Abstract

We extend the analytical theory explaining the trapping of normally dispersive waves by a Raman soliton in an axially uniform optical fiber to include axially nonuniform fibers. It is shown how a changing group velocity in such a fiber leads to the same trapping mechanism as for a decelerating Raman soliton in a uniform fiber. In contrast to this latter case, where the trapping always leads to a blueshift of the confined radiation, the additional design flexibility inherent in the nonuniform geometry permits the redshift of dispersive waves trapped by an accelerating soliton, which itself may blueshift as a result of the associated spectral recoil.

© 2010 Optical Society of America

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References

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  1. D. Tran, G. Sigel, and B. Bendow, “Heavy metal fluoride glasses and fibers: A review,” J. Lightwave Technol. 2, 566–586 (1984).
    [CrossRef]
  2. J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Cryst. Solids 256–257, 6–16 (1999).
    [CrossRef]
  3. B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.
  4. Z. Chen, A. J. Taylor, and A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009).
    [CrossRef] [PubMed]
  5. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. M. de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009).
    [CrossRef]
  6. J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers,” Opt. Express 18, 6722–6739 (2010).
    [CrossRef] [PubMed]
  7. P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
    [CrossRef]
  8. N. Nishizawa and T. Goto, “Pulse trapping by ultrashort soliton pulses in optical fibers across zero-dispersion wavelength,” Opt. Lett. 27, 152–154 (2002).
    [CrossRef]
  9. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibres,” Nature Photon. 1, 653–657 (2007).
    [CrossRef]
  10. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
    [CrossRef]
  11. D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
    [CrossRef] [PubMed]
  12. T. Schreiber, T. V. Andersen, D. Schimpf, J. Limpert, and A. Tünnermann, “Supercontinuum generation by femtosecond single and dual wavelength pumping in photonic crystal fibers with two zero dispersion wavelengths,” Opt. Express 13, 9556–9569 (2005).
    [CrossRef] [PubMed]
  13. T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
    [CrossRef]
  14. X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air–silica microstructure fiber,” Opt. Lett. 26, 358–360 (2001).
    [CrossRef]
  15. J. C. Travers and J. R. Taylor, “Soliton trapping of dispersive waves in tapered optical fibers,” Opt. Lett. 34, 115–117 (2009).
    [CrossRef] [PubMed]
  16. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006).
    [CrossRef] [PubMed]
  17. Z. Chen, A. J. Taylor, and A. Efimov, “Soliton dynamics in non-uniform fiber tapers: analytical description through an improved moment method,” J. Opt. Soc. Am. B 27, 1022–1030 (2010).
    [CrossRef]
  18. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  19. J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  22. R. J. Kobliska and S. A. Solin, “Temperature dependence of the Raman spectrum and the depolarization spectrum of amorphous As2S3,” Phys. Rev. B 8, 756–768 (1973).
    [CrossRef]
  23. O. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003).
    [CrossRef]
  24. A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006).
    [CrossRef] [PubMed]
  25. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
    [CrossRef] [PubMed]
  26. S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express 17, 13588–13600 (2009).
    [CrossRef] [PubMed]
  27. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  28. A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008).
    [CrossRef] [PubMed]

2010 (2)

2009 (4)

2008 (1)

2007 (3)

J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
[CrossRef] [PubMed]

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibres,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

2006 (3)

2005 (1)

2003 (2)

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[CrossRef] [PubMed]

O. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61–68 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (2)

1999 (1)

J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Cryst. Solids 256–257, 6–16 (1999).
[CrossRef]

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

1987 (1)

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

1986 (1)

1984 (1)

D. Tran, G. Sigel, and B. Bendow, “Heavy metal fluoride glasses and fibers: A review,” J. Lightwave Technol. 2, 566–586 (1984).
[CrossRef]

1973 (1)

R. J. Kobliska and S. A. Solin, “Temperature dependence of the Raman spectrum and the depolarization spectrum of amorphous As2S3,” Phys. Rev. B 8, 756–768 (1973).
[CrossRef]

Aggarwal, I.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Aggarwal, I. D.

J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers,” Opt. Express 18, 6722–6739 (2010).
[CrossRef] [PubMed]

J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Cryst. Solids 256–257, 6–16 (1999).
[CrossRef]

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

Andersen, T. V.

Andrés, P.

Bang, O.

Beaud, P.

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

Bendow, B.

D. Tran, G. Sigel, and B. Bendow, “Heavy metal fluoride glasses and fibers: A review,” J. Lightwave Technol. 2, 566–586 (1984).
[CrossRef]

Birks, T. A.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Chandalia, J. K.

Chen, Z.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

de Sterke, C. M.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Efimov, A.

Eggleton, B. J.

Ferrando, A.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

George, A. K.

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibres,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006).
[CrossRef] [PubMed]

Gordon, J. P.

Goto, T.

Hill, S.

Hodel, W.

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

Holzlohner, R.

Hu, J.

Judge, A. C.

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

Knight, J. C.

Knox, W. H.

Kobliska, R. J.

R. J. Kobliska and S. A. Solin, “Temperature dependence of the Raman spectrum and the depolarization spectrum of amorphous As2S3,” Phys. Rev. B 8, 756–768 (1973).
[CrossRef]

König, F.

Kosinski, S. G.

Kudlinski, A.

Kuhlmey, B. T.

Kuklewicz, C. E.

Kung, F.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Laegsgaard, J.

Leonhardt, U.

Limpert, J.

Liu, X.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[CrossRef] [PubMed]

Mägi, E. C.

Menyuk, C. R.

Miret, J. J.

Nguyen, V.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Nishizawa, N.

Pant, R.

Popov, S. V.

Rulkov, A. B.

Russel, P. S. J.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[CrossRef] [PubMed]

Russell, P. S. J.

Sanghera, J.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Sanghera, J. S.

J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers,” Opt. Express 18, 6722–6739 (2010).
[CrossRef] [PubMed]

J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Cryst. Solids 256–257, 6–16 (1999).
[CrossRef]

Schimpf, D.

Schreiber, T.

Shaw, B.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Shaw, L. B.

Sigel, G.

D. Tran, G. Sigel, and B. Bendow, “Heavy metal fluoride glasses and fibers: A review,” J. Lightwave Technol. 2, 566–586 (1984).
[CrossRef]

Silvestre, E.

Sinkin, O.

Skryabin, D. V.

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibres,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006).
[CrossRef] [PubMed]

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[CrossRef] [PubMed]

Solin, S. A.

R. J. Kobliska and S. A. Solin, “Temperature dependence of the Raman spectrum and the depolarization spectrum of amorphous As2S3,” Phys. Rev. B 8, 756–768 (1973).
[CrossRef]

Stone, J. M.

Taylor, A. J.

Taylor, J. R.

Thielen, P.

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

Tran, D.

D. Tran, G. Sigel, and B. Bendow, “Heavy metal fluoride glasses and fibers: A review,” J. Lightwave Technol. 2, 566–586 (1984).
[CrossRef]

Travers, J. C.

Tünnermann, A.

Voronin, A. A.

Wadsworth, W. J.

Weber, H.

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

Windeler, R. S.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Xu, C.

Zheltikov, A. M.

Zweck, J.

Zysset, B.

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Beaud, W. Hodel, B. Zysset, and H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[CrossRef]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

J. Lightwave Technol. (2)

J. Non-Cryst. Solids (1)

J. S. Sanghera and I. D. Aggarwal, “Active and passive chalcogenide glass optical fibers for IR applications: a review,” J. Non-Cryst. Solids 256–257, 6–16 (1999).
[CrossRef]

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

Nature Photon. (1)

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic crystal fibres,” Nature Photon. 1, 653–657 (2007).
[CrossRef]

Opt. Express (7)

J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers,” Opt. Express 18, 6722–6739 (2010).
[CrossRef] [PubMed]

J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
[CrossRef] [PubMed]

T. Schreiber, T. V. Andersen, D. Schimpf, J. Limpert, and A. Tünnermann, “Supercontinuum generation by femtosecond single and dual wavelength pumping in photonic crystal fibers with two zero dispersion wavelengths,” Opt. Express 13, 9556–9569 (2005).
[CrossRef] [PubMed]

S. Hill, C. E. Kuklewicz, U. Leonhardt, and F. König, “Evolution of light trapped by a soliton in a microstructured fiber,” Opt. Express 17, 13588–13600 (2009).
[CrossRef] [PubMed]

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006).
[CrossRef] [PubMed]

Z. Chen, A. J. Taylor, and A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009).
[CrossRef] [PubMed]

A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006).
[CrossRef] [PubMed]

Opt. Lett. (7)

Phys. Rev. A (2)

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. J. Kobliska and S. A. Solin, “Temperature dependence of the Raman spectrum and the depolarization spectrum of amorphous As2S3,” Phys. Rev. B 8, 756–768 (1973).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Science (1)

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[CrossRef] [PubMed]

Other (1)

B. Shaw, P. Thielen, F. Kung, V. Nguyen, J. Sanghera, and I. Aggarwal, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Advanced Solid-State Photonics (Optical Society of America, 2005), paper TuC5.

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Figures (4)

Fig. 1
Fig. 1

Variation of the dispersion and nonlinearity with scaling M of the fiber diameter according to the simple model described in Subsection 3A for the parameter values λ 0 = 2.5 μ m , t 0 = 80   fs , and δ = 0.1 . (a) Δ k , (b) k 1 (solid line) and k 2 (dashed line), and (c) σ 1 (solid line) and σ 2 (dashed line). (d) Variation of the DW frequency Ω d with M according to Eq. (20) with Ω s = 0 . Dotted lines indicate a reference value of zero.

Fig. 2
Fig. 2

Solutions (blue solid lines) of Eq. (31) using the asymptotic form in Eq. (33), with the full potential (red dotted lines) from Eq. (32) overlaid. Parameters are as in Fig. 1 with Ω s = 0 , ζ = 0 , and (a) Δ M = 0 and (b) Δ M = 0.05 , Δ ζ = 200 .

Fig. 3
Fig. 3

Spectral evolution of a soliton and trapped DW according to Eqs. (12, 13) (left column) and the temporal domain representation (right column) of the final forms of | u 1 | 2 (red dotted lines) and | u 2 | 2 (blue solid lines) at ζ = 200 . (a) and (b) Initial pulse and parameters as in Fig. 2a with E 2 = 0.1 . (c) and (d) Initial pulse and parameters as in Fig. 2b with E 2 = 0.1 . (e) and (f) Initial pulse and parameters as in Fig. 2b with E 2 = 0.35 . The predictions of Eqs. (35, 37) are overlaid on the evolving spectra as white dashed lines.

Fig. 4
Fig. 4

Comparison of the evolution of Ω s and Ω d (black solid lines) for the case shown in Fig. 3e according to the numerical solution to the coupled equations (12, 13) with the linear approximations Ω s ( l ) and Ω d ( l ) (red dotted lines), and the nonlinear approximations Ω s ( nl ) and Ω d ( nl ) (blue dashed-dotted lines). The subscripts s and d refer to the soliton and DW, respectively.

Equations (52)

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

τ = t t 0 ,
Ω = ω t 0 ,
ζ = z L D ,
k j ( ζ ) = β 1 ( z , ω j ) t 0 | β 2 ( 0 , ω 1 ) | ,
k j ( ζ ) = β 2 ( z , ω j ) | β 2 ( 0 , ω 1 ) | ,
σ j ( ζ ) = γ ( z , ω j ) γ ( 0 , ω 1 ) ,
u j ( ζ , τ ) = L D γ ( 0 , ω 1 ) U j ( z , t ) ,
χ R ( τ ) = t 0 h R ( t ) ,
ζ u 1 = k 1 τ u 1 i k 1 2 τ 2 u 1 + i ( 1 f R ) σ 1 u 1 [ | u 1 | 2 + 2 | u 2 | 2 ] + i 2 3 f R σ 1 u 1 d τ χ R ( τ τ ) [ | u 1 ( τ ) | 2 + | u 2 ( τ ) | 2 ] ,
ζ u 2 = k 2 τ u 2 i k 2 2 τ 2 u 2 + i ( 1 f R ) σ 2 u 2 [ | u 2 | 2 + 2 | u 1 | 2 ] + i 2 3 f R σ 2 u 2 d τ χ R ( τ τ ) [ | u 2 ( τ ) | 2 + | u 1 ( τ ) | 2 ] .
θ = τ ζ d x [ k 1 ( x ) | k 1 ( x ) | Ω s ( x ) ] ,
ζ u 1 = | k 1 | Ω s θ u 1 + i | k 1 | 2 θ 2 u 1 + i ( 1 f R ) σ 1 u 1 [ | u 1 | 2 + 2 | u 2 | 2 ] + i 2 3 f R σ 1 u 1 d θ χ R ( θ θ ) [ | u 1 ( θ ) | 2 + | u 2 ( θ ) | 2 ] ,
ζ u 2 = [ | k 1 | Ω s + Δ k ] θ u 2 i | k 2 | 2 θ 2 u 2 + i ( 1 f R ) σ 2 u 2 [ | u 2 | 2 + 2 | u 1 | 2 ] + i 2 3 f R σ 2 u 2 d θ χ R ( θ θ ) [ | u 2 ( θ ) | 2 + | u 1 ( θ ) | 2 ] ,
u 1 = ψ ( ζ , θ ) exp ( i Ω s ( ζ ) θ + i ζ d x [ | k 1 ( x ) | Ω s 2 ( x ) 2 + κ 1 ] ) ,
u 2 = ϕ ( ζ , θ ) exp ( i Ω d ( ζ ) θ i ζ d x [ | k 2 ( x ) | Ω d 2 ( x ) 2 κ 2 ] ) .
ζ θ d = | k 2 | Ω d | k 1 | Ω s Δ k ,
θ d = 1 E 2 d θ θ | u 2 | 2 ,
Ω d = 1 E 2 d Ω 2 π Ω | u ̃ 2 | 2 ,
E j = d Ω 2 π | u ̃ j ( ζ , Ω ) | 2 ,     j = 1 , 2.
Ω d = | k 1 | Ω s + Δ k | k 2 | .
ζ ψ = i κ 1 ψ + i | k 1 | 2 θ 2 ψ + i [ ζ Ω s ] θ ψ + i ( 1 f R ) σ 1 ψ [ | ψ | 2 + 2 | ϕ | 2 ] + i 2 3 f R σ 1 ψ d θ χ R ( θ θ ) [ | ψ ( θ ) | 2 + | ϕ ( θ ) | 2 ] ,
ζ ϕ = i κ 2 ϕ i | k 2 | 2 θ 2 ϕ + i [ ζ Ω d ] θ ϕ + i ( 1 f R ) σ 2 ϕ [ | ϕ | 2 + 2 | ψ | 2 ] + i 2 3 f R σ 2 ϕ d θ χ R ( θ θ ) [ | ϕ ( θ ) | 2 + | ψ ( θ ) | 2 ] .
ψ = ψ ( 0 ) + ϵ ψ ( 1 ) + ϵ 2 ψ ( 2 ) + ,
ϕ = ϵ ϕ ( 1 ) + ϵ 2 ϕ ( 2 ) + ,
| k 1 | 2 θ 2 ψ ( 0 ) + ( 1 η f R ) σ 1 | ψ ( 0 ) | 2 ψ ( 0 ) κ 1 ψ ( 0 ) = 0 ,
ψ ( 0 ) = [ ( 1 η f R ) σ 1 | k 1 | ] 1 / 2 sech ( ( 1 η f R ) σ 1 | k 1 | θ ) ,
κ 1 = ( 1 η f R ) 2 σ 1 2 2 | k 1 | .
[ | k 1 | 2 θ 2 + 3 ( 1 η f R ) σ 1 ψ ( 0 ) 2 κ 1 ] ϵ ψ ( 1 ) = [ ζ Ω s ] θ ψ ( 0 ) 2 3 f R σ 1 ψ ( 0 ) d θ χ R ( θ θ ) ψ ( 0 ) 2 ( θ ) .
ζ Ω s ( l ) = π 2 6 f R ( 1 η f R ) 2 | k 1 | σ 1 d Ω 2 π 2   Im [ χ ̃ R ( Ω ) ] Ω 3 sinh 2 ( π | k 1 | Ω 2 ( 1 η f R ) σ 1 ) .
Ω s = 1 E 1 d Ω 2 π Ω | u ̃ 1 | 2 .
[ | k 2 | 2 θ 2 + [ κ 2 V ] ] ϕ ( 1 ) = 0 ,
V = 2 ( 1 η f R ) σ 2 ψ ( 0 ) 2 + [ ζ Ω d ] θ .
ψ ( 0 ) 2 [ ( 1 η f R ) σ 1 | k 1 | ] 1 / 2 exp ( μ ( 1 η f R ) σ 1 | k 1 | θ )
ζ Ω d ( nl ) = E 1 σ 2 E 2 σ 1 [ ζ Ω s ( nl ) ζ Ω s ( l ) ] .
Ω d ( nl ) = E 1 σ 2 E 2 σ 1 [ Ω s ( nl ) Ω s ( l ) ] .
Ω d ( nl ) = | k 1 | Ω s ( nl ) + Δ k | k 2 | ,
Ω s ( nl ) = Ω s ( l ) + σ 1 E 2 Δ k σ 2 E 1 | k 2 | 1 σ 1 E 2 | k 1 | σ 2 E 1 | k 2 | ,
Ω s ( nl ) = Ω s ( l ) + N 2 Δ k N 1 | k 2 | 1 N 2 | k 1 | N 1 | k 2 | ,
Ω d ( nl ) = | k 1 | Ω s ( l ) + Δ k | k 2 | ( 1 N 2 | k 1 | N 1 | k 2 | ) ,
E ¯ 1 = N 1 ( Ω 1 + Ω s ( nl ) ) ,
E ¯ 2 = N 2 ( Ω 2 + Ω d ( nl ) ) ,
ζ [ E ¯ 1 + E ¯ 2 E R ] = 0 ,
δ = β 3 ( 0 , ω 0 ) β 2 ( 0 , ω 1 ) t 0 .
β 2 ( z , ω ) M β 2 ( 0 , M ω ) ,
k 1 = M ( Ω 0 ( M 1 ) δ M ) ,
k 2 = M ( Ω 0 ( M 1 ) δ + M ) ,
Δ k = 2 M 2 Ω 0 ( M 1 ) .
σ 1 = 1 M 2 ,     σ 2 = Ω 2 Ω 1 σ 1 = Ω 0 δ + 1 M 2 ( Ω 0 δ 1 ) .
M ( ζ ) = 1 + Δ M Δ ζ ζ ,
ζ Ω d = 2 ( 1 f R ) σ 2 E 2 d t | u 2 | 2 θ | u 1 | 2 ,
Ω r = Ω 0 + 2 δ ,
M 1 1 + 1 2 δ Ω 0 ,

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