
    [CrossRef] [PubMed]
  4. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
    [CrossRef] [PubMed]
  5. O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.
  6. M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.
  7. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.
  8. M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
    [CrossRef]
  9. D. K. Serkland and P. Kumar, Opt. Lett. 24, 92 (1999).
    [CrossRef]
  10. I. H. Agha, Y. Okawachi, and A. L. Gaeta, Opt. Express 17, 16209 (2009).
    [CrossRef] [PubMed]
  11. Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  12. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
    [CrossRef]
  13. L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
    [CrossRef]
  14. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
    [CrossRef]
  15. A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
    [CrossRef]
  16. I. V. Barashenkov and Yu. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
    [CrossRef]
  17. D. J. Kaup and A. C. Newell, Proc. R. Soc. A 361, 413 (1978).
    [CrossRef]
  18. G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
    [CrossRef]

2011

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef] [PubMed]

2010

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

2009

2008

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

2007

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

2005

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

2000

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[CrossRef]

1999

1996

I. V. Barashenkov and Yu. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[CrossRef]

1990

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

1989

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
[CrossRef]

1978

D. J. Kaup and A. C. Newell, Proc. R. Soc. A 361, 413 (1978).
[CrossRef]

Agha, I. H.

Arcizet, O.

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

Barashenkov, I. V.

I. V. Barashenkov and Yu. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[CrossRef]

Chembo, Y. K.

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Chen, L.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Del’Haye, P.

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef] [PubMed]

Ferdous, F.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Foster, M. A.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Gaeta, A. L.

I. H. Agha, Y. Okawachi, and A. L. Gaeta, Opt. Express 17, 16209 (2009).
[CrossRef] [PubMed]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Hasegawa, A.

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[CrossRef]

Haus, H. A.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef] [PubMed]

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

Ilchenko, V. S.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

Kaup, D. J.

D. J. Kaup and A. C. Newell, Proc. R. Soc. A 361, 413 (1978).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef] [PubMed]

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

Kumar, P.

Kuzucu, O.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Leaird, D. E.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Levy, J. S.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Liang, W.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

Lipson, M.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Maleki, L.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

Matsko, A. B.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

McLaughlin, D. W.

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

Miao, H.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Nakazawa, M.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
[CrossRef]

Newell, A. C.

D. J. Kaup and A. C. Newell, Proc. R. Soc. A 361, 413 (1978).
[CrossRef]

Okawachi, Y.

Overman, E. A.

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

Pearlstein, A. J.

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

Saha, K.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

Savchenkov, A. A.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

Schliesser, A.

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

Seidel, D.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

Serkland, D. K.

Smirnov, Yu. S.

I. V. Barashenkov and Yu. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[CrossRef]

Solomatine, I.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

Srinivasan, K.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
[CrossRef]

Terrones, G.

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

Varghese, L. T.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Wang, J.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Weiner, A. M.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Yu, N.

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

IEEE J. Quantum Electron.

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 6, 1161 (2000).
[CrossRef]

Nature

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Phys. Rev. E

I. V. Barashenkov and Yu. S. Smirnov, Phys. Rev. E 54, 5707 (1996).
[CrossRef]

Phys. Rev. Lett.

P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef] [PubMed]

Proc. IEEE Int. Freq. Control Symp.

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, W. Liang, D. Seidel, and A. B. Matsko, Proc. IEEE Int. Freq. Control Symp. 1, 558 (2010).
[CrossRef]

Proc. R. Soc. A

D. J. Kaup and A. C. Newell, Proc. R. Soc. A 361, 413 (1978).
[CrossRef]

Science

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science 332, 555 (2011).
[CrossRef] [PubMed]

SIAM J. Appl. Math.

G. Terrones, D. W. McLaughlin, E. A. Overman, and A. J. Pearlstein, SIAM J. Appl. Math. 50, 791 (1990).
[CrossRef]

Other

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, in Proceedings of 7th Symposium of Frequency Standards and Metrology, L.Maeki, ed. (World Scientific, 2009), 539.
[CrossRef]

O. Arcizet, A. Schliesser, P. Del’Haye, R. Holzwarth, and T. J. Kippenberg, in Practical Applications of Microresonators in Optics and Photonics, A.B.Matsko, ed. (CRC, 2009), Chap. 11.

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, http://arxiv.org/abs/1102.0326.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, http://arxiv.org/abs/1103.2330.

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Figures (3)

Fig. 1
Fig. 1

Duration and the relative peak power of the pulses leaving the resonator. The duration is determined as T 0 = | β 2 Σ | / ( 2 δ 0 B 2 ) . The relative pulse peak power is calculated as the ratio of the power of the pulse at t = 0 and the cw background power.

Fig. 2
Fig. 2

The cw background | ψ 0 | 2 and normalized pump power h found from the solution of Eqs. (9, 10) as functions of the power of external pumping P 0 of the resonator. The values | ψ | 2 1 / 6 and α ˜ / 2 determine the stability boundaries for the cw background. The values h + and h thr determine some (not all) stability boundaries for the mode-locked solution.

Fig. 3
Fig. 3

Solution of Eq. (11) for α ˜ = 0.01 . The lower branch is stable until point C ( 2 / 27 , 1 / 6 ) . Sector BC is dynamically unstable with respect to fluctuations of the external pump power [ B ( 1 / 16 , 1 / 4 ) ] . Sector AB as well as the top branch are modulationally unstable [ A ( α ˜ 2 / 2 , 1 / 2 ) ] . Static solutions ψ ( x ) localized in x space are unstable at the top branch starting from the point A. Solitons with monotonic asymptotes are allowed for α ˜ / 2 < | ψ 0 | 2 < 1 / 6 . In accordance with Fig. 2, the evaluated cw background is located below point C so that the solution is stable.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

τ 0 A τ + i 2 β 2 Σ 2 A t 2 i γ Σ | A | 2 A = ( α Σ + T c 2 + i δ 0 ) A + T c A in ,
i ψ t ˜ + 2 ψ x 2 ψ + 2 | ψ | 2 ψ = i α ˜ ψ h ,
ψ ± ( x ) = ψ 0 ( 1 + 2 sinh 2 β 1 ± cosh β cosh ( B x ) ) ,
h = 2 cosh 2 β ( 1 + 2 cosh 2 β ) 3 / 2 .
A out = A in [ 1 T c + i 2 δ 0 T c γ Σ P 0 ψ ± ( t 2 δ 0 | β 2 Σ | ) ] .
Δ E 1 = T c [ ( P 0 2 δ 0 γ Σ ψ 0 2 ) τ 0 2 B 2 δ 0 | β 2 Σ | γ Σ ] ,
Δ E 2 = T c [ 2 δ 0 γ Σ ψ 0 2 τ 0 + 2 B 2 δ 0 | β 2 Σ | γ Σ ] .
2 δ 0 γ Σ ψ 0 2 τ 0 + 2 B 2 δ 0 | β 2 Σ | γ Σ = 1 2 P 0 τ 0 .
cosh 4 β ( 1 + 2 cosh 2 β ) 3 = T c γ Σ P 0 4 δ 0 3 ,
δ 0 τ 0 1 + 2 cosh 2 β + 4 δ 0 | β 2 Σ | sinh β 1 + 2 cosh 2 β = 1 2 γ Σ P 0 τ 0 .
| ψ 0 | 2 [ 4 | ψ 0 | 2 ( | ψ 0 | 2 1 ) + 1 + α ˜ 2 ] = h 2 ,
tan θ = α ˜ / ( 1 | ψ 0 | 2 ) .

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