Understanding Gap Vortex Solitons in Periodic Media

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

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

4. The definition and secret of stability of solitons • Important characters of solitons: • The solitons are dynamically locallizednonlienear structures. • Thesolitonsmaintain their shapes while travel at a constant speed. • Thesolitonsrecover their shapes aftercollision. Dispersion ut+uxxx= 0 DispersionDiffraction Nonlinearity ut+ uux= 0 Stability of solitons is the result of balance between dispersionandnonlinearity. ut- 6uux+uxxx= 0

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

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

7. Typical solitons II ? 2D solitons： 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. ω Nonlinear material Bragg Soliton K a b Gap Gap Soliton What is a Gap Soliton? Essential idea: Balance of Nonlinearity and Gap confinement. V. A. BRAZHNYI and V. V. KONOTOP, Modern Physics Letters B, Vol. 18, No. 14 1-25 (2004)

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

11. The previous works The observation of discrete vortex solitons Analysis of discrete vortex solitons Vortex solitons and the instability D. N. Neshev, et al., Phys. Rev. Lett. 92, 123903 (2004); J. W. Fleischer, et al., Phys. Rev. Lett. 92, 123904 (2004). W. J. Firth,et al., Phys Rev. Lett. 79, 2450 (1997). Z. Xu, et al., Phys. Rev. E 71, 016616 (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, 123903 (2004) J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev,J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123904 (2004)

14. Analysis of discrete vortex solitons Z. Xu, Y. K. Kartashov, L.-C. Crasovan, D. Mihalache,and L. Torner, Phys. Rev. E 71, 016616 (2005).

15. Rate of progress The previous works Our work Introduction of solitons 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, 037111 (2007).

18. Nonlinear analysis 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, 1372-1378 (2009)

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

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

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, 021601R (2005)

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