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

nonlinear fiber 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.

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

Two solitons' interference Then we can get The other solitons incident angle can be read out, and the nonlinear

parameter g will be given 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 Schrdinger Equation Model; Vladimir N. Serkin and Akira Hasegawa Phys. 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).

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

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

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

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

Dark Solitons solution by Hirota's bilinearization method 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

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

For bright soliton Dark Solitons solution by Hirota's bilinearization method Then we have one dark soliton solution corresponding to the

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

Dark Solitons solution by Hirota's bilinearization method From the above bilinear equations, we obtain the dark soliton soliution as : Dark Solitons solution by Hirota's bilinearization method Dynamic description of one dark soliton

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.)

Nonautonomous dark Solitons in optical fiber Nonautonomous dark Solitons in optical fiber 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.) Nonautonomous Solitons in a graded-index waveguide Nonautonomous Solitons in a graded-index waveguide

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

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

Observe New year wave in 1995, North sea D.H.Peregrine, Water waves, nonlinear Schrdinger 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 Forced and damped nonlinear Schrdinger equation

M. Onorato, D. Proment, Phys. Lett. A 376, 3057-3059(2012). 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).

Experimental observation(optical fiber and water tank B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010). Scientific Reports .

2.463(2012) .In optical fiber A. Chabchoub, N. P. H offmann, et al., Phys. Rev. Lett. 106, 204502 (2011).

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

Optical rogue wave in a graded-index waveguide 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). 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, 311

9-3127 (2012) Two rogue wave Rogue wave in Three-mode fiber One rogue wave in three-mode fiber

Rogue wave of four-petaled flower Eye-shaped rogue wave Rogue wave in Three-mode fiber Two rogue wave in three-mode fiber

Rogue wave in Three-mode fiber Three rogue wave in three-mode fiber Rogue wave in Three-mode fiber The interaction of three rogue wave Thanks!

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