Abstract

We demonstrate the existence and stability of bright vortex solitons sustained by a ring-shaped parametric gain, for both focusing and defocusing Kerr nonlinearities in lossy optical media. With the defocusing nonlinearity, the vortices are stable at all values of the detuning parameter, while under the focusing nonlinearity their stability region is limited to some positive values of the detuning. Unstable vortices in the focusing medium transform into stable rotating azimuthons.

© 2013 Optical Society of America

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    [CrossRef]
  2. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
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  3. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
    [CrossRef]
  4. V. I. Kruglov, Y. A. Logvin, and V. M. Molkov, J. Mod. Opt. 39, 2277 (1992).
    [CrossRef]
  5. L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
    [CrossRef]
  6. W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
    [CrossRef]
  7. C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
    [CrossRef]
  8. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
    [CrossRef]
  9. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
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  10. S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
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  11. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, Opt. Lett. 35, 628 (2010).
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  15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
    [CrossRef]
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    [CrossRef]
  17. Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
    [CrossRef]
  18. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
    [CrossRef]
  19. O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
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  20. V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
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  21. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Phys. Rev. Lett. 95, 053903 (2005).
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  22. L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
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  24. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
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  26. I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
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  29. T. Kapitula and B. Sandstede, Physica D 124, 58 (1998).
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  30. V. Alexeeva, I. V. Barashenkov, and G. P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000).
    [CrossRef]
  31. J. N. Kutz, SIAM Rev. 48, 629 (2006).
    [CrossRef]
  32. E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
    [CrossRef]
  33. F. Ye, C. Huang, Y. V. Kartashov, and B. Malomed, Opt. Lett. 38, 480 (2013).
    [CrossRef]

2013

2012

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, Phys. Rev. A 85, 023814 (2012).
[CrossRef]

2011

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
[CrossRef]

2010

2009

J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortes, and N. Devine, Opt. Express 17, 4236 (2009).
[CrossRef]

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

2007

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

2006

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

2005

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

2001

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

2000

C. Law, X. Zhang, and G. Swartzlander, Opt. Lett. 25, 55 (2000).
[CrossRef]

L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

V. Alexeeva, I. V. Barashenkov, and G. P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000).
[CrossRef]

1998

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

1997

M. Quiroga-Teixeiro and H. Michinel, J. Opt. Soc. Am. B 14, 2004 (1997).
[CrossRef]

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

1995

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

1993

1992

V. I. Kruglov, Y. A. Logvin, and V. M. Molkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

1991

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

1985

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Akhmediev, N.

Alexeeva, V.

V. Alexeeva, I. V. Barashenkov, and G. P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000).
[CrossRef]

Bang, O.

Barashenkov, I. V.

V. Alexeeva, I. V. Barashenkov, and G. P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000).
[CrossRef]

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Bogdan, M. M.

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Bondila, M.

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

Borovkova, O. V.

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, Phys. Rev. A 85, 023814 (2012).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
[CrossRef]

Briedis, D.

Buryak, A. V.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef]

Crasovan, L. C.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Cross, M. C.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Desyatnikov, A. S.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Devine, N.

Edmundson, D.

Fedorov, S. V.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Firth, W. J.

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Hu, B.

Huang, C.

Kapitula, T.

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

Kartashov, Y. V.

F. Ye, C. Huang, Y. V. Kartashov, and B. Malomed, Opt. Lett. 38, 480 (2013).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, Phys. Rev. A 85, 023814 (2012).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, Opt. Lett. 35, 628 (2010).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

Kath, W. L.

Kenig, E.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Kivshar, Y.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Kivshar, Y. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Korobov, V. I.

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Krolikowski, W.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Molkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Kumar, P.

Kutz, J. N.

Law, C.

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Li, R. D.

Lifshitz, R.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Lobanov, V. E.

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, Phys. Rev. A 85, 023814 (2012).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Molkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Malomed, B.

Malomed, B. A.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Mejia-Cortes, C.

Michinel, H.

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Molkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Molkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Petersen, D.

Petrov, D. V.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Quiroga-Teixeiro, M.

Rosanov, N. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Rotschild, C.

C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef]

Saffman, M.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

Sammut, R. A.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

Sandstede, B.

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

Segev, M.

C. Rotschild, O. Cohen, O. Manela, and M. Segev, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef]

Shatsev, A. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Phys. Rev. Lett. 95, 053903 (2005).
[CrossRef]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Skupin, S.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef]

Soto-Crespo, J. M.

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef]

Swartzlander, G.

Torner, L.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 37, 1799 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, Phys. Rev. A 85, 023814 (2012).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 84, 035602 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 36, 3088 (2011).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, Opt. Lett. 35, 628 (2010).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Towers, I.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, Phys. Lett. A 288, 292 (2001).
[CrossRef]

Tsironis, G. P.

V. Alexeeva, I. V. Barashenkov, and G. P. Tsironis, Phys. Rev. Lett. 84, 3053 (2000).
[CrossRef]

Vlasov, R. A.

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Ye, F.

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Zhang, X.

Electron. Lett.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Europhys. Lett.

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

J. Mod. Opt.

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Figures (3)

Fig. 1.
Fig. 1.

Distributions of the absolute value of the field (top) and phase (bottom) for vortex solitons with (a) m=1, k=1.3, a=4.9, (b) m=2, k=3, a=4.7, and (c) m=1, k=3, a=7. Panels (a) and (b) correspond to the focusing medium, while (c) represents the defocusing one. Thin black circles indicate the radius at which the parametric gain has its maximum.

Fig. 2.
Fig. 2.

Energy flow for (a) m=1 and (b) m=2 vortex solitons in the focusing (curves labeled Uf) and defocusing (curves labeled Ud) media versus the gain strength, a. Panels (a) and (b) correspond to k=3 and k=1.8, respectively. The left, central, and right circles indicate, respectively, the threshold gain ath (left), the value ag corresponding to gain-guiding in the linear medium (center), and critical gain acr (right). The black and red segments represent stable and unstable solution branches. The threshold and critical gain are shown versus k for m=1 in (c), and for m=2 in (d). The black and red lines show ath in the focusing and defocusing media, respectively. The solitary vortices in the focusing medium are stable at ath<a<acr.

Fig. 3.
Fig. 3.

Evolution of profiles of the absolute value of the field in the course of the propagation in the focusing medium. (a) The breakup of a perturbed unstable vortex with m=1 at k=1.3, a=5.5; (b) the breakup of an unstable vortex with m=2 at k=1.8, a=4.5; (c) stable propagation of perturbed vortex with m=1 at k=1.3, a=4.9 (left and center), and of the one with m=2 at k=1.8, a=3.6 (right).

Equations (2)

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iqξ=12(2qη2+2qζ2)(k+iγ)q+σ|q|2qar2mexp(2imφr2/d2)q*.
12(2wη2+2wζ2)(k+iγ)w+σ|w|2war2mexp(r2/d2)w*=0,

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