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

Outline soliton Introduction of optical soliton Two solitons' interference Nonautonomous Solitons rogue wave Introduction of optical 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. G.P. Agrawal, Nonlinear Fiber Optics, Acdemic press (2007). Temporal soliton in nonlinear fiber Spatial soliton in a waveguide

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

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

History of Nonautonomous Solitons Novel Soliton Solutions of the Nonlinear Schrödinger 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, (2007). Analytical Light Bullet Solutions to the Generalized(3 +1 )-Dimensional Nonlinear Schrodinger Equation. Wei-Ping Zhong. Phys. Rev. Lett. 101, (2008). 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 ） 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 ） Reason:

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

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

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

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

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

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 χ

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

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, (2011), J. Opt. Soc. Am. B 28, 236 (2011) ， Z. Y. Yang, L.C.Zhao et.al.)

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, (2010), Optic s Commu nications 283 (2010) Z. Y. Yang, L.C.Zhao et.al.)

Nonautonomous Solitons in a graded-index waveguide

Without the grating, we get

Nonautonomous Solitons in a graded-index waveguide

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

Introduction of rogue wave Observe “New year” wave in 1995, North sea 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

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

B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010). 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, (2007)

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

Optical rogue wave in a graded-index waveguide Long-life rogue wave Classical 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, (2012). B.L. Guo, L.M. Ling, Chin. Phys. Lett. 28, (2011).

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

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