Abstract

We show that one-dimensional parametric spatial solitons undergo temporal instability that leads to their breakup into spatiotemporal patterns with a characteristic snakelike shape.

© 1997 Optical Society of America

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References

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  1. Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).
  2. A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995); L. Torner, C. M. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
    [CrossRef] [PubMed]
  3. H. T. Tran, Opt. Commun. 118, 581 (1995).
    [CrossRef]
  4. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef] [PubMed]
  5. R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
    [CrossRef]
  6. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  7. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  8. A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
    [CrossRef]
  9. S. K. Turitsyn, JETP Lett. 61, 469 (1995).
  10. E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
    [CrossRef]
  11. A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
    [CrossRef]
  12. S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).
    [CrossRef] [PubMed]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

1996 (3)

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

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
[CrossRef]

1995 (5)

S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).
[CrossRef] [PubMed]

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

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

S. K. Turitsyn, JETP Lett. 61, 469 (1995).

1994 (1)

1986 (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

1981 (1)

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

1974 (1)

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
[CrossRef]

Buryak, A. V.

Ferro, P.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

Hagan, D. J.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Kivshar, Yu. S.

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Pelinovsky, D. E.

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

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
[CrossRef]

Schiek, R.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
[CrossRef]

Stegeman, G. I.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

Torner, L.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Tran, H. T.

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

Trillo, S.

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

S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).
[CrossRef] [PubMed]

Turitsyn, S. K.

S. K. Turitsyn, JETP Lett. 61, 469 (1995).

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
[CrossRef]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Europhys. Lett. 35, 25 (1996).
[CrossRef]

JETP Lett. (2)

S. K. Turitsyn, JETP Lett. 61, 469 (1995).

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Opt. Commun. (1)

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rep. (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).
[CrossRef]

Phys. Rev. E (1)

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 113 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

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]

Physica D (1)

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Other (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1990), p. 377.

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Figures (3)

Fig. 1
Fig. 1

Spectral features of the instability gain g for σ=2, β1=σβ2=1, and α=1 (dashed curve), α=5 (solid curve), α=10 (dotted curve). The asymptotic dependence for α=1.0 and σ=2 is given by Eq.  (7).

Fig. 2
Fig. 2

Spatial profiles of the (a) real and (b) imaginary parts of the unstable eigenfunctions 1s (solid curves), 1a* (dashed curves), 2s (dotted–dashed curves), for α=10, σ=2, β1=σβ2=1. We fix a common arbitrary phase so that the eigenfunction with the largest modulus is real at its peak.

Fig. 3
Fig. 3

Spatiotemporal evolutions of a perturbed 1D spatial soliton: (a) input field at Z=0, output fields at (b) Z=20, (c) Z=30, (d) Z=33.

Equations (7)

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iu1Z+122u1X2-β122u1T2+u2u1*=0, iu2Z+12σ2u1X2-β222u2T2+δku2+u122=0,
u1Z, X, T=μσx1μX+a1Z, X, TexpiμZ,
u2Z, X, T=μx2μX+a2Z, X, Texpi2µZ.
ia1ζ+122ξ2-β12τ2a1-a1+x1a2+x2a1*=0, iσa2ζ+122ξ2-β2σ2τ2a2-αa2+x1a1=0,
aj=js expiΩτ+ja exp-iΩτexpiλζ.
[L1x2x10-x2-L10-x1x10L200-x10-L2]-λI=0,
g=Ω2β1+β2/151/2,

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