Abstract

A new numerical scheme for computing self-localized states—or solitons—of nonlinear waveguides is proposed. The idea behind the method is to transform the underlying equation governing the soliton, such as a nonlinear Schrödinger-type equation, into Fourier space and determine a nonlinear nonlocal integral equation coupled to an algebraic equation. The coupling prevents the numerical scheme from diverging. The nonlinear guided mode is then determined from a convergent fixed point iteration scheme. This spectral renormalization method can find wide applications in nonlinear optics and related fields such as Bose–Einstein condensation and fluid mechanics.

© 2005 Optical Society of America

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  1. G. Stegeman and M. Segev, Science 286, 1518 (1999).
    [CrossRef] [PubMed]
  2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
    [CrossRef]
  7. V. I. Petviashvili, Sov. J. Plasma Phys. 2, 257 (1976).
  8. B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).
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    [CrossRef]
  10. M. J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998)
    [CrossRef]
  11. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
    [CrossRef]
  12. M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
    [CrossRef]
  13. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 67, 025601(R) (2003).
    [CrossRef]
  14. Z. H. Musslimani and J. Yang, J. Opt. Soc. Am. B 21, 973 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Lett. 29, 1530 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
    [CrossRef]
  20. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
    [CrossRef]
  21. Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
    [CrossRef]

2005 (1)

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

2004 (3)

2003 (6)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 67, 025601(R) (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

2001 (1)

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

1999 (1)

G. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

1998 (3)

M. J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998)
[CrossRef]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
[CrossRef]

1997 (1)

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

1991 (1)

1976 (1)

V. I. Petviashvili, Sov. J. Plasma Phys. 2, 257 (1976).

1970 (1)

B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 67, 025601(R) (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

M. J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998)
[CrossRef]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Elsevier, 2001).

Assanto, G.

Biondini, G.

Brzdakiewicz, K. A.

Christodoulides, D. N.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Cohen, O.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

Coskun, T. H.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Crasovan, L.

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Darmanyan, S.

S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

Fleischer, J. W.

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

Fratalocchi, A.

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Kadomtsev, B. B.

B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).

Karpierz, M. A.

Kartashov, Y. V.

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Kobyakov, A.

S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
[CrossRef]

Ladouceur, F.

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
[CrossRef]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

Mihalache, D.

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Min, Y.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Mitchell, D. J.

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani and J. Yang, J. Opt. Soc. Am. B 21, 973 (2004).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 67, 025601(R) (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

Peschel, U.

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

Petviashvili, V. I.

V. I. Petviashvili, Sov. J. Plasma Phys. 2, 257 (1976).

B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).

Polodian, L.

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Schwartz, T.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Segev, M.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

G. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Snyder, A. W.

Sohler, W.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Stegeman, G.

G. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Stegeman, G. I.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Torner, L.

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Xu, Z.

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Yang, J.

J. Opt. Soc. Am. B (1)

Nature (2)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. E (4)

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Rev. E 57, 2344 (1998).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. E 67, 025601(R) (2003).
[CrossRef]

Z. Xu, Y. V. Kartashov, L. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, 016616 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

N. K. Efremidis, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, Phys. Rev. Lett. 79, 4990 (1997).
[CrossRef]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

Physica D (1)

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

Science (1)

G. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Sov. J. Plasma Phys. (1)

V. I. Petviashvili, Sov. J. Plasma Phys. 2, 257 (1976).

Sov. Phys. Dokl. (1)

B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970).

Other (2)

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Elsevier, 2001).

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Figures (3)

Fig. 1
Fig. 1

Fundamental lattice soliton obtained by iterating Eq. (10) for I 0 = 1 and μ = 0.8 .

Fig. 2
Fig. 2

Fundamental harmonic A for W 0 = 2 and ν = 2 .

Fig. 3
Fig. 3

Second harmonic B for W 0 = 2 and ν = 2 .

Equations (16)

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

i U z + 2 U V ( x ) U + N ( U 2 ) U = 0 ,
μ u + 2 u V ( x ) u + N ( u 2 ) u = 0 .
u ̂ ( k ) = F [ u ( x ) ] = + + u ( x ) exp [ i ( k x x + k y y ) ] d x ,
u ̂ ( k ) = F [ N ( u 2 ) u ] μ + k 2 .
w ̂ ( k ) = F [ N ( λ 2 w 2 ) w ] μ + k 2 Q λ [ w ̂ ( k ) ] .
G ( λ ) + w ̂ ( k ) 2 d k + w ̂ * ( k ) Q λ [ w ̂ ( k ) ] d k = 0 ,
w ̂ m + 1 ( k ) = F [ N ( λ m 2 w m 2 ) w m ] μ + k 2
u ̂ ( k ) = ( r + μ ) u ̂ r + k 2 F [ V u ] F [ N ( u 2 ) u ] r + k 2 R [ u ̂ ( k ) ] .
+ w ̂ m ( k ) 2 d k + w ̂ m * ( k ) R [ λ m w ̂ m ( k ) ] d k = 0 .
w ̂ m + 1 ( k ) = ( r + μ ) r + k 2 w ̂ m F [ V w m ] r + k 2 + 1 r + k 2 F [ w m 1 + λ m 2 w m 2 ] ,
ν A + W ( x ) A + 2 A + A B = 0 ,
4 ν B + 4 W ( x ) B + 2 B + A 2 2 = 0 ,
ψ ̂ m + 1 = F [ W ψ m ] + λ 2 m F [ ψ m ϕ m ] ν + k 2 ,
ϕ ̂ m + 1 = 4 F [ W ϕ m ] + ( λ 1 m 2 λ 2 m ) F [ ψ m 2 2 ] 4 ν + k 2 .
λ 2 m = + d k [ ( ν + k 2 ) ψ ̂ m 2 ψ ̂ m * F [ W ψ m ] ] + d k ψ ̂ m * F [ ψ m ϕ m ] ,
λ 1 m 2 = λ 2 m + d k [ ( 4 ν + k 2 ) ϕ ̂ m 2 4 ϕ ̂ m * F [ W ϕ m ] ] + d k ϕ ̂ m * F [ ψ m 2 2 ] .

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