Matter and rogue waves of some generalized Gross-Pitaevskii equations with varying potentials and nonlinearities

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1 Matter and rogue waves of some generalized Gross-Pitaevskii equations with varying potentials and nonlinearities Zhenya Yan Key Lab of Mathematics Mechanization, AMSS, Chinese Academy of Sciences (joint works with Konotop, Akhmediev, Bludov, Jiang) Symmetry Methods, Applications, and Related Fields, UBC, May. 14, 2014 Celebrating the work of Prof. G. Bluman.

2 *I was a PIMS Postdoc under the supervision of Prof. G. W. Bluman from [G. W. Bluman, Zhenya Yan, Nonclassical potential solutions of PDEs, Euro. J. Appl. Math. 16 (2005) 239.] *Aug. 2008: Conf. on Symmetries, UBC *May 2014: Conf. on Symmetries, UBC 2

3 Outline 1. Introduction 2. 1D Gross-Pitaevskii /NLS equation 3. 2D Gross-Pitaevskii equation on a domain 4. 3D Gross-Pitaevskii equation 5. Conclusions 3

4 Bose-Einstein Condensates Predicting the BEC BEC was observed BEC is a new state of matter Nobel Prize at the coldest (2001) temperatures (about absolute zero, C) BEC is regarded as the fifth state of matter (after solid, liquid, gas, plasma)

5 Gross-Pitaevskii (GP) equation External potential Interaction: Feshbach resonance Harmonic potential: Optical lattice potential: Double-well potential: 5

6 Nonlinear optics NLS equation Optical soliton A. Hasegawa 6

7 7

8 Harmonic potential Gaussian interaction 8

9 2. 1D GP equation 9

10 Transformation: chemical potential Determining system: 10

11 2.1 van der Waals potential Ex. : n=3, van der Waals potential 11

12 1D GP equation Choosing: 12

13 1D GP equation c=1 p=-1/2, van der Waals law: 13

14 Repulsive interaction G>0 Dark surface mode 14

15 Repulsive interaction G>0 Dark soliton 15

16 Attractive interaction G<0 Dark surface mode: 16

17 Attractive interaction G<0 Bright soliton: 17

18 2.2 Double-well potential and Gaussian interaction Double-well potential: Gaussian potential: Z. Y. Yan, D. M. Jiang, Phys. Rev. E 85, (2012) 18

19 Repulsive interaction G>0 n=1 n=2 19

20 Attractive interaction G<0 Belmonte-Beitia, et al., Phys. Rev. Lett. 98, (2007) 20

21 Attractive interaction G<0 n=1 n=2 21

22 Attractive interaction G<0 n=1 n=2 22

23 2.3 Rogue waves of the NLS equation with varying coefficient ( Rogue wave (freak wave): a special type of solitary waves appear from nowhere and disappear without a trace 23

24 Rogue wave: definition C. Kharif, et al., Rogue Waves in the Ocean (Springer, 2009) 24

25 Rogue wave 1964, Draper: first presented freak wave Rogue wave, monster wave, killer wave, extreme wave, abnormal wave, giant wave, huge wave, etc. 25

26 Rogue wave Draupner platform North Sea, Norway Jan. 1, 1995 New Year s Waves 26

27 Oceanic rogue waves Deep water: NLS equation Peregrine soliton, 1983 D. H Peregrine ( ) 27

28 NLS equation: self-focusing Modified Darboux transformation: multi-rogue waves Nail Akhmediev, et al., Phys. Rev. E 80 (2009) Phys. Lett. A 373 (2009)

29 NLS equation with varying coefficient Z. Y. Yan, Phys. Lett. A 374 (2010)

30 First-order rogue waves 30

31 Second-order rogue waves 31

32 3. 2D GP equation Dirichlet problem 32

33 Conformal mapping: 33

34 34

35 Cauchy-Riemann equation 35

36 36

37 37

38 4. 3D generalized GP equation g_q=0, p=3 : Cubic GP equation p=3, q=5 : Cubic-Quintic GP equation. 38

39 4.1 3D generalized GP equation 39

40 Determining system 40

41 4.1 Stationary case 41

42 Stationary case Attractive case (G_3<0) Bright solitary wave 42

43 Stationary case Repulsive case (G_3>0) Dark solitary wave 43

44 Stationary case: cubic-quintic model

45 3.2 Time-dependent case 45

46 3.2 Time-dependent case Attractive case (G_3<0) Bright solitary wave 46

47 3.2 Time-dependent case Repulsive case (G_3>0) Dark solitary wave 47

48 4.2 3D GP equation 48

49 49

50 Phase: 50

51 Soliton pairs and interactions Z. Y. Yan, C. Hang, Phys. Rev. A 81(2009)

52 3D First-order Rogue wave solutions 52

53 3D Second-order Rogue wave solutions 53

54 5. Conclusions 1. We found matter-wave and rogue wave solutions of the some GP/NLS equations with different potentials in 1D, 2D and 3D spaces. 2. These equations may admit exact matter-wave solutions for other external potentials. 3. These solutions will be useful to excite new physical phenomena in BECs, nonlinear optics, or other related fields.

55 Thanks for your attention!