Abstract

We analyze the effect of phase-matched third-harmonic generation on the existence and stability of (1+1)-dimensional bright and dark spatial solitary waves in optical media with a cubic (or Kerr) nonlinear response. We demonstrate that parametric coupling of the fundamental beam with the third harmonic leads to the existence of two-color solitary waves resembling those in a χ(2) medium and that it can modify drastically the properties of solitary waves due to effective non-Kerr nonlinearities. In particular, we find a power threshold for the existence of two-frequency parametric bright solitons and also reveal the soliton multistability in a Kerr medium that becomes possible owing to a higher-order nonlinear phase shift caused by cascaded third-order processes. We also analyze dark solitary waves and their stabilities. We show that, in a certain parameter domain, parametric χ(3) dark solitons may become unstable owing to the modulational instability of the supporting background or to other instability mechanisms caused by the parametric coupling between the harmonics.

© 1998 Optical Society of America

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References

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  1. D. L. Mills, Nonlinear Optics: Basic Concepts (Springer-Verlag, Berlin, 1991); R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).
  2. I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
    [Crossref]
  3. See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [Crossref]
  4. Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981);R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993);A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994);L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
    [Crossref] [PubMed]
  5. A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
    [Crossref]
  6. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [Crossref] [PubMed]
  7. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [Crossref] [PubMed]
  8. A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
    [Crossref]
  9. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
    [Crossref] [PubMed]
  10. S. Saltiel, S. Tanev, and A. D. Boardman, Opt. Lett. 22, 148 (1997).
    [Crossref] [PubMed]
  11. See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
    [Crossref]
  12. N. A. Ansari, R. A. Sammut, and H. T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996);J. Opt. Soc. Am. B 14, 298 (1997).
    [Crossref]
  13. This exact solution has been presented previously in the papers by K. Hayata and M. Koshiba, Opt. Lett. 19, 1717 (1994), andY. Chen, Phys. Rev. A 50, 5145 (1994). However, unlike the theory of solitary waves in quadratic media, in the case of third-harmonic generation the exact hyperbolic-secant-like solution does not give any useful information about the families of proper (stable) two-wave solitary waves.
    [Crossref] [PubMed]
  14. A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985);IEEE J. Quantum Electron. 21, 1538 (1985).
    [Crossref] [PubMed]
  15. A. W. Snyder, D. J. Mitchell, and A. V. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).
    [Crossref]
  16. Yu. S. Kivshar and W. Krolikowski, Opt. Lett. 20, 1527 (1995);Yu. S. Kivshar and V. V. Afanasjev, Opt. Lett. 21, 1135 (1996);D. E. Pelinovsky, Yu. S. Kivshar, and V. V. Afanasjev, Phys. Rev. E 54, 2015 (1996).
    [Crossref] [PubMed]
  17. See the review paper by Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons:physics and applications,” Phys. Rep. (to be published).
  18. A. V. Buryak, Phys. Rev. E 52, 1156 (1995); “Solitons due to second harmonic generation,” Ph.D. dissertation (The Australian National University, Canberra, Australia, 1996).
    [Crossref]
  19. A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995).
    [Crossref]

1997 (2)

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[Crossref]

S. Saltiel, S. Tanev, and A. D. Boardman, Opt. Lett. 22, 148 (1997).
[Crossref] [PubMed]

1996 (5)

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

N. A. Ansari, R. A. Sammut, and H. T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996);J. Opt. Soc. Am. B 14, 298 (1997).
[Crossref]

A. W. Snyder, D. J. Mitchell, and A. V. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[Crossref] [PubMed]

See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[Crossref]

1995 (6)

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Yu. S. Kivshar and W. Krolikowski, Opt. Lett. 20, 1527 (1995);Yu. S. Kivshar and V. V. Afanasjev, Opt. Lett. 21, 1135 (1996);D. E. Pelinovsky, Yu. S. Kivshar, and V. V. Afanasjev, Phys. Rev. E 54, 2015 (1996).
[Crossref] [PubMed]

A. V. Buryak, Phys. Rev. E 52, 1156 (1995); “Solitons due to second harmonic generation,” Ph.D. dissertation (The Australian National University, Canberra, Australia, 1996).
[Crossref]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995).
[Crossref]

1994 (1)

1985 (1)

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985);IEEE J. Quantum Electron. 21, 1538 (1985).
[Crossref] [PubMed]

1976 (2)

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[Crossref]

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981);R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993);A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994);L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[Crossref] [PubMed]

Afanasjev, V. V.

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

Ansari, N. A.

Boardman, A. D.

Buryak, A. V.

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[Crossref] [PubMed]

A. W. Snyder, D. J. Mitchell, and A. V. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).
[Crossref]

A. V. Buryak, Phys. Rev. E 52, 1156 (1995); “Solitons due to second harmonic generation,” Ph.D. dissertation (The Australian National University, Canberra, Australia, 1996).
[Crossref]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995).
[Crossref]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

Hagan, D. J.

See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Hayata, K.

Kaplan, A. E.

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985);IEEE J. Quantum Electron. 21, 1538 (1985).
[Crossref] [PubMed]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981);R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993);A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994);L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[Crossref] [PubMed]

Kivshar, Yu. S.

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[Crossref]

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[Crossref] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[Crossref]

Yu. S. Kivshar and W. Krolikowski, Opt. Lett. 20, 1527 (1995);Yu. S. Kivshar and V. V. Afanasjev, Opt. Lett. 21, 1135 (1996);D. E. Pelinovsky, Yu. S. Kivshar, and V. V. Afanasjev, Phys. Rev. E 54, 2015 (1996).
[Crossref] [PubMed]

See the review paper by Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons:physics and applications,” Phys. Rep. (to be published).

Koshiba, M.

Krolikowski, W.

Love, J. D.

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

Luther-Davies, B.

See the review paper by Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons:physics and applications,” Phys. Rep. (to be published).

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Micallef, R. W.

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

Mills, D. L.

D. L. Mills, Nonlinear Optics: Basic Concepts (Springer-Verlag, Berlin, 1991); R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

Mitchell, D. J.

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

Richardson, M. C.

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[Crossref]

Saltiel, S.

Sammut, R. A.

Snyder, A. W.

Stegeman, G. I.

See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981);R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993);A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994);L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[Crossref] [PubMed]

Tanev, S.

Tomov, I. V.

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[Crossref]

Torner, L.

See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Tran, H. T.

Trillo, S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[Crossref] [PubMed]

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

See, for example, the recent review paper on cascading in χ(2)  materials, G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[Crossref]

Phys. Lett. A (1)

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[Crossref]

Phys. Rev. A (1)

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41 (1995).
[Crossref]

Phys. Rev. E (2)

A. V. Buryak, Phys. Rev. E 52, 1156 (1995); “Solitons due to second harmonic generation,” Ph.D. dissertation (The Australian National University, Canberra, Australia, 1996).
[Crossref]

See, e.g., R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996) and references therein.
[Crossref]

Phys. Rev. Lett. (5)

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[Crossref] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995);R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996);R. A. Furst, D. M. Baboiu, B. L. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
[Crossref] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985);IEEE J. Quantum Electron. 21, 1538 (1985).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976); A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981);R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993);A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994);L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[Crossref] [PubMed]

Other (2)

See the review paper by Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons:physics and applications,” Phys. Rep. (to be published).

D. L. Mills, Nonlinear Optics: Basic Concepts (Springer-Verlag, Berlin, 1991); R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

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Figures (12)

Fig. 1
Fig. 1

Variation of the normalized total power, Ptot, versus the dimensionless mismatch parameter α for the three distinct families of solitary-wave solutions of Eqs. (8) and (9). The dashed curve corresponds to the asymptotic expansion found analytically in the cascading limit. Lower curves merge at the bifurcation point O (α=9). Points A–D indicate the particular examples presented in Figs. 2(a)–2(d). Filled circle corresponds to the exact solution (14).

Fig. 2
Fig. 2

Examples of the fundamental (thin solid curve) and third-harmonic (thick solid curve) profiles for several one-hump solitary wave solutions, which belong to different families. Profiles (a)–(d) correspond to points A–D in Fig. 1.

Fig. 3
Fig. 3

Examples of the fundamental (thin solid curve) and third-harmonic (thick solid curve) profiles for several multihump (higher-order) solitary wave solutions belonging to the family that also includes the solution (14) given by filled circle in Fig. 1.

Fig. 4
Fig. 4

Switching dynamics in the bistable region. The figure shows what happens when a beam is launched on the unstable branch at α8.4 (point C of Fig. 1) when its amplitude is initially (a) decreased or (b) increased.

Fig. 5
Fig. 5

Variation of the normalized complementary power, P˜tot, versus the dimensionless mismatch parameter α for solitary-wave solutions of Eqs. (17) and (18). Thick solid and dashed curves show the families of one- and two-frequency dark solitary waves, respectively. Two thin dashed–dotted curves correspond to the asymptotic expansions of the cascading limit. Note that for pointlike solutions (given by open circles) we scale P˜tot(α) by the factor 1/2. Points A–D indicate the particular examples presented in Figs. 7(a)–7(d). Filled circle at ∝=-1 corresponds to the exact solution (23).

Fig. 6
Fig. 6

Examples of the fundamental (thin solid curve) and third-harmonic (thick solid curve) profiles for several dark solitary-wave solutions belonging to the family that also includes the solution (23) given by filled circle in Fig. 5.

Fig. 7
Fig. 7

Examples of some resonant dark-soliton structures for negative values of α: two-soliton states (a) α=-24.3172, (b) α=-9.8142, (c) α=-4.9254, and three-soliton state (d) α=-9.5423. Profiles (a)–(d) correspond to points A–D in Fig. 5.

Fig. 8
Fig. 8

Variation of normalized intensity of the three branches of PW solutions as functions of α. The normalized complementary power of dark solitons in the cascading limit is also shown as the dotted curves.

Fig. 9
Fig. 9

Bands of modulation frequencies, which induce instability of PW background waves. The diagonally striped region applies to the lower (dashed curve) branch of PW solutions in Fig. 8 while the horizontally striped region applies to the intermediate (solid curve) branch.

Fig. 10
Fig. 10

Maximum growth rate of modulational instability for each value of α. The dashed and solid portions of the curves for Ωmax relate to the corresponding branches of PW solutions in Fig. 8.

Fig. 11
Fig. 11

Numerical demonstration of the modulational instability of the approximate two-frequency dark soliton solution at α=-80.

Fig. 12
Fig. 12

Numerical demonstration of the drift instability of the dark soliton at α=-9 [presented in Fig. 6(b)]. In this case, the background is modulationally stable but the soliton itself oscillates and then moves off-axis and transforms into a gray soliton.

Equations (57)

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

2Ex2+2Ez2-n2c2 2Et2=4πc2 2PNLt2,
E=12 {E1 exp[i(k1z-ωt)]+E3 exp[i(k3z-3ωt)]}+c.c.,
PNL(ω)=χ(3)8 [3|E1|2E1+6|E3|2E1+3E1*2E3 exp(-iδkz)]exp[i(k1z-ωt)],
PNL(3ω)=χ(3)8 [3|E3|2E3+6|E1|2E3+E13 exp(iδkz)]exp[i(k3z-3ωt)],
2ik1 E1z+2E1x2±χ[(|E1|2+2|E3|2)E1
+E1*2E3 exp(-iδkz)]=0,
2ik3 E3z+2E3x2±9χ[(|E3|2+2|E1|2)E3
+13 E13 exp(iδkz)]=0,
i UZ+2UX2±19 |U|2+2|W|2U+13 U*2W
=0,
iσ WZ+2WX2-ΔσW±(9|W|2+2|U|2)W+19 U3
=0,
U=3(k1x02χ)1/2E1,
W=(k1x02χ)1/2 exp(-iδkz)E3.
U=uβ exp(iβZ),W=wβ exp(i3βZ),
i uz+2ux2-u+19 |u|2+2|w|2u+13 u*2w=0,
iσ wz+2wx2-αw+(9|w|2+2|u|2)w+19 u3=0,
u(x)=32cosh x 1-22-1cosh2 x-2270-29cosh2 x-12cosh4 x+O(3),
w(x)=62cosh3 x 1-31-10cosh3 x+O(3).
Ptot=-(|u|2+3σ|w|2)dx,
Ptot()=36-192-270725 2+O(3),α-1,
w(x)=2α3 sech(αx),u(x)=0,
us(x)=a sech x,ws(x)=bus(x),
63b3-3b2+17b+1=0,
a2=18(18b2+3b+1).
U=uβ exp(-iβZ),W=wβ exp(-i3βZ).
i uz+2ux2+u-19 |u|2+2|w|2u-13 u*2w=0,
iσ wz+2wx2-αw-(9|w|2+2|u|2)w-19 u3=0.
u(ξ)=3 tanh ξ+92 tanh ξ+3 tanh ξcosh2 ξ-9ξ2 cosh2 ξ+2-6758 tanh ξ-273 tanh ξ2 cosh2 ξ+18 tanh ξcosh4 ξ+1395ξ8 cosh2 ξ-27ξ2 cosh4 ξ-27ξ2 tanh ξ4 cosh2 ξ+O(3),
w(ξ)=-3 tanh3 ξ+2812 tanh ξ-189 tanh ξ2 cosh2 ξ+45 tanh ξcosh4 ξ+27ξ tanh2 ξ2 cosh2 ξ+O(3),
P˜tot=-[(|u0|2+3σ|w0|2)-(|u|2+3σ|w|2)]dx,
w(x)=|α|3 tanh|α|2 x,u(x)=0,
P˜tot=218+69-2312120 2+O(3).
us(x)=a2 tanhx2,ws(x)=bus(x),
1-19 u02-2w02-13 u0w0=0,
αw0+(9w02+2u02)w0+19 u03=0.
u0=w0(α+15-21w02)(7w02-1),
882w06-(21α+945)w04+(α2+27α+324)w02-9
=0,
w0=u09 98u04+(49α-819)u02-(513α+1944)8α2+126α+405,
98u06+(105α-567)u04+(36α2-729α-3645)u02
-(324α2+2916α+6561)=0.
u=u0+(ur+iui)cos(κx)exp(Ωz),
w=w0+(wr+iwi)cos(κx)exp(Ωz).
A11=-A22=Ω,A33=-A44=σΩ,
A12=1-κ2-u02/9+2u0w0/3-2w02,
A13=A24=A31=A42=0,
A14=A32=-u02/3,
A21=1-κ2-u02/3-2u0w0/3-2w02,
A23=A41=-(4u0w0+u02/3),
A34=-(α+κ2+2u02+9w02),
A43=-(α+κ2+2u02+27w02).
Ω4+2BΩ2+D=0,
B=12 A12A21+1σ A23A32+12σ2 A34A43,
D=1σ2 (A142-A12A34)(A232-A21A43).
u(ξ; V)=u(ξ; 0)+i 3V2 [1+(2-tanh2 ξ)]+O(2, V2),
w(ξ; V)=w(ξ; 0)-i 9V2  tanh2 ξ+O(2, V2),

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