Abstract

We present an approach for instabilities of solitons that is based on the spectrum of a fourth-order linearized operator. Unlike the standard approach which is based on the slope (Vakhitov–Kolokolov) condition, this approach provides the quantitative value of the instability rate and the qualitative nature of the instability dynamics.

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  1. M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
    [CrossRef]
  2. Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
    [CrossRef]
  3. Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
    [CrossRef]
  4. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
    [CrossRef]
  5. Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
    [CrossRef]
  6. M. I. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
    [CrossRef]
  7. M. I. Weinstein, Commun. Pure Appl. Math. 39, 51 (1986).
    [CrossRef]
  8. M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
    [CrossRef] [PubMed]
  9. J. Yang, J. Comput. Phys. 228, 7007 (2009).
    [CrossRef]
  10. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
    [CrossRef] [PubMed]
  11. J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
    [CrossRef]
  12. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 31, 2595 (2006).
    [CrossRef] [PubMed]

2009

J. Yang, J. Comput. Phys. 228, 7007 (2009).
[CrossRef]

2008

Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
[CrossRef]

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

2006

2005

M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

1999

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

1986

M. I. Weinstein, Commun. Pure Appl. Math. 39, 51 (1986).
[CrossRef]

1985

M. I. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

1973

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Ablowitz, M. J.

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Barad, S.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

Chen, Z.

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Crasovan, L. -C.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Efremidis, N. K.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Fibich, G.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
[CrossRef]

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

Ilan, B.

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 31, 2595 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Musslimani, Z. H.

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Sivan, Y.

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
[CrossRef]

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

Torner, L.

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 31, 2595 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Vysloukh, V. A.

Weinstein, M. I.

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

M. I. Weinstein, Commun. Pure Appl. Math. 39, 51 (1986).
[CrossRef]

M. I. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

Yang, J.

J. Yang, J. Comput. Phys. 228, 7007 (2009).
[CrossRef]

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Commun. Pure Appl. Math.

M. I. Weinstein, Commun. Pure Appl. Math. 39, 51 (1986).
[CrossRef]

J. Comput. Phys.

J. Yang, J. Comput. Phys. 228, 7007 (2009).
[CrossRef]

Nonlinearity

Y. Sivan, G. Fibich, N. K. Efremidis, and S. Barad, Nonlinearity 21, 509 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. E

Y. Sivan, G. Fibich, and B. Ilan, Phys. Rev. E 77, 045601(R) (2008).
[CrossRef]

Y. Sivan, B. Ilan, G. Fibich, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Phys. Rev. Lett.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 2726 (1999).
[CrossRef]

Radiophys. Quantum Electron.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

SIAM J. Math. Anal.

M. I. Weinstein, SIAM J. Math. Anal. 16, 472 (1985).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

On-axis amplitude of the perturbed unstable ground state of the supercritical NLS (blue, solid line) agrees with the prediction (12) with Ω max 2.9 (red, dashed line) both for (a) a focusing perturbation ε = 10 2 and (b) a defocusing perturbation ε = 10 2 .

Fig. 2
Fig. 2

(a) Frequency-power curve for solitons of (14) centered at a lattice maximum, (b) the negative eigenvalues of L L + , and (C) the instability rates computed from (b) using (10) [same line types as in (b)]. The shapes in (c) are delineated for Fig. 3. Dotted curve is 100 | P ( μ ) | .

Fig. 3
Fig. 3

Eigenmodes for the shapes in Fig. 2c. (a) Asymmetric drift eigenmode corresponding to μ = 2 ( ) , (b) asymmetric drift eigenmode (dashed curve) and symmetric amplitude eigenmode (solid curve) corresponding to μ = 5 .

Equations (18)

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i ψ t ( x , t ) + Δ ψ + F ( | ψ | 2 ) ψ = 0 , Δ k = 1 d x k 2 ,
μ R + Δ R + F ( R 2 ) R = 0 .
h t = i { [ Δ + μ F ( R 2 ) ] h R 2 F ( R 2 ) h * } ,
h ( x , t ) = [ u ( x ) + i v ( x ) ] e Ω t ,
L u = Ω v , L Δ + μ F ( R 2 ) ,
L + v = Ω u , L + L 2 R 2 F ( R 2 ) .
L L + v = λ v , λ Ω 2 .
L + v = λ L 1 v , v R .
λ min inf v R , v 2 = 1 ( v , L + v ) ( v , L 1 v ) = ( v min , L + v min ) ( v min , L 1 v min ) .
sgn λ min = sgn α L + , α L + inf v R , v 2 = 1 ( v , L + v ) .
Ω max λ min .
P ( μ ) = μ ( c σ , d 1 ) c σ , d R ( · , μ = 1 ) 2 2 , c σ , d 2 σ d 2 σ .
ψ 0 ( x ) = R ( x ; μ ) + ε u ( x ) , | ε | 1 ,
ψ ( x , t ) [ R + ε 2 ( u + i v ) e Ω max t + ε 2 ( u i v ) e Ω max t ] e i μ t .
| ψ ( 0 , t ) | | R ( 0 ) + ε [ u ( 0 ) cosh ( Ω max t ) + i v ( 0 ) sinh ( Ω max t ) ] | .
i ψ t ( x , t ) + Δ ψ + F ( | ψ | 2 ) ψ V ( x ) ψ = 0 , x R d .
L Δ + μ + V F ( R V 2 ) , L + L 2 R V 2 F ( R V 2 ) ,
i ψ t ( t , x ) + ψ x x 2 cos ( 2 π x ) ψ + | ψ | 4 ψ = 0 ,

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