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Soliton and related problems in nonlinear physics

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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

nonlinearfiber

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 soliton’s 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 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).

Nonautonomous Solitons

Engineering integrable nonautonomous nonlinear Schrödinger 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

Dynamic description

we obtain

Finally, we obtain the solution as

Dark Solitons solution by Hirota's bilinearizationmethod

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, 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-indexwaveguide

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

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

Forced and damped nonlinear Schrödinger 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). ScientificReports. 2.463(2012) .In optical fiber

A. Chabchoub, N. P. Hoffmann, 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, 3119-3127 (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

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