Abstract

The propagation of solitons in low-birefringent optical fiber is considered when four-wave-mixing leads to energy exchange between polarization components. The stability of stationary solutions is addressed with a linear stability analysis expressed as an eigenvalue equation. It is shown that in the vicinity of the bifurcation point the eigenvalues must be either pure real (stable) or pure imaginary (unstable). The transition between these (zero eigenvalue) lends itself to analytical solutions, in spite of the nonintegrability of the original system of partial differential equations. It is demonstrated that the marginally stable perturbations of phase shift and temporal shift also apply to mixed-mode stationary solutions. The bifurcation point is found exactly along with the corresponding eigenfunction. This analysis provides the elliptically polarized stationary solution just above the bifurcation point. It is also shown that the fast-mode soliton just below the bifurcation point and the elliptically polarized soliton just above the bifurcation point are stable.

© 1999 Optical Society of America

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References

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  1. C. R. Menyuk, IEEE J. Quantum Electron. 25, 2674 (1989).
    [CrossRef]
  2. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
    [CrossRef]
  3. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear pulses and beams (Chapman & Hall, London 1997).
  4. D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, J. Opt. Soc. Am. B 14, 869 (1997).
    [CrossRef]
  5. D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
    [CrossRef]
  6. S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).
  7. G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
    [CrossRef] [PubMed]
  8. D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
    [CrossRef]
  9. K. J. Blow, N. J. Doran, and D. Wood, Opt. Lett. 12, 202 (1987).
    [CrossRef] [PubMed]
  10. N. Akhmediev and J. M. Soto-Crespo, Phys. Rev. E 49, 5742(1994); N. Akhmediev, A. Buryak, and J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994).
    [CrossRef]
  11. Y. Barad and Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997).
    [CrossRef]
  12. N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, J. Opt. Soc. Am. B 12, 434 (1995).
    [CrossRef]
  13. Y. Chen and J. Atai, Phys. Rev. E 52, 3102 (1995).
    [CrossRef]
  14. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988).
    [CrossRef] [PubMed]
  15. Y. Chen, Phys. Rev. E 57, 3542 (1998).
    [CrossRef]
  16. G. Rowlands, J. Inst. Math. Appl. 13, 367 (1974).
    [CrossRef]
  17. T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
    [CrossRef]
  18. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, London 1965).
  19. J. W. Evans, Indiana Univ. Math. J. 21, 877 (1972); 22, 75 (1972); 24, 1169 (1975).
    [CrossRef]
  20. M. J. Ablowitz and B. M. Herbst, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 50, 339 (1990).
    [CrossRef]
  21. B. M. Herbst and M. J. Ablowitz, Phys. Rev. Lett. 62, 2065 (1989).
    [CrossRef] [PubMed]
  22. H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
    [CrossRef]

1998 (4)

D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
[CrossRef]

D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
[CrossRef]

Y. Chen, Phys. Rev. E 57, 3542 (1998).
[CrossRef]

T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
[CrossRef]

1997 (2)

1995 (2)

1992 (2)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

1990 (1)

M. J. Ablowitz and B. M. Herbst, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 50, 339 (1990).
[CrossRef]

1989 (2)

B. M. Herbst and M. J. Ablowitz, Phys. Rev. Lett. 62, 2065 (1989).
[CrossRef] [PubMed]

C. R. Menyuk, IEEE J. Quantum Electron. 25, 2674 (1989).
[CrossRef]

1988 (1)

1987 (1)

1986 (1)

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

1974 (2)

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

G. Rowlands, J. Inst. Math. Appl. 13, 367 (1974).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and B. M. Herbst, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 50, 339 (1990).
[CrossRef]

B. M. Herbst and M. J. Ablowitz, Phys. Rev. Lett. 62, 2065 (1989).
[CrossRef] [PubMed]

Aitchison, J. S.

D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, J. Opt. Soc. Am. B 14, 869 (1997).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, J. Opt. Soc. Am. B 12, 434 (1995).
[CrossRef]

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

Andersen, D. R.

Ankiewicz, A.

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

Arnold, J. M.

D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
[CrossRef]

D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, J. Opt. Soc. Am. B 14, 869 (1997).
[CrossRef]

Atai, J.

Y. Chen and J. Atai, Phys. Rev. E 52, 3102 (1995).
[CrossRef]

Barad, Y.

Y. Barad and Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997).
[CrossRef]

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

Blow, K. J.

Buryak, A. V.

Chen, Y.

Y. Chen, Phys. Rev. E 57, 3542 (1998).
[CrossRef]

Y. Chen and J. Atai, Phys. Rev. E 52, 3102 (1995).
[CrossRef]

Christodoulides, D. N.

Doran, N. J.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

Gregori, G.

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

Herbst, B. M.

M. J. Ablowitz and B. M. Herbst, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 50, 339 (1990).
[CrossRef]

B. M. Herbst and M. J. Ablowitz, Phys. Rev. Lett. 62, 2065 (1989).
[CrossRef] [PubMed]

Hutchings, D. C.

D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
[CrossRef]

D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, J. Opt. Soc. Am. B 14, 869 (1997).
[CrossRef]

Joseph, R. I.

Kapitula, T.

T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
[CrossRef]

Manakov, S. V.

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

Menyuk, C. R.

C. R. Menyuk, IEEE J. Quantum Electron. 25, 2674 (1989).
[CrossRef]

Mitchell, J. D.

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

Parker, D. F.

D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
[CrossRef]

Rowlands, G.

G. Rowlands, J. Inst. Math. Appl. 13, 367 (1974).
[CrossRef]

Sandstede, B.

T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
[CrossRef]

Silberberg, Y.

Y. Barad and Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997).
[CrossRef]

Soto-Crespo, J. M.

Tran, H. T.

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

Wabnitz, S.

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

Wood, D.

IEEE J. Quantum Electron. (1)

C. R. Menyuk, IEEE J. Quantum Electron. 25, 2674 (1989).
[CrossRef]

J. Inst. Math. Appl. (1)

G. Rowlands, J. Inst. Math. Appl. 13, 367 (1974).
[CrossRef]

J. Lightwave Technol. (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, J. Lightwave Technol. 10, 28 (1992).
[CrossRef]

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

Opt. Commun. (1)

H. T. Tran, J. D. Mitchell, N. N. Akhmediev, and A. Ankiewicz, Opt. Commun. 93, 227 (1992).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

D. C. Hutchings, J. M. Arnold, and J. S. Aitchison, Opt. Quantum Electron. 30, 771 (1998).
[CrossRef]

Phys. Rev. E (3)

Y. Chen, Phys. Rev. E 57, 3542 (1998).
[CrossRef]

D. C. Hutchings, J. M. Arnold, and D. F. Parker, Phys. Rev. E 58, 6649 (1998).
[CrossRef]

Y. Chen and J. Atai, Phys. Rev. E 52, 3102 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

B. M. Herbst and M. J. Ablowitz, Phys. Rev. Lett. 62, 2065 (1989).
[CrossRef] [PubMed]

Y. Barad and Y. Silberberg, Phys. Rev. Lett. 78, 3290 (1997).
[CrossRef]

Physica D (1)

T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

M. J. Ablowitz and B. M. Herbst, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 50, 339 (1990).
[CrossRef]

Sov. Phys. JETP (1)

S. V. Manakov, Sov. Phys. JETP 38, 248 (1974).

Other (4)

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear pulses and beams (Chapman & Hall, London 1997).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, London 1965).

J. W. Evans, Indiana Univ. Math. J. 21, 877 (1972); 22, 75 (1972); 24, 1169 (1975).
[CrossRef]

N. Akhmediev and J. M. Soto-Crespo, Phys. Rev. E 49, 5742(1994); N. Akhmediev, A. Buryak, and J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Calculated amplitude profiles for the elliptically polarized soliton stationary solutions taking δ=1/3 and γ=10-4. Solid curve, slow component (u); dashed curve, fast component (v); (a) Ω=1.6×10-4, (b) Ω=2.0×10-4, (c) Ω=4.0×10-4.

Fig. 2
Fig. 2

Eigenfunction for Xu at the bifurcation point (solid curve). Also shown for comparison is the hyperbolic secant envelope of the fast soliton stationary solution (dashed curve).

Equations (35)

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iuz+122ut2+γu+[uu*+(1-δ)vv*]u+δu*v2=0,
ivz+122vt2-γv+[(1-δ)uu*+vv*]v+δu2v*=0.
12d2Udt2-(Ω-γ)U+[U2+(1-2δ)V2]U=0,
12d2Vdt2-(Ω+γ)V+[(1-2δ)U2+V2]V=0.
U(t)Cu exp-[2(Ω-γ)]1/2|t|,
V(t)Cv exp-[2(Ω+γ)]1/2|t|.
j=½ [Xj(x)+Yj(x)]exp(iλz)+½ [Xj(x)-Yj(x)]* exp(-iλ*z),
L1Y=λX,
L2X=λY,
X=XuXv.
L1=12d2dt2-(Ω-γ)+U02+V02-2δU0V0-2δU0V012d2dt2-(Ω+γ)+U02+V02,
L2=12d2dt2-(Ω-γ)+3U02+(1-2δ)V022(1-2δ)U0V02(1-2δ)U0V012d2dt2-(Ω+γ)+(1-2δ)U02+3V02.
YU0V0,
XdU0/dtdV0/dt
Pνμ(x)=1Γ(1-µ)1+x1-xμ/2×F-ν, ν+1; 1-µ;1-x2
12d2dt2-(Ω-γ)+(1-2δ)V02Xu=0.
ν(ν+1)=2(1-2δ),
μ2=1-4a2γ=Ω-γΩ+γ.
F(a, b; c; 1)=Γ(c)Γ(c-a-b)Γ(c-a)Γ(c-b),
F(a, b; c; 0)=1.
Xu(t)=Pν-ν(tanh t/a)=1Γ(ν+1)12sechtaν.
U(t)=Cu sechνta,
V(t)=1asechta,
Y, X=-YXdt.
Y, L1Y=λY, X,
X, L2X=λX, Y.
Y, L1YX, L2X=λ2|Y, X|2.
L1=L1u00L1v,
L2=L2u00L2v.
12d2dt2+12a2ν(ν+1)sech2ta+ζ0,
ζq=ζ0+(ν-q)22a2,
F, L1uF:indefinitesign(onepositiveeigenvalue),
F, L1vF0(onezeroeigenvalueforanyp),
F, L2uF<0,
F, L2vF:indefinitesign(onepositiveeigenvalue).

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