Soliton and related problems in nonlinear physics

1. Soliton and related problems in nonlinear physics Zhan-Ying Yang , Li-Chen Zhao and Chong Liu Department of Physics, Northwest University

2. Outline Introduction of optical soliton soliton Two solitons' interference Nonautonomous Solitons Introduction of optical rogue wave rogue wave Nonautonomous rogue wave Rogur wave in two and three mode nonlinearfiber

3. Introduction of soliton Solitons, whose first known description in the scientific literature, in the form of ‘‘a large solitary elevation, a rounded, smooth, and well-defined heap of water,’’ goes back to the historical observation made in a chanal near Edinburgh by John Scott Russell in the 1830s.

4. Introduction of optical soliton Zabusky and Kruskal introduced for the first time the soliton concept to characterize nonlinear solitary waves that do not disperse and preserve their identity during propagation and after a collision. (Phys. Rev. Lett. 15, 240 (1965) ) Optical solitons. A significant contribution to the experimental and theoretical studies of solitons was the identification of various forms of robust solitary waves in nonlinear optics.

5. Introduction of optical soliton Optical solitons can be subdivided into two broad categories—spatial and temporal. Temporal soliton in nonlinear fiber Spatial soliton in a waveguide G.P. Agrawal, Nonlinear Fiber Optics, Acdemic press (2007).

6. Two solitons' interference We study continuous wave optical beams propagating inside a planar nonlinear waveguide

7. Two solitons' interference Then we can get The other soliton’s incident angle can be read out, and the nonlinear parameter g will be given

8. History of Nonautonomous Solitons Reason: A: The test of solitons in nonuniform media with time-dependent density gradients .（spatial soliton） B: The test of the core medium of the real fibers, which cannot be homogeneous, fiber loss is inevitable, and dissipation weakens the nonlinearity.（temporal soliton） Novel Soliton Solutions of the Nonlinear Schrödinger Equation Model; Vladimir N. Serkin and Akira HasegawaPhys. Rev. Lett. 85, 4502 (2000) . Nonautonomous Solitons in External Potentials; V. N. Serkin, Akira Hasegawa,and T. L. Belyaeva Phys. Rev. Lett. 98, 074102 (2007). Analytical Light Bullet Solutions to the Generalized(3 +1 )-Dimensional Nonlinear Schrodinger Equation. Wei-Ping Zhong. Phys. Rev. Lett. 101, 123904 (2008).

9. Nonautonomous Solitons Engineering integrable nonautonomous nonlinear Schrödinger equations , Phys. Rev. E. 79, 056610 (2009), Hong-Gang Luo, et.al.)

10. Bright Solitons solution by Darboux transformation Under the integrability condition We get Dynamics of a nonautonomous soliton in a generalized nonlinear Schrodinger equation ,Phys. Rev. E. 83, 066602 (2011) , Z. Y. Yang, et.al.)

11. Nonautonomous bright Solitons under the compatibility condition We obtain the developing equation.

12. Nonautonomous bright Solitons the Darboux transformation can be presented as we can derive the evolution equation of Q as follows:

13. Nonautonomous bright Solitons Dynamic description we obtain Finally, we obtain the solution as

14. Dark Solitons solution by Hirota's bilinearizationmethod

15. Dark Solitons solution by Hirota's bilinearization method We assume the solution as Where g(x,t) is a complex function and f(x,t) is a real function

16. Dark Solitons solution by Hirota's bilinearization method by Hirota's bilinearization method, we reduce Eq.(6) as For dark soliton For bright soliton

17. Dark Solitons solution by Hirota's bilinearization method Then we have one dark soliton solution corresponding to the different powers of χ

18. Dark Solitons solution by Hirota's bilinearization method Two dark soliton solution corresponding to the different powers of χ

19. Dark Solitons solution by Hirota's bilinearization method From the above bilinear equations, we obtain the dark soliton soliution as :

20. Dark Solitons solution by Hirota's bilinearization method Dynamic description of one dark soliton

21. Nonautonomous bright Solitons in optical fiber Dynamics of a nonautonomous soliton in a generalized nonlinear Schrodinger equation ,Phys. Rev. E. 83, 066602 (2011) ,J. Opt. Soc. Am. B 28 , 236 (2011)， Z. Y. Yang, L.C.Zhao et.al.)

22. Nonautonomous dark Solitons in optical fiber

23. Nonautonomous dark Solitons in optical fiber

24. Nonautonomous Solitons in a graded-index waveguide Snakelike nonautonomous solitons in a graded-index grating waveguide , Phys. Rev. A 81 , 043826 (2010), Optic s Commu nications 283 (2010) 3768 . Z. Y. Yang, L.C.Zhao et.al.)

25. Nonautonomous Solitons in a graded-index waveguide

26. Nonautonomous Solitons in a graded-index waveguide Without the grating , we get

27. Nonautonomous Solitons in a graded-indexwaveguide

28. Nonautonomous Solitons in a graded-index waveguide

29. Introduction of rogue wave Mysterious freak wave, killer wave Oceannography Vol.18，No.3，Sept. 2005。

30. Introduction of rogue wave D.H.Peregrine, Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B25,1643 (1983); Wave appears from nowhere and disappears without a trace, N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373 (2009) 675 Observe “New year” wave in 1995, North sea

31. Forced and damped nonlinear Schrödinger equation M. Onorato, D. Proment, Phys. Lett. A 376,3057-3059(2012).

32. Experimental observation(optical fiber) As rogue waves are exceedingly difficult to study directly, the relationship between rogue waves and solitons has not yet been definitively established, but it is believed that they are connected. Optical rogue waves. Nature 450,1054-1057 (2007) B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010).

33. Experimental observation(optical fiber and water tank) B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010). ScientificReports. 2.463(2012) .In optical fiber A. Chabchoub, N. P. Hoffmann, et al., Phys. Rev. Lett. 106, 204502 (2011).

34. Optical rogue wave in a graded-index waveguide Classical rogue wave Long-life rogue wave

35. Optical rogue wave in a graded-index waveguide

36. Rogue wave in Two-mode fiber F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, Phys. Rev. Lett. 109, 044102 (2012). B.L. Guo, L.M. Ling, Chin. Phys. Lett. 28, 110202 (2011).

37. Bright rogue wave and dark rogue wave Rogue wave of four-petaled flower Eye-shaped rogue wave L.C.Zhao, J. Liu, Joun. Opt. Soc. Am. B 29, 3119-3127 (2012) Two rogue wave

38. Rogue wave in Three-mode fiber One rogue wave in three-mode fiber Rogue wave of four-petaled flower Eye-shaped rogue wave

39. Rogue wave in Three-mode fiber Two rogue wave in three-mode fiber

40. Rogue wave in Three-mode fiber Three rogue wave in three-mode fiber

41. Rogue wave in Three-mode fiber The interaction of three rogue wave

42. Thanks!