Abstract

We study the existence, formation, and stability of quasi-one-dimensional (stripes) and two-dimensional (bullets) spatio-temporal soliton solutions for a (2+1)-dimensional modified nonlinear Schrödinger equation, in the presence of the self-steepening effect. These solutions, which are on top of a continuous-wave background, are either dark or antidark, the latter being supported by the self-steepening effect. We show that there exist stable small-amplitude lump solitons, which, in the context of nonlinear optics, constitute novel light bullets.

© 2001 Optical Society of America

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References

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  1. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  2. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).
  3. J. R. de Oliveira and M. A. de Moura, “Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation,” Phys. Rev. E 57, 4751–4756 (1998).
    [CrossRef]
  4. V. E. Vekslerchik, “Dark soliton of the generalized nonlinear Schrödinger equation,” Phys. Lett. A 153, 195–198 (1991).
    [CrossRef]
  5. D. J. Frantzeskakis, “Small-amplitude solitary structuresfor an extended nonlinear Schrödinger equation,” J. Phys. A Math. Nucl. Gen. 29, 3631–3639 (1996).
    [CrossRef]
  6. Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 1181–1197 (1998).
    [CrossRef]
  7. E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Zh. Eksp. Teor. Fiz. 94, 119–129 (1988) [Sov. Phys. JETP 67, 1583–1588 (1988)].
  8. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University, London, 1990).
  9. D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
    [CrossRef]
  10. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1284 (1990).
    [CrossRef] [PubMed]
  11. Y. Chen and J. Atai, “Dark optical bullets in light self-trapping,” Opt. Lett. 20, 133–135 (1995).
    [CrossRef] [PubMed]
  12. B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
    [CrossRef]
  13. G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
    [CrossRef]
  14. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
    [CrossRef]
  15. M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
    [CrossRef] [PubMed]
  16. T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl. 55, 1–35 (1974).
    [CrossRef]
  17. Yu. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
    [CrossRef] [PubMed]
  18. G. Huang and M. G. Velarde, “Head-on collisions of dark solitons near the zero-dispersion point in optical fibers,” Phys. Rev. A 54, 3048–3051 (1996).
  19. H. E. Nistazakis, D. J. Frantzeskakis, P. S. Balourdos, A. Tsigopoulos, and B. A. Malomed, “Dynamics of anti-dark and dark solitons in (2+1)-dimensional generalized nonlinear Schrödinger equation,” Phys. Lett. A (to be published).
  20. V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in the Atmosphere (Gordon and Breach, Philadelphia, 1992).
  21. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

2000

M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
[CrossRef] [PubMed]

1999

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
[CrossRef]

1998

G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
[CrossRef]

J. R. de Oliveira and M. A. de Moura, “Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation,” Phys. Rev. E 57, 4751–4756 (1998).
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 1181–1197 (1998).
[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

1997

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

1996

G. Huang and M. G. Velarde, “Head-on collisions of dark solitons near the zero-dispersion point in optical fibers,” Phys. Rev. A 54, 3048–3051 (1996).

D. J. Frantzeskakis, “Small-amplitude solitary structuresfor an extended nonlinear Schrödinger equation,” J. Phys. A Math. Nucl. Gen. 29, 3631–3639 (1996).
[CrossRef]

1995

1991

V. E. Vekslerchik, “Dark soliton of the generalized nonlinear Schrödinger equation,” Phys. Lett. A 153, 195–198 (1991).
[CrossRef]

Yu. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

1990

1988

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Zh. Eksp. Teor. Fiz. 94, 119–129 (1988) [Sov. Phys. JETP 67, 1583–1588 (1988)].

1974

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl. 55, 1–35 (1974).
[CrossRef]

Afanasjev, V. V.

Yu. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

Anderson, D.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Atai, J.

Berntson, A.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Blaauboer, M.

M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
[CrossRef] [PubMed]

Chen, Y.

de Moura, M. A.

J. R. de Oliveira and M. A. de Moura, “Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation,” Phys. Rev. E 57, 4751–4756 (1998).
[CrossRef]

de Oliveira, J. R.

J. R. de Oliveira and M. A. de Moura, “Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation,” Phys. Rev. E 57, 4751–4756 (1998).
[CrossRef]

Drummond, P.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Frantzeskakis, D. J.

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

D. J. Frantzeskakis, “Small-amplitude solitary structuresfor an extended nonlinear Schrödinger equation,” J. Phys. A Math. Nucl. Gen. 29, 3631–3639 (1996).
[CrossRef]

Gottwald, G.

G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
[CrossRef]

Grimshaw, R.

G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
[CrossRef]

He, H.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Hizanidis, K.

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

Huang, G.

G. Huang and M. G. Velarde, “Head-on collisions of dark solitons near the zero-dispersion point in optical fibers,” Phys. Rev. A 54, 3048–3051 (1996).

Kivshar, Yu. S.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 1181–1197 (1998).
[CrossRef]

Yu. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

Kurizki, G.

M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
[CrossRef] [PubMed]

Kuznetsov, E. A.

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Zh. Eksp. Teor. Fiz. 94, 119–129 (1988) [Sov. Phys. JETP 67, 1583–1588 (1988)].

Lisak, M.

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Liu, X.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 1181–1197 (1998).
[CrossRef]

Malomed, B. A.

M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
[CrossRef] [PubMed]

G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
[CrossRef]

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Polymilis, C.

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
[CrossRef]

Silberberg, Y.

Taniuti, T.

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl. 55, 1–35 (1974).
[CrossRef]

Turitsyn, S. K.

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Zh. Eksp. Teor. Fiz. 94, 119–129 (1988) [Sov. Phys. JETP 67, 1583–1588 (1988)].

Vekslerchik, V. E.

V. E. Vekslerchik, “Dark soliton of the generalized nonlinear Schrödinger equation,” Phys. Lett. A 153, 195–198 (1991).
[CrossRef]

Velarde, M. G.

G. Huang and M. G. Velarde, “Head-on collisions of dark solitons near the zero-dispersion point in optical fibers,” Phys. Rev. A 54, 3048–3051 (1996).

Wise, F. W.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
[CrossRef]

J. Phys. A Math. Nucl. Gen.

D. J. Frantzeskakis, “Small-amplitude solitary structuresfor an extended nonlinear Schrödinger equation,” J. Phys. A Math. Nucl. Gen. 29, 3631–3639 (1996).
[CrossRef]

Opt. Lett.

Phys. Lett. A

D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, and C. Polymilis, “Stable antidark light bullets supported by the third-order dispersion,” Phys. Lett. A 248, 203–207 (1998).
[CrossRef]

V. E. Vekslerchik, “Dark soliton of the generalized nonlinear Schrödinger equation,” Phys. Lett. A 153, 195–198 (1991).
[CrossRef]

G. Gottwald, R. Grimshaw, and B. A. Malomed, “Stable two-dimensional parametric solitons in fluid systems,” Phys. Lett. A 248, 208–218 (1998).
[CrossRef]

Phys. Rep.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 1181–1197 (1998).
[CrossRef]

Phys. Rev. A

Yu. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

G. Huang and M. G. Velarde, “Head-on collisions of dark solitons near the zero-dispersion point in optical fibers,” Phys. Rev. A 54, 3048–3051 (1996).

Phys. Rev. E

J. R. de Oliveira and M. A. de Moura, “Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation,” Phys. Rev. E 57, 4751–4756 (1998).
[CrossRef]

B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Phys. Rev. Lett.

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82, 4631–4633 (1999).
[CrossRef]

M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-induced transparency media,” Phys. Rev. Lett. 84, 1906–1909 (2000).
[CrossRef] [PubMed]

Prog. Theor. Phys. Suppl.

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Prog. Theor. Phys. Suppl. 55, 1–35 (1974).
[CrossRef]

Sov. Phys. JETP

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Zh. Eksp. Teor. Fiz. 94, 119–129 (1988) [Sov. Phys. JETP 67, 1583–1588 (1988)].

Other

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge University, London, 1990).

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

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).

H. E. Nistazakis, D. J. Frantzeskakis, P. S. Balourdos, A. Tsigopoulos, and B. A. Malomed, “Dynamics of anti-dark and dark solitons in (2+1)-dimensional generalized nonlinear Schrödinger equation,” Phys. Lett. A (to be published).

V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in the Atmosphere (Gordon and Breach, Philadelphia, 1992).

I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

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Equations (34)

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i uz+12 2ux2-α 2ut2+σ|u|2u+iσβ t(|u|2u)=0.
φz-σ|u0|2ρ+12 φx2-αφt2
-12ρ-1/22x2(ρ1/2)-α 2t2(ρ1/2)+σβ|u0|2ρ φt=0,
ρz+x ρ φx-α t ρ φt+3σβ|u0|2ρ ρt=0.
X=x,Z=3/2z,T=1/2(t-Cz).
ρ=1+ρ1+2ρ2+,
φ=σ|u0|2z+1/2φ1+3/2φ2+,
(βσ|u0|2-C) φ1T-σ|u0|2ρ1=0,
(3βσ|u0|2-C) ρ1T-α 2φ1T2=0.
(βσ|u0|2-C)(3βσ|u0|2-C)-ασ|u0|2=0,
CC±=2βσ|u0|2±|u0|(β2|u0|2+ασ)1/2.
φ1Z+α4 2ρ1T2-α2 φ1T2+βσ|u0|2ρ1 φ1T
+(βσ|u0|2-C) φ2T-σ|u0|2ρ2=0,
ρ1Z+2φ1X2-α T ρ1 φ1T+3βσ|u0|2ρ1 ρ1T
+(3βσ|u0|2-C) ρ2T-α 2φ2T2=0.
T ρ1Z+Aρ1 ρ1T+B 3ρ1T3+Γ 2ρ1X2=0,
A=32|u0|2[β2|u0|2-σ(α+βC)](2βσ|u0|2-C)-1,
B=18α2(2βσ|u0|2-C)-1,
Γ=12σ|u0|2(2βσ|u0|2-C)-1.
ρ1(X, Z, T)=12Bκ12A sech2κ1σT+κ23BΓ1/2X-B(4κ12+3σκ22)Z+δ,
ρ1(X, Z, T)=24BA (-ξ2+μ22ζ2+μ2-2)(ξ2+μ22ζ2+μ2-2)2,
ξ=-T+μ13BΓ1/2X+3B(μ12-μ22)Z,
ζ=3BΓ1/2X+6μ1BZ,
-ασβ2<|u0|2<-95 ασβ2,
|u0|2>-95 ασβ2,
uu01+12ρ1exp[i(σ|u0|2z+1/2φ1)],
kz=2ωσβ|u0|2±(ωσβ|u0|2)2+(kx2-αω2)-σ|u0|2+14(kx2-αω2)1/2.
dφds=1K K1ρ-μ+32σβν|u0|2ρ,
dρds2=-βν|u0|2K2(ρ4+C3ρ3+C2ρ2+C1ρ+C0),
C3=4σβ2ν2|u0|2(K-βμν),
C2=4β2ν2|u0|4(μ2-σβνK1|u0|2),
C1=8K2K2β2ν2|u0|4,C0=4K12β2ν2|u0|4,
ρd(s)=d+aκ tanh2 s1+κ tanh2 s,
ρad(s)=d+aη tanh2 s1+η tanh2 s,

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