Abstract

The dynamics of nonlinear pulse propagation in arrays of linearly coupled optical waveguides is considered. For an array of three waveguides, a simplified model based on a variational method is able to predict the steady-state solutions and the strong coupling dynamics with a good degree of accuracy and to furnish a necessary condition for the existence of localized states, as confirmed by numerical solutions of the governing equations. The pulse–radiation interaction is also studied, revealing that additional, long-period oscillations overlap the linear coupling frequency. In the regime of initial conditions that lead to strong localization dynamics, we present a theoretical result that predicts to a good approximation the asymptotic localized state.

© 1997 Optical Society of America

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  1. A. V. Buryak and N. N. Akhmediev, IEEE J. Quantum Electron. 31, 682 (1995).
    [Crossref]
  2. C. Paré and M. Florjańczyk, Phys. Rev. A 41, 6287 (1990).
    [Crossref]
  3. E. Cagliotti, S. Trillo, S. Wabnitz, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. B 7, 374 (1990).
    [Crossref]
  4. P. L. Chu, B. A. Malomed, and G. D. Peng, J. Opt. Soc. Am. B 10, 1379 (1993).
    [Crossref]
  5. I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).
  6. F. Kh. Abdullaev, R. M. Abrarov, and S. A. Darmanyan, Opt. Lett. 14, 131 (1989).
    [Crossref] [PubMed]
  7. Y. S. Kivshar, Opt. Lett. 18, 7 (1993).
    [Crossref] [PubMed]
  8. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
    [Crossref]
  9. A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
    [Crossref] [PubMed]
  10. A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, Opt. Lett. 19, 332 (1994).
    [Crossref] [PubMed]
  11. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Opt. Lett. 19, 329 (1994).
    [Crossref] [PubMed]
  12. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
    [Crossref]
  13. N. Finlayson and G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
    [Crossref]
  14. J. M. Soto-Crespo and E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
    [Crossref]
  15. N. N. Akhmediev and A. V. Buryak, J. Opt. Soc. Am. B 11, 804 (1994).
    [Crossref]
  16. A. B. Aceves and M. Santagiustina, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 11 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper NME11.
  17. P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.
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    [Crossref] [PubMed]
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    [Crossref]
  20. A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, Opt. Lett. 19, 1186 (1994).
    [Crossref] [PubMed]
  21. A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
    [Crossref]
  22. A. V. Buryak and N. N. Akhmediev, Opt. Commun. 110, 287 (1994).
    [Crossref]
  23. M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 (1996).
    [Crossref]
  24. G. R. Hadley, Opt. Lett. 16, 624 (1991); IEEE J. Quantum Electron. 28, 363 (1992).
    [Crossref] [PubMed]
  25. N. Akhmediev and A. Ankiewicz, J. Electr. Electron. Eng. Austral. 15, 277–287 (1995); Phys. Rev. Lett. 70, 2395 (1993).
    [Crossref]
  26. D. B. Mortimore and J. W. Arkwright, Appl. Opt. 30, 650 (1991).
    [Crossref] [PubMed]
  27. W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
    [Crossref]
  28. A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
    [Crossref]

1996 (2)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 (1996).
[Crossref]

1995 (7)

N. Akhmediev and A. Ankiewicz, J. Electr. Electron. Eng. Austral. 15, 277–287 (1995); Phys. Rev. Lett. 70, 2395 (1993).
[Crossref]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[Crossref]

A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
[Crossref]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. V. Buryak and N. N. Akhmediev, IEEE J. Quantum Electron. 31, 682 (1995).
[Crossref]

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

1994 (5)

1993 (2)

1991 (3)

1990 (3)

N. Finlayson and G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[Crossref]

C. Paré and M. Florjańczyk, Phys. Rev. A 41, 6287 (1990).
[Crossref]

E. Cagliotti, S. Trillo, S. Wabnitz, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. B 7, 374 (1990).
[Crossref]

1989 (1)

1988 (2)

1980 (1)

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Abdullaev, F. Kh.

Abrarov, R. M.

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, Opt. Lett. 19, 1186 (1994).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Opt. Lett. 19, 329 (1994).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, Opt. Lett. 19, 332 (1994).
[Crossref] [PubMed]

A. B. Aceves and M. Santagiustina, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 11 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper NME11.

Aitchinson, J. S.

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, J. Electr. Electron. Eng. Austral. 15, 277–287 (1995); Phys. Rev. Lett. 70, 2395 (1993).
[Crossref]

Akhmediev, N. N.

A. V. Buryak and N. N. Akhmediev, IEEE J. Quantum Electron. 31, 682 (1995).
[Crossref]

N. N. Akhmediev and A. V. Buryak, J. Opt. Soc. Am. B 11, 804 (1994).
[Crossref]

A. V. Buryak and N. N. Akhmediev, Opt. Commun. 110, 287 (1994).
[Crossref]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, J. Electr. Electron. Eng. Austral. 15, 277–287 (1995); Phys. Rev. Lett. 70, 2395 (1993).
[Crossref]

Arkwright, J. W.

Buryak, A. V.

A. V. Buryak and N. N. Akhmediev, IEEE J. Quantum Electron. 31, 682 (1995).
[Crossref]

N. N. Akhmediev and A. V. Buryak, J. Opt. Soc. Am. B 11, 804 (1994).
[Crossref]

A. V. Buryak and N. N. Akhmediev, Opt. Commun. 110, 287 (1994).
[Crossref]

Cagliotti, E.

Christodoulides, D. N.

Chu, P. L.

Crosignani, B.

Darmanyan, S. A.

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, Opt. Lett. 19, 1186 (1994).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Opt. Lett. 19, 329 (1994).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, S. Trillo, and S. Wabnitz, Opt. Lett. 19, 332 (1994).
[Crossref] [PubMed]

Di Porto, P.

Finlayson, N.

N. Finlayson and G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[Crossref]

Florjanczyk, M.

C. Paré and M. Florjańczyk, Phys. Rev. A 41, 6287 (1990).
[Crossref]

Goldman, M. V.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Gölles, M.

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

Hadley, G. R.

Hafizi, B.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Joseph, R. I.

Kang, J. U.

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

Kath, W. L.

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[Crossref]

Kivshar, Y. S.

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

Y. S. Kivshar, Opt. Lett. 18, 7 (1993).
[Crossref] [PubMed]

Kuznetsov, A. Z.

A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
[Crossref]

Lederer, F.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

Luther, G. G.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, Opt. Lett. 19, 1186 (1994).
[Crossref] [PubMed]

Malomed, B. A.

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

P. L. Chu, B. A. Malomed, and G. D. Peng, J. Opt. Soc. Am. B 10, 1379 (1993).
[Crossref]

Mikhailov, A. V.

A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
[Crossref]

Millar, P.

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

Mortimore, D. B.

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

Paré, C.

C. Paré and M. Florjańczyk, Phys. Rev. A 41, 6287 (1990).
[Crossref]

Peng, G. D.

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

Rubenchik, A. M.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, Opt. Lett. 19, 1186 (1994).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Opt. Lett. 19, 329 (1994).
[Crossref] [PubMed]

Rypdal, K.

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Santagiustina, M.

A. B. Aceves and M. Santagiustina, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 11 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper NME11.

Shimokhin, I. A.

A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
[Crossref]

Smyth, N. F.

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo and E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[Crossref]

Stegeman, G. I.

N. Finlayson and G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[Crossref]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
[Crossref]

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

Trillo, S.

Turitsyn, S. K.

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Opt. Lett. 19, 329 (1994).
[Crossref] [PubMed]

Uzunov, I. M.

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

Villeneuve, A.

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

Wabnitz, S.

Weinstein, M. I.

M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 (1996).
[Crossref]

Wright, E. M.

J. M. Soto-Crespo and E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[Crossref]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 672 (1988).
[Crossref]

Yeary, B.

M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 (1996).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

N. Finlayson and G. I. Stegeman, Appl. Phys. Lett. 56, 2276 (1990).
[Crossref]

IEEE J. Quantum Electron. (1)

A. V. Buryak and N. N. Akhmediev, IEEE J. Quantum Electron. 31, 682 (1995).
[Crossref]

J. Appl. Phys. (1)

J. M. Soto-Crespo and E. M. Wright, J. Appl. Phys. 70, 7240 (1991).
[Crossref]

J. Electr. Electron. Eng. Austral. (1)

N. Akhmediev and A. Ankiewicz, J. Electr. Electron. Eng. Austral. 15, 277–287 (1995); Phys. Rev. Lett. 70, 2395 (1993).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

A. V. Buryak and N. N. Akhmediev, Opt. Commun. 110, 287 (1994).
[Crossref]

Opt. Lett. (8)

Phys. Fluids (1)

M. V. Goldman, K. Rypdal, and B. Hafizi, Phys. Fluids 23, 945 (1980).
[Crossref]

Phys. Lett. A (1)

M. I. Weinstein and B. Yeary, Phys. Lett. A 222, 157 (1996).
[Crossref]

Phys. Rev. A (2)

I. M. Uzunov, R. Muschall, M. Gölles, Y. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. A 51, 2527 (1995).

C. Paré and M. Florjańczyk, Phys. Rev. A 41, 6287 (1990).
[Crossref]

Phys. Rev. E (2)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[Crossref]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[Crossref]

Phys. Rev. Lett. (1)

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, Phys. Rev. Lett. 75, 73 (1995).
[Crossref] [PubMed]

Physica D (2)

A. Z. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, Physica D 87, 201 (1995).
[Crossref]

A. B. Aceves, C. De Angelis, G. G. Luther, A. M. Rubenchik, and S. K. Turitsyn, Physica D 87, 262 (1995).
[Crossref]

Other (2)

A. B. Aceves and M. Santagiustina, in Nonlinear Optics: Materials, Fundamentals, and Applications, Vol. 11 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper NME11.

P. Millar, J. S. Aitchinson, J. U. Kang, G. I. Stegeman, and A. Villeneuve, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper SaA4, p. 125.

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Figures (10)

Fig. 1
Fig. 1

Bifurcation diagram of a for the fixed points of Eqs. (7) as a function of k=K/E2. Solid (dashed) curves represent the stable (unstable) critical points. The stable fixed point, which never bifurcates, has phase p=0; those for a=1 have cos-1[-1/(8πk)]; those with a=0 have cos-1[-1/(64πk)]; all others have p=π.

Fig. 2
Fig. 2

Phase portrait of the orbits in the phase plane [a cos(p), a sin(p)] for k=0.05 (from the variational approach). The separatrices are indicated by large dots; small dots represent other orbits. Note that switching is the only allowed dynamics.

Fig. 3
Fig. 3

Phase portrait of the orbits in the phase plane [a cos(p), a sin(p)] for k=0.025 (from the variational approach). Note the region close to the boundary, which implies the existence of localized states.

Fig. 4
Fig. 4

Pulse energy as a function of propagation distance. The solid curve represent the total pulse energy, the dashed curve the central WG pulse energy, and the dotted curve the energy in the lateral WG. The initial conditions are q(n)=An sech(Ant), with A2=3.025, A1=A3=0, and K=1.

Fig. 5
Fig. 5

Same as for Fig. 4 but with the initial conditions A2 =2.75, A1=A3=0, and K=1.

Fig. 6
Fig. 6

Steady-state solution |q|=|q(n0)| as a result of the numerical solution of the coupled partial differential equations (3) with N=3. The initial conditions were A2=1.25, A1=A3 =0.875, pn=0, and k=1/36. The amplitude of the output steady-state pulse is 1.15, and that predicted by the variational approach is 1.149.

Fig. 7
Fig. 7

Steady-state solution |v|=|q(n0±1)| as a result of the numerical solution of the coupled partial differential equations (3). The initial conditions are the same as in Fig. 6. The amplitude of the output steady-state pulse is 0.9, and that predicted by the variational approach is 0.9255.

Fig. 8
Fig. 8

Example of creation of a localized state for an array composed of three WG’s. The initial condition are the same as for Fig. 1, i.e., q(n0)=2.5 sech(t) and q(n0±1)=0.25 sech(t), with n0=2.

Fig. 9
Fig. 9

Example of creation of a localized state for an array composed of nine WG’s. The initial conditions are the same as for Fig. 8 but with n0=5 and q(n)=0 for |n-n0|>1. Note that because the initial transverse distribution was close to the localized state in Fig. 8 the result is a similar final asymptotic state.

Fig. 10
Fig. 10

Fourier transform of the amplitude waveform as a function of the propagation distance z as a result of a numerical solution. The initial conditions are q(n0)=2 sech(t), q(n0±1) =0.5 sech(t), and k=5/36.

Tables (2)

Tables Icon

Table 1 Comparison of Output Peak Amplitude of the Pulse as Predicted by Analytic Formula (17) and Results from Numerical Solutions with Three or Nine WG’sa

Tables Icon

Table 2 Comparison of Output Peak Amplitude of the Pulse as Predicted by Analytic Formula (17) and Results from Numerical Solutions with Three WG’sa

Equations (20)

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

H=Kn=1N+1 -|q(n)-q(n-1)|2dt-12n=1N-[|q(n)|4-|qt(n)|2]=K n=1N+1 Hnint+n=1NHn.
E=n=1N -|q(n)(t, z)|2dt.
zLqz(n)*+tLqt(n)*-Lq(n)*=0,
L=n=1N Ln+n=1N+1 Lnint,
Ln=i2[q(n)*qz(n)-q(n)qz(n)*]-12|qt(n)|2+12|q(n)|4,
Lnint=-K|q(n)-q(n-1)|2.
dAdZ=hp=(A2-1)sin(p)I(A),
dpdZ=-hA=β(3+10A+3A2)+ddA[(A2-1)cos(p)I(A)],
h(A, p)=β(3A+5A2+A3)+(A2-1)cos(p)I(A);
I(A)=-+ sech(A1t)sech(A2t)dt
|q(n0)|=η sech(ηt),
|q(n0±1)|=12η{exp(ηt)ln[1+exp(-2ηt)]+exp(-ηt)ln[1+exp(2ηt)]},
θ=n=1N -(n2+t2)|q(n)(t, z)|2dt.
θzz=4H+2K2E+34n=1N-|q(n)|4dt-K n=1N-1-(2n+1)(|q(n+1)|2-|q(n)|2)×[q(n+1)q(n)*+q(n+1)*q(n)]dt-4K n=1N-1-|q(n+1)-q(n)|2dt-2K2 n=1N-2-[q(n+2)q(n)*+q(n+2)*q(n)]dt,
θC0η+C2η3=F1(1/η),
θzzF2(1/η)ηz2-1η2F1(1/η)ηzz,
F2=2F1η3+F1η4
ηzz=-η2F1-F2ηz2+4 n=1N Hn+2K2E+η3,
ηe-4 n=1N Hn-2K2E1/3.
4 n=1N Hn=4H-4K n=1N+1 Hnint.

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