Abstract

The concept of soliton internal mode is introduced to explain quantitatively the long-lived oscillations of self-guided beams, or breathing spatial solitons. Cubic–quintic nonlinearity is considered in detail, and it is shown that the existence of the internal mode affects strongly the beam propagation in non-Kerr media, leading to oscillatory dependence of the output width of the beam versus its input power.

© 1997 Optical Society of America

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References

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  1. J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  2. See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
    [CrossRef] [PubMed]
  3. A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
    [CrossRef]
  4. D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
    [CrossRef]
  5. A. W. Snyder and D. J. Mitchell, Opt. Lett. 22161997. Gaussons of the logarithmic NLSE were first predicted by I. Bialynicki-Birula and J. Mycielski, Phys. Scr. 20, 539 (1979).
  6. The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
    [CrossRef]
  7. R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
    [CrossRef]
  8. D. J. Kaup, J. Math. Anal. Appl. 54, 849 (1976); Phys. Rev. A 42, 5689 (1990).
    [CrossRef]

1997

1996

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

1995

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

1988

See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

1983

The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
[CrossRef]

1976

D. J. Kaup, J. Math. Anal. Appl. 54, 849 (1976); Phys. Rev. A 42, 5689 (1990).
[CrossRef]

1974

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Afanasjev, V. V.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

Campbell, D. K.

The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
[CrossRef]

Dessailly, R.

See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Froehly, C.

See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Hewlett, S. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

Kaup, D. J.

D. J. Kaup, J. Math. Anal. Appl. 54, 849 (1976); Phys. Rev. A 42, 5689 (1990).
[CrossRef]

Kivshar, Yu. S.

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

Love, J. D.

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

Maneuf, S.

See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Micallef, R. W.

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

Mitchell, D. J.

Pelinovsky, D. E.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Schonfeld, J. F.

The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
[CrossRef]

Snyder, A. W.

Wingate, C. A.

The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

J. Math. Anal. Appl.

D. J. Kaup, J. Math. Anal. Appl. 54, 849 (1976); Phys. Rev. A 42, 5689 (1990).
[CrossRef]

Opt. Commun.

See, e.g., S. Maneuf, R. Dessailly, and C. Froehly, Opt. Commun. 65, 193 (1988); W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. E

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

R. W. Micallef, V. V. Afanasjev, Yu. S. Kivshar, and J. D. Love, Phys. Rev. E 54, 2936 (1996).
[CrossRef]

Physica D

The concept of internal mode was introduced earlier for kinks, topological solitons of the Klein–Gordon models; e.g., D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Physica D 9, 1 (1983).It was also used recently to explain oscillations of two-wave χ2 solitary waves; see C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Yu. S. Kivshar, Phys. Rev. E 54, 4321 (1996).
[CrossRef]

Progr. Theor. Phys. Suppl.

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Example of the soliton profile Φx (thick solid curve) and the localized eigenfunctions Ux (thin solid curve) and Wx (dashed curve) for the solitary-wave equation  (2) of the cubic–quintic nonlinearity σ=+1 at β=5.

Fig. 2
Fig. 2

(a) Dependence of the frequency of the soliton internal mode Ω0 on the propagation constant β shown as the ratio Ω0/β versus β. (b) Ratio of the mode width wmode and soliton width ws as a function of β.

Fig. 3
Fig. 3

Dynamics of a perturbed soliton in the cubic–quintic nonlinear Schrödinger model: (a) σ=+1 and β=5 (the mode exists, and its width is of the order of the soliton width); (b) σ=-1 and β=0.8 (the mode does not exist). The initially perturbed beam is taken as a scaled soliton with the amplitude increased by 10% of its value. The dashed lines in (a) and (b) show the corresponding unperturbed values of the peak intensity.

Fig. 4
Fig. 4

Output beam width wout at z=10, normalized to the input beam width w0, versus the input beam power P, normalized to the soliton power P0. Thin curve, the soliton internal mode exists (σ=+1, β=2); thick curve, the soliton internal mode does not exist (σ=-1, β=0.6).

Equations (4)

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

iψz+2ψx2+Fψ2ψ=0,
Φx=β1+σβ cosh2βx+1-1/2.
ψx, z=Φx+Ux-WxexpiΩz+U*x+W*xexp-iΩzexpiβz.
-d2Wdx2+βW-FΦ2W=ΩU, -d2Wdx2+βU-FΦ2U-2Φ2FΦ2U=ΩW.

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