Abstract

The interaction of two colored solitons was analyzed in the framework of a particle-like model, derived from a soliton perturbation theory. From “energy” considerations, a soliton capture threshold and the re-coloring of the escaping solitons were derived. The results were compared to the spectral boundaries of a second order soliton as well as to previous reports. The capture of colored solitons was shown to be impractical without additional means. This particle-like model was further generalized to apply also for non-equal intensity colored solitons. Detailed calculations—beyond the particle-like approximation, exhibited additional mechanisms, namely dissipation and friction-like forces, which served as sources for the relaxation of the solitons oscillations within the captured state, thus enhancing the capture phenomenon.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1999 (3)

C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[CrossRef]

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[CrossRef]

1998 (1)

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[CrossRef]

1997 (1)

1996 (3)

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[CrossRef]

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[CrossRef]

1995 (1)

1994 (1)

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[CrossRef]

1993 (2)

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[CrossRef] [PubMed]

H. Avramopoulos and A. Whitaker, “Addressable fiber-loop memory,” Opt. Lett. 18, 22–24 (1993).
[CrossRef] [PubMed]

1992 (4)

1991 (1)

B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[CrossRef] [PubMed]

1989 (1)

1987 (1)

1983 (1)

Adams, L.E.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Agarwal, A.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Agrawal, G.P.

G.P. Agrawal, Nonlinear Fiber Optics, 2’nd Ed, (Academic Press, NY, 1995), Chap. 2,7.

Anderson, D

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[CrossRef]

Arbel, D.

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[CrossRef]

Avramopoulos, H.

Barry, R. A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Chabat, M.W.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[CrossRef]

Chan, V. W. S.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Chbat, M.W.

Doerr, C.R.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Etrich, C.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

Evangelides, S.G.

Feigenbaum, E.

E. Feigenbaum, “Multi-colored optical storage rings – solitons interaction,” Thesis (EE Department -Technion, Israel, 2003)

E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

Finn, S.G.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Gill, T.L.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[CrossRef]

Glesk, I.

Gordon, J.P.

Hall, K. L.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Haner, M.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Haus, H. A.

Haus, H.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[CrossRef]

A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991).
[CrossRef]

H.A. Haus, “Lecture 11” in Opical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz, ed. (Springer, NY,1999).

Höök, A

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[CrossRef]

Ippen, E.P.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Islam, M.N.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[CrossRef]

M.N. Islam, “Ultrafast all-optical logic gates based on soliton trapping in fibers,” Opt. Lett. 14,1257–1259 (1989)
[CrossRef] [PubMed]

M.N. Islam, Ultrafast Fiber Switching Devices and Systems ,(Cambridge, GB, 1992).

Karlsson, M.

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[CrossRef]

Kaup, D.J.

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[CrossRef] [PubMed]

Khatri, F.I.

Kintzer, E.S.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

kubota, H.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[CrossRef]

Kumar, P.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Lai, Y.

Lederer, F.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

Lisak, M.

M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta 50, 265–270 (1994).
[CrossRef]

Mahmood, M.F.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[CrossRef]

Malomed, B.A.

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[CrossRef]

D.J. Kaup and B.A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993)
[CrossRef] [PubMed]

B.A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[CrossRef] [PubMed]

Manakov, S.V.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Meccozi, A.

Mecozzi, A.

Mel’nikov, I. V.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

Menyuk, C.R.

Mihalache, D.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

Mollenauer, L.F.

Moors, J.D.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

A. Meccozi, J.D. Moors, H.A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992); L.F. Mollenauer, S.G. Evangelides, and J.P. Gordon, “Wavelength division with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362––367 (1991).
[CrossRef]

Nakazawa, M.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[CrossRef]

Novikov, S.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Orenstein, M.

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[CrossRef]

E. Feigenbaum and M. Orenstein, “Mutual Capture of Colored Solitons Assisted by Matched Modulator,” Submitted

Panoiu, N. C.

N. C. Panoiu, I. V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E 60, 4868–4876 (1999).
[CrossRef]

Pitaevskii, L.P.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Prucnal, P.R.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[CrossRef]

Pruncnal, P.R.

Rauschenbach, K.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Shen, Y.R.

Y.R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), 40.

Soccolich, C.E.

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[CrossRef]

Su, Y.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Suzuki, K.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[CrossRef]

Swanson, E.A.

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Tasgal, R.S.

B.A. Malomed and R.S. Tasgal , “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998)
[CrossRef]

Wang, L.

A. Agarwal, L. Wang, Y. Su, and P. Kumar, “All-optical erasable storage buffer based on parametric nonlinearity in fiber,” OFC 2001, Th-H5 (2001)

Whitaker, A.

Wong, W. S.

Wong, W.S.

H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996)
[CrossRef]

R. A. Barry, V. W. S. Chan, K. L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium – ultrafast TDM netwarks,” J. Selected Areas In Com. 14, 999–1012 (1996).
[CrossRef]

Yamada, E.

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[CrossRef]

Zachary, W.W.

M.F. Mahmood, W.W. Zachary, and T.L. Gill, “Polarization dynamics of vector solitons in an elliptically low-birefringent Kerr medium,” Opt. Eng. 35, 1844–1846 (1996)
[CrossRef]

Zakharov, V.E.

S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of solitons – The inverse scattering method, (Plenum Press, New York ,1984), 68–79

Elec. Lett. (1)

M. Nakazawa, H. kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. 28, 1099–1100 (1992).
[CrossRef]

IEEE J. of Quant. Elec. (1)

D. Arbel and M. Orenstein “Self stabilization of dense soliton trains is passively mode-locked ring laser,” IEEE J. of Quant. Elec. 35, 977–982 (1999)
[CrossRef]

IEEE Phot. Tech. Lett. (1)

C.E. Soccolich, M.W. Chabat, M.N. Islam, and P.R. Prucnal, “Cascade of ultrafast soliton-dragging and trapping logic gates,” IEEE Phot. Tech. Lett. 4, 1043–1046 (1992)
[CrossRef]

J. Eng. Math. (1)

C.R. Menyuk, “Application of multiple-length-scale method to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[CrossRef]

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

J. Selected Areas In Com. (1)

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Supplementary Material (3)

» Media 1: AVI (1064 KB)     
» Media 2: AVI (999 KB)     
» Media 3: AVI (828 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Perturbation calculation of escaping colored solitons. p0=0.145×2π, τ0=5. (a) Center (τ) of escaping soliton (blue) vs. the center of non-perturbed one (black). (b) Solitons carrier (red) vs. center accumulated perturbation (i.e. difference of the curves depicted at figure 1(a)) (blue).

Fig. 2.
Fig. 2.

The soliton trace in the energy plane. The red and blue curves are for p0=-0.137×2π and -0.16×2π respectively with τ0=0.

Fig. 3.
Fig. 3.

Simulation (XPM coupled NLSEs) of equal intensity colored solitons interaction. The intensity envelope of one soliton as the two solitons propagate simultaneously in the fiber for: (a) (1MB) Escape (p0=-0.20×2π), (b) (1MB) Intermediate (p0=-0.15×2π) and (c) (0.83MB) Capture (p0=-0.07×2π). τ0=0.

Fig. 4.
Fig. 4.

Solitons re-coloring vs. initial frequency (τ0=0).

Fig. 5.
Fig. 5.

Trajectories of re-coloring in the energy plane.

Fig. 6.
Fig. 6.

Temporal amplitude calculated by XPM simulations for one of the captured solitons (contour) and the soliton center calculated using Eq. (9) (bold curve). τ0=0, θ1= θ2=0, p0=-0.07×2π.

Fig. 7.
Fig. 7.

Comparison of solitons capture using full fledged NLSE simulation (contours) and the perturbation calculations (bold curves) curves: (a) soliton center, (b) center frequency. p0=0.05×2π, θ12=0, τ0=0.

Fig. 8.
Fig. 8.

A comparison of a second order soliton spectrum at its temporal propagation peaks and the combined spectrum of two co-centered colored solitons at capture threshold. The two colored solitons parameters: Dτ=0, W=1, Dp=2×0.193×2π, Dθ=0.

Fig. 9.
Fig. 9.

Re-coloring (pf/ p0 vs. p0) within particle-like model predictions compared to full fledged simulation. W2=1.5, W1=1, τ0=0.

Equations (51)

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z u = j { 1 2 β " 2 T + δ u 2 } u ,
u = W sech { ε W ( T τ ) } exp { j ( pT + θ ) } ,
z τ = β " p ,
z θ = 1 2 ( δ W 2 + β " p 2 ) ,
z Δ W = S W ,
z Δ τ = S τ + β " Δ p ,
z Δ p = S p ,
z Δ θ = S θ + δ W ΔW ,
S m Im dT { f m * s ( z , T ) exp { j 2 W 2 z } } .
z W = S W ,
z τ = S τ + β " p ,
z p = S p ,
z θ = S θ + δW ( S W dz ) + 1 2 ( δ W 2 + β " p 2 ) .
u 1 = W 1 sech { ε W 1 ( T τ 1 ) } exp { j ( p 1 T + θ 1 ) } ,
u 2 = W 2 sech { ε W 2 ( T τ 2 ) } exp { j ( p 2 T + θ 2 ) } ,
z u 1 = j ( 1 2 β " 2 T + δ ( u 1 2 + 2 u 2 2 ) ) ,
z u 2 = j ( 1 2 β " 2 T + δ ( u 2 2 + 2 u 1 2 ) ) .
δ ( u 2 ) 2 u 1 * + δ ( u 1 ) 2 u 2 * .
s XPM = 2 δ u 2 2 u 1 .
S W XPM = 0 ,
S τ XPM = 0 ,
S p XPM = 2 δ ε 2 W 1 2 W 2 2 dT { tanh ( ε W 1 ( T τ 1 ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
S θ XPM = 2 δ ε W 1 W 2 2 dT { ( 1 ε W 1 ( T τ 1 ) tanh ( ε W 1 ( T τ 1 ) ) ) ·
· sech 2 ( ε W 1 ( T τ 1 ) ) sech 2 ( ε W 2 ( T τ 2 ) ) } ,
F ( ) = ( 2 δ β " ε W 3 ) + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( ξ ) } ,
z ( β " p ) = F ( 2 τ ) = F ( 2 τ ) ,
z τ = β " p ,
2 z τ = F ( 2 τ ) .
Δ τ ( z : + ) = εW p 0 2 ,
2 z τ ( 16 15 δ 2 W 4 ) F L = 2 F L τ .
E k = 1 2 ( β " ) 2 p 2 .
E p = 2 ( β " ε W ) 2 1 ( 2 ε W τ ) coth { 2 ε W τ } sinh 2 { 2 ε W τ } .
E p min = 2 3 ( β " ε W ) 2 .
p 0 TH = 4 3 ε W .
RC = p f p 0 = 1 4 3 ( ε W ) 2 p 0 2 ,
E k 0 = 1 1 RC 2 E p 0 .
2 z τ = 2 F L τ V z τ .
z ( β " p k ) = 2 ( δ W k W 3 k ) 2 + { tanh ( ε W k ξ ) sech 2 ( ε W k ξ ) sech 2 ( ε W 3 k ( ξ ( τ 3 k τ k ) ) ) } ,
z τ k = β " p k ; k∈ 1,2 .
m 2 m 1 t 2 r 1 t 2 r 2 = + { tanh { ε W 1 ξ } sech 2 { ε W 1 ξ } sech 2 { ε W 2 ( ξ ) } } , + { tanh { ε W 2 ξ } sech 2 { ε W 2 ξ } sech 2 { ε W 1 ( ξ ) } } , = W 2 W 1 .
m k = W k
z ( β " p k ) = F m k ,
F = 2 β " δ ε ( W 1 W 2 ) 2 + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 ) } .
μ 2 t r = F .
E K = 1 2 μ ( t r ) 2 = 2 W 1 W 2 W 1 + W 2 ( β " p 0 ) 2 ,
E p = 2 β " δ ε ( W 1 W 2 ) 2 Δ τ + dr { + { tanh ( ξ ) sech 2 ( ξ ) sech 2 ( W 2 W 1 ξ ε W 2 r ) } } .
E pmin = 2 β " δ ε W 1 2 W 2 + { tanh ( ξ ) sech 2 ( ξ ) tanh ( W 2 W 1 ξ ) } .
p 0 TH = 1 3 ε ( W 1 + W 2 ) ; W 2 W 1 .
p o TH ( W 1 , W 2 ) = 1 2 { p o TH ( W 1 , W 1 ) + p o TH ( W 2 , W 2 ) } .
Dp = ( D p 0 ) 2 4 3 ε 2 ( W 1 + W 2 ) 2 ; W 1 W 2 ,

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