1. Gap vortex solitons in periodic media with quadratic nonlinearity
Chao Hang, Vladimir V. Konotop, and Boris A. Malomed
Centro de Fisica Teorica e Computacional (CFTC),
Universidade de Lisboa, Complexo Interdisciplinar

2. Contents 1 Introduction of solitons 2 The previous works 3 Our work 4
Conclusion and expectation

3. Rate of progress Introduction of solitons The previous works Our work
Conclusion

4. The definition and secret of stability of solitons
Important characters of solitons:
The solitons are dynamically locallized nonlienear structures.
The solitons maintain their shapes while travel at a constant speed.
The solitons recover their shapes after collision.
Dispersion
ut + uxxx = 0
Dispersion Diffraction
Nonlinearity
ut + uux = 0
Stability of solitons is the result of balance between dispersion and nonlinearity.
ut - 6uux + uxxx = 0

5. Solitons in nature Soliton in sea (Hawii)
Solitons on a branch (Norway)
Soliton in river (Australia)
Soliton in atmosphere

6. Pulse soliton Kink soliton Envelope soliton Typical solitons I
1D solitons：
KdV eq.
Pulse soliton
SG eq.
Kink soliton
NLS eq.
Envelope
soliton

7. Typical solitons II ? 2D solitons： 3D solitons ： donut soliton
Vortex soliton
3D solitons ：
donut soliton
potato soliton

8. The grating-induced bandgap
Photonic Crystals: Periodic Dielectric Structures
Photonic Band Gap:
Prohibited Frequency Region
The first gap
Eigenvalue problem:
The semi-infinite gap

9. Essential idea: Balance of Nonlinearity and Gap confinement.
What is a Gap Soliton?
Gap Soliton K
Gap ω
Bragg Soliton b a
Nonlinear material
Essential idea: Balance of Nonlinearity and Gap confinement.
V. A. BRAZHNYI and V. V. KONOTOP, Modern Physics Letters B, Vol. 18, No (2004)

10. Rate of progress Introduction of solitons The previous works Our work
Conclusion

11. The previous works Vortex solitons and the instability
The observation
of discrete
vortex solitons
Analysis of discrete
vortex solitons
D. N. Neshev, et al., Phys. Rev.
Lett. 92, (2004); J. W. Fleischer, et al., Phys. Rev. Lett. 92,
(2004).
W. J. Firth, et al., Phys Rev. Lett. 79, 2450 (1997).
Z. Xu, et al., Phys. Rev. E 71, (2005).

12. Vortex solitons and the instability
W. J. Firth and D. V. Skryabin, Phys Rev. Lett. 79, 2450 (1997).

13. Experimental observation of the discrete vortex solitons
D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Yu. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Phys. Rev. Lett. 92, (2004)
J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, (2004)

14. Analysis of discrete vortex solitons
Z. Xu, Y. K. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, Phys. Rev. E 71, (2005).

15. Rate of progress Introduction of solitons The previous works Our work
Conclusion

16. Model
Coupled evolution equations for complex amplitudes of the FF and SH fields in the spatial domain:
with the Hamiltonian:

17. Linear analysis We are interested in steady state: Band structure:
We arrive the eigenvalue problem:
V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, (2007).

18. Nonlinear analysis The properties of the Gap Vortex solitons
The configuration of the soliton
The stability of the soliton
The properties of the Gap Vortex solitons
The generation of the soliton
The delocalization transition
of the soliton

19. The configuration of the gap vortex soliton
Intensity profiles and phase distributions of gap-vortex solitons.
Initial condition:

20. The stability of the gap vortex soliton
In our case stability can be determined by Vatkhitov Kolokolov (VK)
criterion:

21. The generation of the gap vortex soliton
SH-generation efficiency:

22. The delocalization transition of the gap vortex soliton
H.A. Cruz, V. A. Brazhnyi, V. V. Konotop, M. Salerno, G. L. Alfimov, "One-dimensional delocalizing transitions of matter waves in optical lattices" Physica D 238, (2009)

23. Rate of progress Introduction of solitons The previous works Our work
Conclusion

24. Conclusion 1
We have studied the bandgap structure induced by the transverse grating. 2
We have demon-
strated that stable gap vortex solitons belonging to one of the finite total gaps. 3
We have explored the spontaneous generation of the gap vortex solitons and the delocalization
transition.

25. Further works Soliton algebra: Three-dimensional gap vortex solitons:
D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, L.-C. Crasovan, Y. V. Kartashov, and L. Torner, Phys. Rev. A 72, R (2005)

26. Muito Obligado!