Abstract

Relaxation of the nonlinear spatiotemporal dynamics of cylindrically symmetric Schrödinger solitons due to their temporal modulation instability leads to soliton breakup into a train of X waves.

© 2008 Optical Society of America

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  1. J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2103 (2002).
    [CrossRef]
  4. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef] [PubMed]
  5. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
    [CrossRef] [PubMed]
  6. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
    [CrossRef] [PubMed]
  7. M. Kolesik, E. M. Wright, and J. V. Moloney, Opt. Express 13, 10729 (2005).
    [CrossRef] [PubMed]
  8. M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
    [CrossRef]
  9. C. Conti, Phys. Rev. E 68, 016606 (2003).
    [CrossRef]
  10. C. Conti, Phys. Rev. E 70, 046613 (2004).
    [CrossRef]
  11. V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).
  12. D. E. Pelinovsky, Math. Comput. Simul. 55, 585 (2001).
    [CrossRef]
  13. A. V. Yulin, D. V. Skryabin, and A. G. Vladimirov, Opt. Express 14, 12347 (2006).
    [CrossRef] [PubMed]
  14. A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997).
    [CrossRef] [PubMed]
  15. N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Opt. Lett. 17, 393 (1992).
    [CrossRef] [PubMed]
  16. S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
    [CrossRef] [PubMed]
  17. L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
    [CrossRef] [PubMed]

2007 (1)

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

2006 (2)

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

A. V. Yulin, D. V. Skryabin, and A. G. Vladimirov, Opt. Express 14, 12347 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (2)

C. Conti, Phys. Rev. E 70, 046613 (2004).
[CrossRef]

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

2003 (3)

C. Conti, Phys. Rev. E 68, 016606 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

2002 (2)

S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2103 (2002).
[CrossRef]

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

2001 (1)

D. E. Pelinovsky, Math. Comput. Simul. 55, 585 (2001).
[CrossRef]

1997 (1)

1992 (2)

N. N. Akhmediev, V. I. Korneev, and R. F. Nabiev, Opt. Lett. 17, 393 (1992).
[CrossRef] [PubMed]

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef] [PubMed]

1974 (1)

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Akhmediev, N. N.

Bergé, L.

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Bragheri, F.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Buryak, A. V.

Conti, C.

C. Conti, Phys. Rev. E 70, 046613 (2004).
[CrossRef]

C. Conti, Phys. Rev. E 68, 016606 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Couairon, A.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

De Rossi, A.

Di Trapani, P.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Dubietis, A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Emplit, Ph.

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

Faccio, D.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Germaschewski, K.

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Gorza, S. P.

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

Grauer, R.

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Greenleaf, J. F.

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef] [PubMed]

Haelterman, M.

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Juul Rasmussen, J.

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Kivshar, Y. S.

Kolesik, M.

Korneev, V. I.

Lu, J. Y.

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef] [PubMed]

Moloney, J. V.

Nabiev, R. F.

Orlov, S.

Parola, A.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, Math. Comput. Simul. 55, 585 (2001).
[CrossRef]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2103 (2002).
[CrossRef]

Porras, M. A.

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Roig, N.

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

Rubenchik, A. M.

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Skryabin, D. V.

Stabinis, A.

Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997).
[CrossRef] [PubMed]

Trull, J.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Vladimirov, A. G.

Wright, E. M.

Yulin, A. V.

Zakharov, V. E.

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef] [PubMed]

Math. Comput. Simul. (1)

D. E. Pelinovsky, Math. Comput. Simul. 55, 585 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, Phys. Rev. A 76, 011803(R) (2007).
[CrossRef]

Phys. Rev. E (2)

C. Conti, Phys. Rev. E 68, 016606 (2003).
[CrossRef]

C. Conti, Phys. Rev. E 70, 046613 (2004).
[CrossRef]

Phys. Rev. Lett. (5)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

S. P. Gorza, N. Roig, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 92, 084101 (2004).
[CrossRef] [PubMed]

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen, Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. M. Rubenchik, Sov. Phys. JETP 38, 494 (1974).

Other (1)

H.E.Hernández-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves (Wiley, 2008).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Profile a ( r ) of the ground state of the NSE with saturable nonlinearity with γ I = 0.1 . (b), (c) Parts ϵ u ( r ) and ϵ v ( r ) of the unstable perturbation δ A at modulation frequency ( k 0 k NL ) 1 2 Ω s = 0.628 , for which κ = k NL ( 0.402 i 0.109 ) . The seed ϵ is such that the perturbation is small (note the vertical scale changes).

Fig. 2
Fig. 2

For the initial condition in (a) and (b), gray scale plots at k NL z = ( c ) , (d) 9 and (e), (f) 18 of A ( r , τ ) 2 (top) and A ( K , Ω ) 2 (bottom). (g) Variation along z of the power in the soliton, perturbation, and newly generated frequencies. (h) Solid curve, expected final X-wave spectrum (4) for the soliton of Fig. 1a ( α = 0.200 k NL ) when modulated at ( k 0 k NL ) 1 2 Ω s = 0.628 . Squares, circles, and triangles place the maxima of the long-distance spectrum for simulations with seeds ϵ ( k NL z = 18 ) , ϵ 2 ( k NL z = 24 ) , and ϵ 4 ( k NL z = 30 ) .

Equations (4)

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z A = i 2 k 0 Δ A i k 0 2 τ 2 A + f ( A 2 ) A ,
δ A = ϵ [ u ( r ) e i Ω s τ e i κ z + v ( r ) e i Ω s τ e i κ z ] e i α z ,
z A = i α A κ I ϵ ϵ A ( κ R Ω s ) τ A ,
K ( Ω ) = { 2 k 0 [ α ( κ R Ω s ) Ω + k 0 Ω 2 2 ] } 1 2 ,

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